$\int_{0}^{9} \int_{0}^{3} \int_{4 y}^{12} \frac{5 \cos \left(x^{2}\right)}{6 \sqrt{z}} d x d y d z$

Step 1 ：Given the triple integral problem: \(\int_{0}^{9} \int_{0}^{3} \int_{4 y}^{12} \frac{5 \cos \left(x^{2}\right)}{6 \sqrt{z}} d x d y d z\)

Step 2 ：First, we perform the innermost integral with respect to x: \(-5\sqrt{2}\sqrt{\pi}\text{fresnelc}(4\sqrt{2}y/\sqrt{\pi})\gamma(1/4)/(48\sqrt{z}\gamma(5/4)) + 5\sqrt{2}\sqrt{\pi}\text{fresnelc}(12\sqrt{2}/\sqrt{\pi})\gamma(1/4)/(48\sqrt{z}\gamma(5/4))\)

Step 3 ：Next, we perform the second integral with respect to y: \(-5\sqrt{2}\sqrt{\pi}\left(-\sqrt{2}\sin(144)\gamma(1/4)/(32\sqrt{\pi}\gamma(5/4)) + 3\text{fresnelc}(12\sqrt{2}/\sqrt{\pi})\gamma(1/4)/(4\gamma(5/4))\right)\gamma(1/4)/(48\sqrt{z}\gamma(5/4)) + 5\sqrt{2}\sqrt{\pi}\text{fresnelc}(12\sqrt{2}/\sqrt{\pi})\gamma(1/4)/(16\sqrt{z}\gamma(5/4))\)

Step 4 ：Finally, we perform the outermost integral with respect to z: \(-5\sqrt{2}\sqrt{\pi}\left(-\sqrt{2}\sin(144)\gamma(1/4)/(32\sqrt{\pi}\gamma(5/4)) + 3\text{fresnelc}(12\sqrt{2}/\sqrt{\pi})\gamma(1/4)/(4\gamma(5/4))\right)\gamma(1/4)/(8\gamma(5/4)) + 15\sqrt{2}\sqrt{\pi}\text{fresnelc}(12\sqrt{2}/\sqrt{\pi})\gamma(1/4)/(8\gamma(5/4))\)

Step 5 ：Thus, the final answer is \(\boxed{-5\sqrt{2}\sqrt{\pi}\left(-\sqrt{2}\sin(144)\gamma\left(\frac{1}{4}\right)/(32\sqrt{\pi}\gamma\left(\frac{5}{4}\right)) + 3\text{fresnelc}(12\sqrt{2}/\sqrt{\pi})\gamma\left(\frac{1}{4}\right)/(4\gamma\left(\frac{5}{4}\right))\right)\gamma\left(\frac{1}{4}\right)/(8\gamma\left(\frac{5}{4}\right)) + 15\sqrt{2}\sqrt{\pi}\text{fresnelc}(12\sqrt{2}/\sqrt{\pi})\gamma\left(\frac{1}{4}\right)/(8\gamma\left(\frac{5}{4}\right))}\)