Integrand size = 18, antiderivative size = 328 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=-\frac {16 b^2}{715 x^{11/6}}+\frac {256 b^4}{45045 x^{7/6}}-\frac {4096 b^6}{675675 \sqrt {x}}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{5 x^{5/2}}+\frac {32 b^2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{715 x^{11/6}}-\frac {512 b^4 \cos ^2\left (a+b \sqrt [3]{x}\right )}{45045 x^{7/6}}+\frac {8192 b^6 \cos ^2\left (a+b \sqrt [3]{x}\right )}{675675 \sqrt {x}}+\frac {32768 b^{15/2} \sqrt {\pi } \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{675675}-\frac {32768 b^{15/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)}{675675}+\frac {8 b \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{65 x^{13/6}}-\frac {128 b^3 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{6435 x^{3/2}}+\frac {2048 b^5 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{225225 x^{5/6}}-\frac {32768 b^7 \cos \left (a+b \sqrt [3]{x}\right ) \sin \left (a+b \sqrt [3]{x}\right )}{675675 \sqrt [6]{x}} \]
-16/715*b^2/x^(11/6)+256/45045*b^4/x^(7/6)-2/5*cos(a+b*x^(1/3))^2/x^(5/2)+ 32/715*b^2*cos(a+b*x^(1/3))^2/x^(11/6)-512/45045*b^4*cos(a+b*x^(1/3))^2/x^ (7/6)+8/65*b*cos(a+b*x^(1/3))*sin(a+b*x^(1/3))/x^(13/6)-128/6435*b^3*cos(a +b*x^(1/3))*sin(a+b*x^(1/3))/x^(3/2)+2048/225225*b^5*cos(a+b*x^(1/3))*sin( a+b*x^(1/3))/x^(5/6)-32768/675675*b^7*cos(a+b*x^(1/3))*sin(a+b*x^(1/3))/x^ (1/6)+32768/675675*b^(15/2)*cos(2*a)*FresnelC(2*x^(1/6)*b^(1/2)/Pi^(1/2))* Pi^(1/2)-32768/675675*b^(15/2)*FresnelS(2*x^(1/6)*b^(1/2)/Pi^(1/2))*sin(2* a)*Pi^(1/2)-4096/675675*b^6/x^(1/2)+8192/675675*b^6*cos(a+b*x^(1/3))^2/x^( 1/2)
Time = 0.43 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\frac {-135135-135135 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+15120 b^2 x^{2/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )-3840 b^4 x^{4/3} \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+4096 b^6 x^2 \cos \left (2 \left (a+b \sqrt [3]{x}\right )\right )+32768 b^{15/2} \sqrt {\pi } x^{5/2} \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )-32768 b^{15/2} \sqrt {\pi } x^{5/2} \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right ) \sin (2 a)+41580 b \sqrt [3]{x} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-6720 b^3 x \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )+3072 b^5 x^{5/3} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )-16384 b^7 x^{7/3} \sin \left (2 \left (a+b \sqrt [3]{x}\right )\right )}{675675 x^{5/2}} \]
(-135135 - 135135*Cos[2*(a + b*x^(1/3))] + 15120*b^2*x^(2/3)*Cos[2*(a + b* x^(1/3))] - 3840*b^4*x^(4/3)*Cos[2*(a + b*x^(1/3))] + 4096*b^6*x^2*Cos[2*( a + b*x^(1/3))] + 32768*b^(15/2)*Sqrt[Pi]*x^(5/2)*Cos[2*a]*FresnelC[(2*Sqr t[b]*x^(1/6))/Sqrt[Pi]] - 32768*b^(15/2)*Sqrt[Pi]*x^(5/2)*FresnelS[(2*Sqrt [b]*x^(1/6))/Sqrt[Pi]]*Sin[2*a] + 41580*b*x^(1/3)*Sin[2*(a + b*x^(1/3))] - 6720*b^3*x*Sin[2*(a + b*x^(1/3))] + 3072*b^5*x^(5/3)*Sin[2*(a + b*x^(1/3) )] - 16384*b^7*x^(7/3)*Sin[2*(a + b*x^(1/3))])/(675675*x^(5/2))
Time = 0.97 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.11, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3897, 3042, 3795, 15, 3042, 3795, 15, 3042, 3795, 15, 3042, 3795, 15, 3042, 3793, 2009}
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.
