Integrand size = 23, antiderivative size = 466 \[ \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}+\frac {b \arctan \left (\sqrt {3}+2 \sqrt [3]{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {\sqrt {3} a^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3} \left (a^2+b^2\right ) d}+\frac {\sqrt {3} a \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )}{2 \left (a^2+b^2\right ) d}+\frac {b \arctan \left (\sqrt [3]{\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 b^{2/3} \left (a^2+b^2\right ) d}-\frac {a \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )}{2 \left (a^2+b^2\right ) d}+\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}-\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{\tan (c+d x)}+\tan ^{\frac {2}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac {a^{5/3} \log (a+b \tan (c+d x))}{2 b^{2/3} \left (a^2+b^2\right ) d}+\frac {a \log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )}{4 \left (a^2+b^2\right ) d} \] Output:
1/2*b*arctan(-3^(1/2)+2*tan(d*x+c)^(1/3))/(a^2+b^2)/d+1/2*b*arctan(3^(1/2) +2*tan(d*x+c)^(1/3))/(a^2+b^2)/d-3^(1/2)*a^(5/3)*arctan(1/3*(a^(1/3)-2*b^( 1/3)*tan(d*x+c)^(1/3))*3^(1/2)/a^(1/3))/b^(2/3)/(a^2+b^2)/d+1/2*3^(1/2)*a* arctan(1/3*(1-2*tan(d*x+c)^(2/3))*3^(1/2))/(a^2+b^2)/d+b*arctan(tan(d*x+c) ^(1/3))/(a^2+b^2)/d-3/2*a^(5/3)*ln(a^(1/3)+b^(1/3)*tan(d*x+c)^(1/3))/b^(2/ 3)/(a^2+b^2)/d-1/2*a*ln(1+tan(d*x+c)^(2/3))/(a^2+b^2)/d+1/4*3^(1/2)*b*ln(1 -3^(1/2)*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3))/(a^2+b^2)/d-1/4*3^(1/2)*b*ln(1 +3^(1/2)*tan(d*x+c)^(1/3)+tan(d*x+c)^(2/3))/(a^2+b^2)/d+1/2*a^(5/3)*ln(a+b *tan(d*x+c))/b^(2/3)/(a^2+b^2)/d+1/4*a*ln(1-tan(d*x+c)^(2/3)+tan(d*x+c)^(4 /3))/(a^2+b^2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.33 \[ \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {5 a \left (2 \sqrt {3} \arctan \left (\frac {1-2 \tan ^{\frac {2}{3}}(c+d x)}{\sqrt {3}}\right )-2 \log \left (1+\tan ^{\frac {2}{3}}(c+d x)\right )+\log \left (1-\tan ^{\frac {2}{3}}(c+d x)+\tan ^{\frac {4}{3}}(c+d x)\right )\right )+30 a \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {2}{3}}(c+d x)+12 b \operatorname {Hypergeometric2F1}\left (\frac {5}{6},1,\frac {11}{6},-\tan ^2(c+d x)\right ) \tan ^{\frac {5}{3}}(c+d x)}{20 \left (a^2+b^2\right ) d} \] Input:
Integrate[Tan[c + d*x]^(5/3)/(a + b*Tan[c + d*x]),x]
Output:
(5*a*(2*Sqrt[3]*ArcTan[(1 - 2*Tan[c + d*x]^(2/3))/Sqrt[3]] - 2*Log[1 + Tan [c + d*x]^(2/3)] + Log[1 - Tan[c + d*x]^(2/3) + Tan[c + d*x]^(4/3)]) + 30* a*Hypergeometric2F1[2/3, 1, 5/3, -((b*Tan[c + d*x])/a)]*Tan[c + d*x]^(2/3) + 12*b*Hypergeometric2F1[5/6, 1, 11/6, -Tan[c + d*x]^2]*Tan[c + d*x]^(5/3 ))/(20*(a^2 + b^2)*d)
Time = 1.54 (sec) , antiderivative size = 425, normalized size of antiderivative = 0.91, number of steps used = 30, number of rules used = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.261, Rules used = {3042, 4057, 3042, 4011, 3042, 4011, 3042, 4021, 3042, 3957, 266, 807, 750, 16, 824, 27, 216, 1142, 25, 1083, 217, 1103, 4117, 60, 60, 68, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^{5/3}}{a+b \tan (c+d x)}dx\) |
| \(\Big \downarrow \) 4057 |
| \(\displaystyle \frac {\int \tan ^{\frac {5}{3}}(c+d x) (a-b \tan (c+d x))dx}{a^2+b^2}+\frac {b^2 \int \frac {\tan ^{\frac {5}{3}}(c+d x) \left (\tan ^2(c+d x)+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {\int \tan (c+d x)^{5/3} (a-b \tan (c+d x))dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {\int \tan ^{\frac {2}{3}}(c+d x) (b+a \tan (c+d x))dx-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \tan (c+d x)^{2/3} (b+a \tan (c+d x))dx-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {\int \frac {b \tan (c+d x)-a}{\sqrt [3]{\tan (c+d x)}}dx+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b \tan (c+d x)-a}{\sqrt [3]{\tan (c+d x)}}dx+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 4021 |
