Integrand size = 16, antiderivative size = 250 \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {8 b^2 \cos \left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {32 b^4 \cos \left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {128 b^6 \cos \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {256 b^{15/2} \sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{675675}-\frac {256 b^{15/2} \sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)}{675675}+\frac {4 b \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {16 b^3 \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {64 b^5 \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {256 b^7 \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}} \]
-2/5*cos(a+b*x^(1/3))/x^(5/2)+8/715*b^2*cos(a+b*x^(1/3))/x^(11/6)-32/45045 *b^4*cos(a+b*x^(1/3))/x^(7/6)+4/65*b*sin(a+b*x^(1/3))/x^(13/6)-16/6435*b^3 *sin(a+b*x^(1/3))/x^(3/2)+64/225225*b^5*sin(a+b*x^(1/3))/x^(5/6)-256/67567 5*b^7*sin(a+b*x^(1/3))/x^(1/6)+256/675675*b^(15/2)*cos(a)*FresnelC(x^(1/6) *b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)-256/675675*b^(15/2)*FresnelS(x ^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)+128/675675*b^6*co s(a+b*x^(1/3))/x^(1/2)
Time = 0.36 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.95 \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\frac {2 \left (-135135 \cos \left (a+b \sqrt [3]{x}\right )+3780 b^2 x^{2/3} \cos \left (a+b \sqrt [3]{x}\right )-240 b^4 x^{4/3} \cos \left (a+b \sqrt [3]{x}\right )+64 b^6 x^2 \cos \left (a+b \sqrt [3]{x}\right )+128 b^{15/2} \sqrt {2 \pi } x^{5/2} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )-128 b^{15/2} \sqrt {2 \pi } x^{5/2} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right ) \sin (a)+20790 b \sqrt [3]{x} \sin \left (a+b \sqrt [3]{x}\right )-840 b^3 x \sin \left (a+b \sqrt [3]{x}\right )+96 b^5 x^{5/3} \sin \left (a+b \sqrt [3]{x}\right )-128 b^7 x^{7/3} \sin \left (a+b \sqrt [3]{x}\right )\right )}{675675 x^{5/2}} \]
(2*(-135135*Cos[a + b*x^(1/3)] + 3780*b^2*x^(2/3)*Cos[a + b*x^(1/3)] - 240 *b^4*x^(4/3)*Cos[a + b*x^(1/3)] + 64*b^6*x^2*Cos[a + b*x^(1/3)] + 128*b^(1 5/2)*Sqrt[2*Pi]*x^(5/2)*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)] - 128* b^(15/2)*Sqrt[2*Pi]*x^(5/2)*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a] + 20790*b*x^(1/3)*Sin[a + b*x^(1/3)] - 840*b^3*x*Sin[a + b*x^(1/3)] + 96*b^5 *x^(5/3)*Sin[a + b*x^(1/3)] - 128*b^7*x^(7/3)*Sin[a + b*x^(1/3)]))/(675675 *x^(5/2))
Time = 1.48 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.09, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.750, Rules used = {3897, 3042, 3778, 25, 3042, 3778, 3042, 3778, 25, 3042, 3778, 3042, 3778, 25, 3042, 3778, 3042, 3778, 25, 3042, 3778, 3042, 3787, 3042, 3785, 3786, 3832, 3833}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3897 |
| \(\displaystyle 3 \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{17/6}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )}{x^{17/6}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle 3 \left (\frac {2}{15} b \int -\frac {\sin \left (a+b \sqrt [3]{x}\right )}{x^{5/2}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{x^{5/2}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{x^{5/2}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{13/6}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )}{x^{13/6}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (\frac {2}{11} b \int -\frac {\sin \left (a+b \sqrt [3]{x}\right )}{x^{11/6}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{x^{11/6}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{x^{11/6}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{3/2}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )}{x^{3/2}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (\frac {2}{7} b \int -\frac {\sin \left (a+b \sqrt [3]{x}\right )}{x^{7/6}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{x^{7/6}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{x^{7/6}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{5/6}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )}{x^{5/6}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (\frac {2}{3} b \int -\frac {\sin \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \int \frac {\sin \left (a+b \sqrt [3]{x}\right )}{\sqrt {x}}d\sqrt [3]{x}-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \left (2 b \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \left (2 b \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \left (2 b \left (\cos (a) \int \frac {\cos \left (b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\sin (a) \int \frac {\sin \left (b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \left (2 b \left (\cos (a) \int \frac {\sin \left (\sqrt [3]{x} b+\frac {\pi }{2}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\sin (a) \int \frac {\sin \left (b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \left (2 b \left (2 \cos (a) \int \cos \left (b x^{2/3}\right )d\sqrt [6]{x}-\sin (a) \int \frac {\sin \left (b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \left (2 b \left (2 \cos (a) \int \cos \left (b x^{2/3}\right )d\sqrt [6]{x}-2 \sin (a) \int \sin \left (b x^{2/3}\right )d\sqrt [6]{x}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \left (2 b \left (2 \cos (a) \int \cos \left (b x^{2/3}\right )d\sqrt [6]{x}-\frac {\sqrt {2 \pi } \sin (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{\sqrt {b}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle 3 \left (-\frac {2}{15} b \left (\frac {2}{13} b \left (-\frac {2}{11} b \left (\frac {2}{9} b \left (-\frac {2}{7} b \left (\frac {2}{5} b \left (-\frac {2}{3} b \left (2 b \left (\frac {\sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{\sqrt {b}}-\frac {\sqrt {2 \pi } \sin (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [6]{x}\right )}{\sqrt {b}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{5 x^{5/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{9 x^{3/2}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}\right )-\frac {2 \sin \left (a+b \sqrt [3]{x}\right )}{13 x^{13/6}}\right )-\frac {2 \cos \left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}\right )\) |
3*((-2*Cos[a + b*x^(1/3)])/(15*x^(5/2)) - (2*b*((-2*Sin[a + b*x^(1/3)])/(1 3*x^(13/6)) + (2*b*((-2*Cos[a + b*x^(1/3)])/(11*x^(11/6)) - (2*b*((-2*Sin[ a + b*x^(1/3)])/(9*x^(3/2)) + (2*b*((-2*Cos[a + b*x^(1/3)])/(7*x^(7/6)) - (2*b*((-2*Sin[a + b*x^(1/3)])/(5*x^(5/6)) + (2*b*((-2*Cos[a + b*x^(1/3)])/ (3*Sqrt[x]) - (2*b*(2*b*((Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x^ (1/6)])/Sqrt[b] - (Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x^(1/6)]*Sin[a]) /Sqrt[b]) - (2*Sin[a + b*x^(1/3)])/x^(1/6)))/3))/5))/7))/9))/11))/13))/15)
3.1.54.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[2/d Subst[Int[Sin[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f }, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[Sin[( d*e - c*f)/d] Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c, d , e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol ] :> Module[{k = Denominator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
Time = 0.46 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(-\frac {2 \cos \left (a +b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{13 x^{\frac {13}{6}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{11 x^{\frac {11}{6}}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{9 x^{\frac {3}{2}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{9}\right )}{11}\right )}{13}\right )}{5}\) | \(180\) |
| default | \(-\frac {2 \cos \left (a +b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{13 x^{\frac {13}{6}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{11 x^{\frac {11}{6}}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{9 x^{\frac {3}{2}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}+\frac {2 b \left (-\frac {\cos \left (a +b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}-\frac {2 b \left (-\frac {\sin \left (a +b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+\sqrt {b}\, \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{9}\right )}{11}\right )}{13}\right )}{5}\) | \(180\) |