| \(\displaystyle \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3897 |
| \(\displaystyle 3 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{17/6}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2}{x^{17/6}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{13/6}}d\sqrt [3]{x}+\frac {8}{195} b^2 \int \frac {1}{x^{13/6}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{13/6}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2}{x^{13/6}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{3/2}}d\sqrt [3]{x}+\frac {8}{99} b^2 \int \frac {1}{x^{3/2}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{3/2}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2}{x^{3/2}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \left (-\frac {16}{35} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/6}}d\sqrt [3]{x}+\frac {8}{35} b^2 \int \frac {1}{x^{5/6}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{35 x^{5/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \left (-\frac {16}{35} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{5/6}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{35 x^{5/6}}-\frac {16 b^2}{105 \sqrt {x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \left (-\frac {16}{35} b^2 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2}{x^{5/6}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{35 x^{5/6}}-\frac {16 b^2}{105 \sqrt {x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \left (-\frac {16}{35} b^2 \left (-\frac {16}{3} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}+\frac {8}{3} b^2 \int \frac {1}{\sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt [6]{x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{35 x^{5/6}}-\frac {16 b^2}{105 \sqrt {x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \left (-\frac {16}{35} b^2 \left (-\frac {16}{3} b^2 \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{\sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt [6]{x}}+\frac {16}{3} b^2 \sqrt [6]{x}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{35 x^{5/6}}-\frac {16 b^2}{105 \sqrt {x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \left (-\frac {16}{35} b^2 \left (-\frac {16}{3} b^2 \int \frac {\sin \left (a+b \sqrt [3]{x}+\frac {\pi }{2}\right )^2}{\sqrt [6]{x}}d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt [6]{x}}+\frac {16}{3} b^2 \sqrt [6]{x}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{35 x^{5/6}}-\frac {16 b^2}{105 \sqrt {x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \left (-\frac {16}{35} b^2 \left (-\frac {16}{3} b^2 \int \left (\frac {\cos \left (2 a+2 b \sqrt [3]{x}\right )}{2 \sqrt [6]{x}}+\frac {1}{2 \sqrt [6]{x}}\right )d\sqrt [3]{x}-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt [6]{x}}+\frac {16}{3} b^2 \sqrt [6]{x}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{35 x^{5/6}}-\frac {16 b^2}{105 \sqrt {x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (-\frac {16}{195} b^2 \left (-\frac {16}{99} b^2 \left (-\frac {16}{35} b^2 \left (-\frac {16}{3} b^2 \left (\frac {\sqrt {\pi } \cos (2 a) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{2 \sqrt {b}}-\frac {\sqrt {\pi } \sin (2 a) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt [6]{x}}{\sqrt {\pi }}\right )}{2 \sqrt {b}}+\sqrt [6]{x}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{3 \sqrt {x}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{3 \sqrt [6]{x}}+\frac {16}{3} b^2 \sqrt [6]{x}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{7 x^{7/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{35 x^{5/6}}-\frac {16 b^2}{105 \sqrt {x}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{11 x^{11/6}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{99 x^{3/2}}-\frac {16 b^2}{693 x^{7/6}}\right )-\frac {2 \cos ^2\left (a+b \sqrt [3]{x}\right )}{15 x^{5/2}}+\frac {8 b \sin \left (a+b \sqrt [3]{x}\right ) \cos \left (a+b \sqrt [3]{x}\right )}{195 x^{13/6}}-\frac {16 b^2}{2145 x^{11/6}}\right )\) |
3*((-16*b^2)/(2145*x^(11/6)) - (2*Cos[a + b*x^(1/3)]^2)/(15*x^(5/2)) + (8* b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(195*x^(13/6)) - (16*b^2*((-16*b^ 2)/(693*x^(7/6)) - (2*Cos[a + b*x^(1/3)]^2)/(11*x^(11/6)) + (8*b*Cos[a + b *x^(1/3)]*Sin[a + b*x^(1/3)])/(99*x^(3/2)) - (16*b^2*((-16*b^2)/(105*Sqrt[ x]) - (2*Cos[a + b*x^(1/3)]^2)/(7*x^(7/6)) + (8*b*Cos[a + b*x^(1/3)]*Sin[a + b*x^(1/3)])/(35*x^(5/6)) - (16*b^2*((16*b^2*x^(1/6))/3 - (2*Cos[a + b*x ^(1/3)]^2)/(3*Sqrt[x]) - (16*b^2*(x^(1/6) + (Sqrt[Pi]*Cos[2*a]*FresnelC[(2 *Sqrt[b]*x^(1/6))/Sqrt[Pi]])/(2*Sqrt[b]) - (Sqrt[Pi]*FresnelS[(2*Sqrt[b]*x ^(1/6))/Sqrt[Pi]]*Sin[2*a])/(2*Sqrt[b])))/3 + (8*b*Cos[a + b*x^(1/3)]*Sin[ a + b*x^(1/3)])/(3*x^(1/6))))/35))/99))/195)