| \(\displaystyle \frac {-a \int \frac {1}{\sqrt [3]{\tan (c+d x)}}dx+b \int \tan ^{\frac {2}{3}}(c+d x)dx+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-a \int \frac {1}{\sqrt [3]{\tan (c+d x)}}dx+b \int \tan (c+d x)^{2/3}dx+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {-\frac {a \int \frac {1}{\sqrt [3]{\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}+\frac {b \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-\frac {3 a \int \frac {\sqrt [3]{\tan (c+d x)}}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}+\frac {3 b \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {-\frac {3 a \int \frac {1}{\tan (c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)}{2 d}+\frac {3 b \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {-\frac {3 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\tan ^{\frac {2}{3}}(c+d x)\right )}{2 d}+\frac {3 b \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {-\frac {3 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 b \int \frac {\tan ^{\frac {4}{3}}(c+d x)}{\tan ^2(c+d x)+1}d\sqrt [3]{\tan (c+d x)}}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {-\frac {3 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 b \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int -\frac {1-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{2 \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \int -\frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}{2 \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )}d\sqrt [3]{\tan (c+d x)}\right )}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {-\frac {3 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 b \left (\frac {1}{3} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{6} \int \frac {1-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {-\frac {3 a \left (\frac {1}{3} \int \left (2-\tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 b \left (-\frac {1}{6} \int \frac {1-\sqrt {3} \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-\frac {3 a \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)-\frac {1}{2} \int \left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )d\tan ^{\frac {2}{3}}(c+d x)\right )+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 b \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {3 a \left (\frac {1}{3} \left (\frac {3}{2} \int 1d\tan ^{\frac {2}{3}}(c+d x)+\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)\right )+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 b \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {3 a \left (\frac {1}{3} \left (\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)-3 \int \frac {1}{-2 \tan ^{\frac {2}{3}}(c+d x)-2}d\left (2 \tan ^{\frac {2}{3}}(c+d x)-1\right )\right )+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 b \left (\frac {1}{6} \left (-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (-\int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-1}d\left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {3 a \left (\frac {1}{3} \left (\frac {1}{2} \int \left (1-2 \tan ^{\frac {2}{3}}(c+d x)\right )d\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )\right )+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 b \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}}{\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}}{\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1}d\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )\right )}{d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {b^2 \int \frac {\tan (c+d x)^{5/3} \left (\tan (c+d x)^2+1\right )}{a+b \tan (c+d x)}dx}{a^2+b^2}+\frac {-\frac {3 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}+\frac {3 b \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {b^2 \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)}d\tan (c+d x)}{d \left (a^2+b^2\right )}+\frac {-\frac {3 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}+\frac {3 b \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {b^2 \left (\frac {3 \tan ^{\frac {5}{3}}(c+d x)}{5 b}-\frac {a \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{a+b \tan (c+d x)}d\tan (c+d x)}{b}\right )}{d \left (a^2+b^2\right )}+\frac {-\frac {3 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}+\frac {3 b \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {b^2 \left (\frac {3 \tan ^{\frac {5}{3}}(c+d x)}{5 b}-\frac {a \left (\frac {3 \tan ^{\frac {2}{3}}(c+d x)}{2 b}-\frac {a \int \frac {1}{\sqrt [3]{\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{b}\right )}{b}\right )}{d \left (a^2+b^2\right )}+\frac {-\frac {3 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}+\frac {3 b \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle \frac {b^2 \left (\frac {3 \tan ^{\frac {5}{3}}(c+d x)}{5 b}-\frac {a \left (\frac {3 \tan ^{\frac {2}{3}}(c+d x)}{2 b}-\frac {a \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{\tan (c+d x)} \sqrt [3]{a}}{\sqrt [3]{b}}+\tan ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\tan (c+d x)}}{2 b}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{\tan (c+d x)}}d\sqrt [3]{\tan (c+d x)}}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b \tan (c+d x))}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\right )}{b}\right )}{d \left (a^2+b^2\right )}+\frac {-\frac {3 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}+\frac {3 b \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {b^2 \left (\frac {3 \tan ^{\frac {5}{3}}(c+d x)}{5 b}-\frac {a \left (\frac {3 \tan ^{\frac {2}{3}}(c+d x)}{2 b}-\frac {a \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{\tan (c+d x)} \sqrt [3]{a}}{\sqrt [3]{b}}+\tan ^{\frac {2}{3}}(c+d x)}d\sqrt [3]{\tan (c+d x)}}{2 b}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b \tan (c+d x))}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\right )}{b}\right )}{d \left (a^2+b^2\right )}+\frac {-\frac {3 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}+\frac {3 b \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {b^2 \left (\frac {3 \tan ^{\frac {5}{3}}(c+d x)}{5 b}-\frac {a \left (\frac {3 \tan ^{\frac {2}{3}}(c+d x)}{2 b}-\frac {a \left (\frac {3 \int \frac {1}{-\tan ^{\frac {2}{3}}(c+d x)-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b \tan (c+d x))}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\right )}{b}\right )}{d \left (a^2+b^2\right )}+\frac {-\frac {3 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}+\frac {3 b \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {3 a \left (\frac {\arctan \left (\frac {2 \tan ^{\frac {2}{3}}(c+d x)-1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+1\right )\right )}{2 d}+\frac {3 a \tan ^{\frac {2}{3}}(c+d x)}{2 d}+\frac {3 b \left (\frac {1}{3} \arctan \left (\sqrt [3]{\tan (c+d x)}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)-\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )-\arctan \left (\sqrt {3}-2 \sqrt [3]{\tan (c+d x)}\right )\right )+\frac {1}{6} \left (\arctan \left (2 \sqrt [3]{\tan (c+d x)}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\tan ^{\frac {2}{3}}(c+d x)+\sqrt {3} \sqrt [3]{\tan (c+d x)}+1\right )\right )\right )}{d}-\frac {3 b \tan ^{\frac {5}{3}}(c+d x)}{5 d}}{a^2+b^2}+\frac {b^2 \left (\frac {3 \tan ^{\frac {5}{3}}(c+d x)}{5 b}-\frac {a \left (\frac {3 \tan ^{\frac {2}{3}}(c+d x)}{2 b}-\frac {a \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{\tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{\tan (c+d x)}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b \tan (c+d x))}{2 \sqrt [3]{a} b^{2/3}}\right )}{b}\right )}{b}\right )}{d \left (a^2+b^2\right )}\) |
Input:
Int[Tan[c + d*x]^(5/3)/(a + b*Tan[c + d*x]),x]
Output:
(b^2*(-((a*(-((a*(-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*Tan[c + d*x]^(1/3))/a^ (1/3))/Sqrt[3]])/(a^(1/3)*b^(2/3))) - (3*Log[a^(1/3) + b^(1/3)*Tan[c + d*x ]^(1/3)])/(2*a^(1/3)*b^(2/3)) + Log[a + b*Tan[c + d*x]]/(2*a^(1/3)*b^(2/3) )))/b) + (3*Tan[c + d*x]^(2/3))/(2*b)))/b) + (3*Tan[c + d*x]^(5/3))/(5*b)) )/((a^2 + b^2)*d) + ((-3*a*(ArcTan[(-1 + 2*Tan[c + d*x]^(2/3))/Sqrt[3]]/Sq rt[3] + Log[1 + Tan[c + d*x]^(2/3)]/3))/(2*d) + (3*b*(ArcTan[Tan[c + d*x]^ (1/3)]/3 + (-ArcTan[Sqrt[3] - 2*Tan[c + d*x]^(1/3)] + (Sqrt[3]*Log[1 - Sqr t[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/2)/6 + (ArcTan[Sqrt[3] + 2* Tan[c + d*x]^(1/3)] - (Sqrt[3]*Log[1 + Sqrt[3]*Tan[c + d*x]^(1/3) + Tan[c + d*x]^(2/3)])/2)/6))/d + (3*a*Tan[c + d*x]^(2/3))/(2*d) - (3*b*Tan[c + d* x]^(5/3))/(5*d))/(a^2 + b^2)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Tan[e + f*x])^m, x], x] + Simp[d/b Int [(b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ[c^ 2 + d^2, 0] && !IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(c^2 + d^2) Int[(a + b*Tan[e + f*x])^m *(c - d*Tan[e + f*x]), x], x] + Simp[d^2/(c^2 + d^2) Int[(a + b*Tan[e + f *x])^m*((1 + Tan[e + f*x]^2)/(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d ^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Time = 0.20 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.73
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {3 \left (-\sqrt {3}\, b -a \right ) \ln \left (1-\sqrt {3}\, \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}\right )}{4}-3 \left (\sqrt {3}\, a +b +\frac {\left (-\sqrt {3}\, b -a \right ) \sqrt {3}}{2}\right ) \arctan \left (-\sqrt {3}+2 \tan \left (d x +c \right )^{\frac {1}{3}}\right )+\frac {3 \left (-\sqrt {3}\, b +a \right ) \ln \left (1+\sqrt {3}\, \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}\right )}{4}+3 \left (\sqrt {3}\, a -b -\frac {\left (-\sqrt {3}\, b +a \right ) \sqrt {3}}{2}\right ) \arctan \left (\sqrt {3}+2 \tan \left (d x +c \right )^{\frac {1}{3}}\right )}{3 a^{2}+3 b^{2}}+\frac {3 \left (-\frac {\ln \left (\tan \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\tan \left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2}}{a^{2}+b^{2}}+\frac {-\frac {3 a \ln \left (1+\tan \left (d x +c \right )^{\frac {2}{3}}\right )}{2}+3 b \arctan \left (\tan \left (d x +c \right )^{\frac {1}{3}}\right )}{3 a^{2}+3 b^{2}}}{d}\) | \(341\) |
| default | \(\frac {\frac {-\frac {3 \left (-\sqrt {3}\, b -a \right ) \ln \left (1-\sqrt {3}\, \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}\right )}{4}-3 \left (\sqrt {3}\, a +b +\frac {\left (-\sqrt {3}\, b -a \right ) \sqrt {3}}{2}\right ) \arctan \left (-\sqrt {3}+2 \tan \left (d x +c \right )^{\frac {1}{3}}\right )+\frac {3 \left (-\sqrt {3}\, b +a \right ) \ln \left (1+\sqrt {3}\, \tan \left (d x +c \right )^{\frac {1}{3}}+\tan \left (d x +c \right )^{\frac {2}{3}}\right )}{4}+3 \left (\sqrt {3}\, a -b -\frac {\left (-\sqrt {3}\, b +a \right ) \sqrt {3}}{2}\right ) \arctan \left (\sqrt {3}+2 \tan \left (d x +c \right )^{\frac {1}{3}}\right )}{3 a^{2}+3 b^{2}}+\frac {3 \left (-\frac {\ln \left (\tan \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\tan \left (d x +c \right )^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x +c \right )^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \tan \left (d x +c \right )^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2}}{a^{2}+b^{2}}+\frac {-\frac {3 a \ln \left (1+\tan \left (d x +c \right )^{\frac {2}{3}}\right )}{2}+3 b \arctan \left (\tan \left (d x +c \right )^{\frac {1}{3}}\right )}{3 a^{2}+3 b^{2}}}{d}\) | \(341\) |
Input:
int(tan(d*x+c)^(5/3)/(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(3/(3*a^2+3*b^2)*(-1/4*(-3^(1/2)*b-a)*ln(1-3^(1/2)*tan(d*x+c)^(1/3)+ta n(d*x+c)^(2/3))-(3^(1/2)*a+b+1/2*(-3^(1/2)*b-a)*3^(1/2))*arctan(-3^(1/2)+2 *tan(d*x+c)^(1/3))+1/4*(-3^(1/2)*b+a)*ln(1+3^(1/2)*tan(d*x+c)^(1/3)+tan(d* x+c)^(2/3))+(3^(1/2)*a-b-1/2*(-3^(1/2)*b+a)*3^(1/2))*arctan(3^(1/2)+2*tan( d*x+c)^(1/3)))+3*(-1/3/b/(a/b)^(1/3)*ln(tan(d*x+c)^(1/3)+(a/b)^(1/3))+1/6/ b/(a/b)^(1/3)*ln(tan(d*x+c)^(2/3)-(a/b)^(1/3)*tan(d*x+c)^(1/3)+(a/b)^(2/3) )+1/3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*tan(d*x+c)^( 1/3)-1)))*a^2/(a^2+b^2)+3/(3*a^2+3*b^2)*(-1/2*a*ln(1+tan(d*x+c)^(2/3))+b*a rctan(tan(d*x+c)^(1/3))))
Result contains complex when optimal does not.