| meijerg | \(\frac {3 \cos \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (b^{2}\right )^{\frac {15}{4}} \left (-\frac {512 \sqrt {2}\, \left (-\frac {64 x^{2} b^{6}}{135135}+\frac {16 x^{\frac {4}{3}} b^{4}}{9009}-\frac {4 x^{\frac {2}{3}} b^{2}}{143}+1\right ) \cos \left (b \,x^{\frac {1}{3}}\right )}{15 \sqrt {\pi }\, x^{\frac {5}{2}} \left (b^{2}\right )^{\frac {15}{4}}}+\frac {1024 \sqrt {2}\, b \left (-64 x^{2} b^{6}+48 x^{\frac {4}{3}} b^{4}-420 x^{\frac {2}{3}} b^{2}+10395\right ) \sin \left (b \,x^{\frac {1}{3}}\right )}{2027025 \sqrt {\pi }\, x^{\frac {13}{6}} \left (b^{2}\right )^{\frac {15}{4}}}+\frac {131072 b^{\frac {15}{2}} \operatorname {C}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{2027025 \left (b^{2}\right )^{\frac {15}{4}}}\right )}{512}-\frac {3 \sin \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, b^{\frac {15}{2}} \left (-\frac {512 \sqrt {2}\, \left (-\frac {128 x^{2} b^{6}}{155925}+\frac {32 x^{\frac {4}{3}} b^{4}}{51975}-\frac {8 x^{\frac {2}{3}} b^{2}}{1485}+\frac {2}{15}\right ) \cos \left (b \,x^{\frac {1}{3}}\right )}{13 \sqrt {\pi }\, x^{\frac {13}{6}} b^{\frac {13}{2}}}-\frac {512 \sqrt {2}\, \left (-64 x^{2} b^{6}+240 x^{\frac {4}{3}} b^{4}-3780 x^{\frac {2}{3}} b^{2}+135135\right ) \sin \left (b \,x^{\frac {1}{3}}\right )}{2027025 \sqrt {\pi }\, x^{\frac {5}{2}} b^{\frac {15}{2}}}+\frac {131072 \,\operatorname {S}\left (\frac {x^{\frac {1}{6}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )}{2027025}\right )}{512}\) | \(261\) |
-2/5*cos(a+b*x^(1/3))/x^(5/2)-4/5*b*(-1/13/x^(13/6)*sin(a+b*x^(1/3))+2/13* b*(-1/11*cos(a+b*x^(1/3))/x^(11/6)-2/11*b*(-1/9/x^(3/2)*sin(a+b*x^(1/3))+2 /9*b*(-1/7/x^(7/6)*cos(a+b*x^(1/3))-2/7*b*(-1/5/x^(5/6)*sin(a+b*x^(1/3))+2 /5*b*(-1/3*cos(a+b*x^(1/3))/x^(1/2)-2/3*b*(-1/x^(1/6)*sin(a+b*x^(1/3))+b^( 1/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelC(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))-s in(a)*FresnelS(x^(1/6)*b^(1/2)*2^(1/2)/Pi^(1/2))))))))))
Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.66 \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\frac {2 \, {\left (128 \, \sqrt {2} \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 128 \, \sqrt {2} \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) - {\left (240 \, b^{4} x^{\frac {11}{6}} - 3780 \, b^{2} x^{\frac {7}{6}} - {\left (64 \, b^{6} x^{2} - 135135\right )} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) + 2 \, {\left (48 \, b^{5} x^{\frac {13}{6}} - 420 \, b^{3} x^{\frac {3}{2}} - {\left (64 \, b^{7} x^{2} - 10395 \, b\right )} x^{\frac {5}{6}}\right )} \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{675675 \, x^{3}} \]
2/675675*(128*sqrt(2)*pi*b^7*x^3*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*x^( 1/6)*sqrt(b/pi)) - 128*sqrt(2)*pi*b^7*x^3*sqrt(b/pi)*fresnel_sin(sqrt(2)*x ^(1/6)*sqrt(b/pi))*sin(a) - (240*b^4*x^(11/6) - 3780*b^2*x^(7/6) - (64*b^6 *x^2 - 135135)*sqrt(x))*cos(b*x^(1/3) + a) + 2*(48*b^5*x^(13/6) - 420*b^3* x^(3/2) - (64*b^7*x^2 - 10395*b)*x^(5/6))*sin(b*x^(1/3) + a))/x^3
\[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\int \frac {\cos {\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {7}{2}}}\, dx \]
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.30 \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\frac {3 \, {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (a\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, i \, b x^{\frac {1}{3}}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b^{7}}{4 \, x^{\frac {1}{6}}} \]
3/4*(((I - 1)*sqrt(2)*gamma(-15/2, I*b*x^(1/3)) - (I + 1)*sqrt(2)*gamma(-1 5/2, -I*b*x^(1/3)))*cos(a) + ((I + 1)*sqrt(2)*gamma(-15/2, I*b*x^(1/3)) - (I - 1)*sqrt(2)*gamma(-15/2, -I*b*x^(1/3)))*sin(a))*sqrt(b*x^(1/3))*b^7/x^ (1/6)
\[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\int { \frac {\cos \left (b x^{\frac {1}{3}} + a\right )}{x^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos \left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\int \frac {\cos \left (a+b\,x^{1/3}\right )}{x^{7/2}} \,d x \]