3.1.60.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
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 = 0.58 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.63
| method | result | size |
| derivativedivides | \(-\frac {1}{5 x^{\frac {5}{2}}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{13 x^{\frac {13}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{11 x^{\frac {11}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{9 x^{\frac {3}{2}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{9}\right )}{11}\right )}{13}\right )}{5}\) | \(207\) |
| default | \(-\frac {1}{5 x^{\frac {5}{2}}}-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{2}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{13 x^{\frac {13}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{11 x^{\frac {11}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{9 x^{\frac {3}{2}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{7 x^{\frac {7}{6}}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{5 x^{\frac {5}{6}}}+\frac {4 b \left (-\frac {\cos \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{3 \sqrt {x}}-\frac {4 b \left (-\frac {\sin \left (2 a +2 b \,x^{\frac {1}{3}}\right )}{x^{\frac {1}{6}}}+2 \sqrt {b}\, \sqrt {\pi }\, \left (\cos \left (2 a \right ) \operatorname {C}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )-\sin \left (2 a \right ) \operatorname {S}\left (\frac {2 x^{\frac {1}{6}} \sqrt {b}}{\sqrt {\pi }}\right )\right )\right )}{3}\right )}{5}\right )}{7}\right )}{9}\right )}{11}\right )}{13}\right )}{5}\) | \(207\) |
-1/5/x^(5/2)-1/5/x^(5/2)*cos(2*a+2*b*x^(1/3))-4/5*b*(-1/13/x^(13/6)*sin(2* a+2*b*x^(1/3))+4/13*b*(-1/11/x^(11/6)*cos(2*a+2*b*x^(1/3))-4/11*b*(-1/9/x^ (3/2)*sin(2*a+2*b*x^(1/3))+4/9*b*(-1/7/x^(7/6)*cos(2*a+2*b*x^(1/3))-4/7*b* (-1/5/x^(5/6)*sin(2*a+2*b*x^(1/3))+4/5*b*(-1/3/x^(1/2)*cos(2*a+2*b*x^(1/3) )-4/3*b*(-1/x^(1/6)*sin(2*a+2*b*x^(1/3))+2*b^(1/2)*Pi^(1/2)*(cos(2*a)*Fres nelC(2*x^(1/6)*b^(1/2)/Pi^(1/2))-sin(2*a)*FresnelS(2*x^(1/6)*b^(1/2)/Pi^(1 /2))))))))))
Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\frac {2 \, {\left (16384 \, \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \cos \left (2 \, a\right ) \operatorname {C}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) - 16384 \, \pi b^{7} x^{3} \sqrt {\frac {b}{\pi }} \operatorname {S}\left (2 \, x^{\frac {1}{6}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (2 \, a\right ) - 2048 \, b^{6} x^{\frac {5}{2}} + 1920 \, b^{4} x^{\frac {11}{6}} - 7560 \, b^{2} x^{\frac {7}{6}} - {\left (3840 \, b^{4} x^{\frac {11}{6}} - 15120 \, b^{2} x^{\frac {7}{6}} - {\left (4096 \, b^{6} x^{2} - 135135\right )} \sqrt {x}\right )} \cos \left (b x^{\frac {1}{3}} + a\right )^{2} + 4 \, {\left (768 \, b^{5} x^{\frac {13}{6}} - 1680 \, b^{3} x^{\frac {3}{2}} - {\left (4096 \, b^{7} x^{2} - 10395 \, b\right )} x^{\frac {5}{6}}\right )} \cos \left (b x^{\frac {1}{3}} + a\right ) \sin \left (b x^{\frac {1}{3}} + a\right )\right )}}{675675 \, x^{3}} \]
2/675675*(16384*pi*b^7*x^3*sqrt(b/pi)*cos(2*a)*fresnel_cos(2*x^(1/6)*sqrt( b/pi)) - 16384*pi*b^7*x^3*sqrt(b/pi)*fresnel_sin(2*x^(1/6)*sqrt(b/pi))*sin (2*a) - 2048*b^6*x^(5/2) + 1920*b^4*x^(11/6) - 7560*b^2*x^(7/6) - (3840*b^ 4*x^(11/6) - 15120*b^2*x^(7/6) - (4096*b^6*x^2 - 135135)*sqrt(x))*cos(b*x^ (1/3) + a)^2 + 4*(768*b^5*x^(13/6) - 1680*b^3*x^(3/2) - (4096*b^7*x^2 - 10 395*b)*x^(5/6))*cos(b*x^(1/3) + a)*sin(b*x^(1/3) + a))/x^3
\[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\int \frac {\cos ^{2}{\left (a + b \sqrt [3]{x} \right )}}{x^{\frac {7}{2}}}\, dx \]
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.27 \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=-\frac {240 \, \sqrt {2} {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, 2 i \, b x^{\frac {1}{3}}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \cos \left (2 \, a\right ) + {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, 2 i \, b x^{\frac {1}{3}}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {15}{2}, -2 i \, b x^{\frac {1}{3}}\right )\right )} \sin \left (2 \, a\right )\right )} \sqrt {b x^{\frac {1}{3}}} b^{7} x^{\frac {7}{3}} + 1}{5 \, x^{\frac {5}{2}}} \]
-1/5*(240*sqrt(2)*((-(I - 1)*sqrt(2)*gamma(-15/2, 2*I*b*x^(1/3)) + (I + 1) *sqrt(2)*gamma(-15/2, -2*I*b*x^(1/3)))*cos(2*a) + (-(I + 1)*sqrt(2)*gamma( -15/2, 2*I*b*x^(1/3)) + (I - 1)*sqrt(2)*gamma(-15/2, -2*I*b*x^(1/3)))*sin( 2*a))*sqrt(b*x^(1/3))*b^7*x^(7/3) + 1)/x^(5/2)
\[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\int { \frac {\cos \left (b x^{\frac {1}{3}} + a\right )^{2}}{x^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^2\left (a+b \sqrt [3]{x}\right )}{x^{7/2}} \, dx=\int \frac {{\cos \left (a+b\,x^{1/3}\right )}^2}{x^{7/2}} \,d x \]