Time = 2.08 (sec) , antiderivative size = 73878, normalized size of antiderivative = 158.54 \[ \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(tan(d*x+c)^(5/3)/(a+b*tan(d*x+c)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {\tan ^{\frac {5}{3}}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \] Input:
integrate(tan(d*x+c)**(5/3)/(a+b*tan(d*x+c)),x)
Output:
Integral(tan(c + d*x)**(5/3)/(a + b*tan(c + d*x)), x)
Timed out. \[ \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(tan(d*x+c)^(5/3)/(a+b*tan(d*x+c)),x, algorithm="maxima")
Output:
Timed out
Time = 0.28 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.93 \[ \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {a^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \tan \left (d x + c\right )^{\frac {1}{3}} \right |}\right )}{a^{3} d + a b^{2} d} + \frac {{\left (\sqrt {3} a + b\right )} \arctan \left (\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}}\right )}{2 \, {\left (a^{2} d + b^{2} d\right )}} - \frac {{\left (\sqrt {3} a - b\right )} \arctan \left (-\sqrt {3} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}}\right )}{2 \, {\left (a^{2} d + b^{2} d\right )}} + \frac {b \arctan \left (\tan \left (d x + c\right )^{\frac {1}{3}}\right )}{a^{2} d + b^{2} d} + \frac {a \log \left (\tan \left (d x + c\right )^{\frac {4}{3}} - \tan \left (d x + c\right )^{\frac {2}{3}} + 1\right )}{4 \, {\left (a^{2} d + b^{2} d\right )}} - \frac {3 \, b \log \left (\sqrt {3} \tan \left (d x + c\right )^{\frac {1}{3}} + \tan \left (d x + c\right )^{\frac {2}{3}} + 1\right )}{4 \, {\left (\sqrt {3} a^{2} d + \sqrt {3} b^{2} d\right )}} + \frac {3 \, b \log \left (-\sqrt {3} \tan \left (d x + c\right )^{\frac {1}{3}} + \tan \left (d x + c\right )^{\frac {2}{3}} + 1\right )}{4 \, {\left (\sqrt {3} a^{2} d + \sqrt {3} b^{2} d\right )}} - \frac {a \log \left (\tan \left (d x + c\right )^{\frac {2}{3}} + 1\right )}{2 \, {\left (a^{2} d + b^{2} d\right )}} - \frac {3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \tan \left (d x + c\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (\sqrt {3} a^{2} b^{2} + \sqrt {3} b^{4}\right )} d} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} a \log \left (\left (-\frac {a}{b}\right )^{\frac {2}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \tan \left (d x + c\right )^{\frac {1}{3}} + \tan \left (d x + c\right )^{\frac {2}{3}}\right )}{2 \, {\left (a^{2} b^{2} + b^{4}\right )} d} \] Input:
integrate(tan(d*x+c)^(5/3)/(a+b*tan(d*x+c)),x, algorithm="giac")
Output:
-a^2*(-a/b)^(2/3)*log(abs(-(-a/b)^(1/3) + tan(d*x + c)^(1/3)))/(a^3*d + a* b^2*d) + 1/2*(sqrt(3)*a + b)*arctan(sqrt(3) + 2*tan(d*x + c)^(1/3))/(a^2*d + b^2*d) - 1/2*(sqrt(3)*a - b)*arctan(-sqrt(3) + 2*tan(d*x + c)^(1/3))/(a ^2*d + b^2*d) + b*arctan(tan(d*x + c)^(1/3))/(a^2*d + b^2*d) + 1/4*a*log(t an(d*x + c)^(4/3) - tan(d*x + c)^(2/3) + 1)/(a^2*d + b^2*d) - 3/4*b*log(sq rt(3)*tan(d*x + c)^(1/3) + tan(d*x + c)^(2/3) + 1)/(sqrt(3)*a^2*d + sqrt(3 )*b^2*d) + 3/4*b*log(-sqrt(3)*tan(d*x + c)^(1/3) + tan(d*x + c)^(2/3) + 1) /(sqrt(3)*a^2*d + sqrt(3)*b^2*d) - 1/2*a*log(tan(d*x + c)^(2/3) + 1)/(a^2* d + b^2*d) - 3*(-a*b^2)^(2/3)*a*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*tan(d *x + c)^(1/3))/(-a/b)^(1/3))/((sqrt(3)*a^2*b^2 + sqrt(3)*b^4)*d) + 1/2*(-a *b^2)^(2/3)*a*log((-a/b)^(2/3) + (-a/b)^(1/3)*tan(d*x + c)^(1/3) + tan(d*x + c)^(2/3))/((a^2*b^2 + b^4)*d)
Time = 10.00 (sec) , antiderivative size = 2137, normalized size of antiderivative = 4.59 \[ \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \] Input:
int(tan(c + d*x)^(5/3)/(a + b*tan(c + d*x)),x)
Output:
symsum(log(root(32*a^2*b^2*d^4*z^4 + 16*b^4*d^4*z^4 + 16*a^4*d^4*z^4 - 16* a*b^2*d^3*z^3 - 16*a^3*d^3*z^3 - 4*b^2*d^2*z^2 + 12*a^2*d^2*z^2 - 4*a*d*z + 1, z, k)^2*(root(32*a^2*b^2*d^4*z^4 + 16*b^4*d^4*z^4 + 16*a^4*d^4*z^4 - 16*a*b^2*d^3*z^3 - 16*a^3*d^3*z^3 - 4*b^2*d^2*z^2 + 12*a^2*d^2*z^2 - 4*a*d *z + 1, z, k)*(root(32*a^2*b^2*d^4*z^4 + 16*b^4*d^4*z^4 + 16*a^4*d^4*z^4 - 16*a*b^2*d^3*z^3 - 16*a^3*d^3*z^3 - 4*b^2*d^2*z^2 + 12*a^2*d^2*z^2 - 4*a* d*z + 1, z, k)^2*(root(32*a^2*b^2*d^4*z^4 + 16*b^4*d^4*z^4 + 16*a^4*d^4*z^ 4 - 16*a*b^2*d^3*z^3 - 16*a^3*d^3*z^3 - 4*b^2*d^2*z^2 + 12*a^2*d^2*z^2 - 4 *a*d*z + 1, z, k)*((6561*tan(c + d*x)^(1/3)*(96*a^4*b^9*d^6 - 8*a^2*b^11*d ^6 + 208*a^6*b^7*d^6 + 96*a^8*b^5*d^6 - 8*a^10*b^3*d^6))/d^8 + (6561*root( 32*a^2*b^2*d^4*z^4 + 16*b^4*d^4*z^4 + 16*a^4*d^4*z^4 - 16*a*b^2*d^3*z^3 - 16*a^3*d^3*z^3 - 4*b^2*d^2*z^2 + 12*a^2*d^2*z^2 - 4*a*d*z + 1, z, k)^2*(64 *a*b^14*d^6 + 192*a^3*b^12*d^6 + 128*a^5*b^10*d^6 - 128*a^7*b^8*d^6 - 192* a^9*b^6*d^6 - 64*a^11*b^4*d^6))/d^6) - (6561*(44*a^2*b^10*d^3 + 20*a^4*b^8 *d^3 - 28*a^6*b^6*d^3 + 60*a^8*b^4*d^3 + 64*a^10*b^2*d^3))/d^6) - (6561*ta n(c + d*x)^(1/3)*(8*a^9*b*d^3 + a^3*b^7*d^3 + 50*a^5*b^5*d^3 - 55*a^7*b^3* d^3))/d^8) + (6561*(a*b^8 + a^3*b^6 + 4*a^7*b^2))/d^6) - (6561*tan(c + d*x )^(1/3)*(a^6*b - a^4*b^3))/d^8)*root(32*a^2*b^2*d^4*z^4 + 16*b^4*d^4*z^4 + 16*a^4*d^4*z^4 - 16*a*b^2*d^3*z^3 - 16*a^3*d^3*z^3 - 4*b^2*d^2*z^2 + 12*a ^2*d^2*z^2 - 4*a*d*z + 1, z, k), k, 1, 4) - (log(tan(c + d*x)^(1/3) + 1...
\[ \int \frac {\tan ^{\frac {5}{3}}(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {\tan \left (d x +c \right )^{\frac {5}{3}}}{a +\tan \left (d x +c \right ) b}d x \] Input:
int(tan(d*x+c)^(5/3)/(a+b*tan(d*x+c)),x)
Output:
int((tan(c + d*x)**(2/3)*tan(c + d*x))/(tan(c + d*x)*b + a),x)