Integrand size = 25, antiderivative size = 275 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx=\frac {7 \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 d^{5/2} f}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 d^{5/2} f}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 d^{5/2} f}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}+\sqrt {d} \tan (e+f x)}\right )}{2 \sqrt {2} a^2 d^{5/2} f}-\frac {7}{6 a^2 d f (d \tan (e+f x))^{3/2}}+\frac {9}{2 a^2 d^2 f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f (d \tan (e+f x))^{3/2} \left (a^2+a^2 \tan (e+f x)\right )} \] Output:
7/2*arctan((d*tan(f*x+e))^(1/2)/d^(1/2))/a^2/d^(5/2)/f-1/4*arctan(1-2^(1/2 )*(d*tan(f*x+e))^(1/2)/d^(1/2))*2^(1/2)/a^2/d^(5/2)/f+1/4*arctan(1+2^(1/2) *(d*tan(f*x+e))^(1/2)/d^(1/2))*2^(1/2)/a^2/d^(5/2)/f-1/4*arctanh(2^(1/2)*( d*tan(f*x+e))^(1/2)/(d^(1/2)+d^(1/2)*tan(f*x+e)))*2^(1/2)/a^2/d^(5/2)/f-7/ 6/a^2/d/f/(d*tan(f*x+e))^(3/2)+9/2/a^2/d^2/f/(d*tan(f*x+e))^(1/2)+1/2/d/f/ (d*tan(f*x+e))^(3/2)/(a^2+a^2*tan(f*x+e))
Time = 2.82 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx=\frac {21 d^{3/2} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )-3 \left (-d^2\right )^{3/4} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt [4]{-d^2}}\right )+3 \left (-d^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt [4]{-d^2}}\right )+\frac {d^2 (20-4 \cot (e+f x)+27 \tan (e+f x))}{\sqrt {d \tan (e+f x)} (1+\tan (e+f x))}}{6 a^2 d^4 f} \] Input:
Integrate[1/((d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^2),x]
Output:
(21*d^(3/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]] - 3*(-d^2)^(3/4)*ArcTan[S qrt[d*Tan[e + f*x]]/(-d^2)^(1/4)] + 3*(-d^2)^(3/4)*ArcTanh[Sqrt[d*Tan[e + f*x]]/(-d^2)^(1/4)] + (d^2*(20 - 4*Cot[e + f*x] + 27*Tan[e + f*x]))/(Sqrt[ d*Tan[e + f*x]]*(1 + Tan[e + f*x])))/(6*a^2*d^4*f)
Time = 1.96 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.17, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.120, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 27, 2030, 3042, 3957, 266, 826, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 27, 73, 216}
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.
| \(\displaystyle \int \frac {1}{(a \tan (e+f x)+a)^2 (d \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \tan (e+f x)+a)^2 (d \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \frac {\int \frac {5 d \tan ^2(e+f x) a^2+7 d a^2-2 d \tan (e+f x) a^2}{2 (d \tan (e+f x))^{5/2} (\tan (e+f x) a+a)}dx}{2 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 d \tan ^2(e+f x) a^2+7 d a^2-2 d \tan (e+f x) a^2}{(d \tan (e+f x))^{5/2} (\tan (e+f x) a+a)}dx}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 d \tan (e+f x)^2 a^2+7 d a^2-2 d \tan (e+f x) a^2}{(d \tan (e+f x))^{5/2} (\tan (e+f x) a+a)}dx}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {2 \int \frac {3 \left (9 a^3 d^3+7 a^3 \tan ^2(e+f x) d^3+2 a^3 \tan (e+f x) d^3\right )}{2 (d \tan (e+f x))^{3/2} (\tan (e+f x) a+a)}dx}{3 a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {9 a^3 d^3+7 a^3 \tan ^2(e+f x) d^3+2 a^3 \tan (e+f x) d^3}{(d \tan (e+f x))^{3/2} (\tan (e+f x) a+a)}dx}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {9 a^3 d^3+7 a^3 \tan (e+f x)^2 d^3+2 a^3 \tan (e+f x) d^3}{(d \tan (e+f x))^{3/2} (\tan (e+f x) a+a)}dx}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {-\frac {2 \int \frac {7 a^4 d^5+9 a^4 \tan ^2(e+f x) d^5+2 a^4 \tan (e+f x) d^5}{2 \sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {7 a^4 d^5+9 a^4 \tan ^2(e+f x) d^5+2 a^4 \tan (e+f x) d^5}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {7 a^4 d^5+9 a^4 \tan (e+f x)^2 d^5+2 a^4 \tan (e+f x) d^5}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan ^2(e+f x)+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {\int \frac {4 a^5 d^5 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}dx}{2 a^2}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan ^2(e+f x)+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+2 a^3 d^5 \int \frac {\tan (e+f x)}{\sqrt {d \tan (e+f x)}}dx}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan ^2(e+f x)+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+2 a^3 d^4 \int \sqrt {d \tan (e+f x)}dx}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+2 a^3 d^4 \int \sqrt {d \tan (e+f x)}dx}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {2 a^3 d^5 \int \frac {\sqrt {d \tan (e+f x)}}{\tan ^2(e+f x) d^2+d^2}d(d \tan (e+f x))}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {4 a^3 d^5 \int \frac {d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {4 a^3 d^5 \left (\frac {1}{2} \int \frac {d^2 \tan ^2(e+f x)+d}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}-\frac {1}{2} \int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \tan ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {-\frac {7 a^4 d^5 \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {7 a^4 d^5 \int \frac {1}{a \sqrt {d \tan (e+f x)} (\tan (e+f x)+1)}d\tan (e+f x)}{f}+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {7 a^3 d^5 \int \frac {1}{\sqrt {d \tan (e+f x)} (\tan (e+f x)+1)}d\tan (e+f x)}{f}+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {14 a^3 d^4 \int \frac {1}{\tan (e+f x)+1}d\sqrt {d \tan (e+f x)}}{f}+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) (d \tan (e+f x))^{3/2}}+\frac {-\frac {-\frac {\frac {14 a^3 d^{9/2} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {4 a^3 d^5 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{f}}{a d^3}-\frac {18 a^2 d^2}{f \sqrt {d \tan (e+f x)}}}{a d^3}-\frac {14 a}{3 f (d \tan (e+f x))^{3/2}}}{4 a^3 d}\) |
Input:
Int[1/((d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^2),x]
Output:
1/(2*d*f*(d*Tan[e + f*x])^(3/2)*(a^2 + a^2*Tan[e + f*x])) + ((-14*a)/(3*f* (d*Tan[e + f*x])^(3/2)) - (-(((14*a^3*d^(9/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/ Sqrt[d]])/f + (4*a^3*d^5*((-(ArcTan[1 - Sqrt[2]*Sqrt[d]*Tan[e + f*x]]/(Sqr t[2]*Sqrt[d])) + ArcTan[1 + Sqrt[2]*Sqrt[d]*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d] ))/2 + (Log[d - Sqrt[2]*d^(3/2)*Tan[e + f*x] + d^2*Tan[e + f*x]^2]/(2*Sqrt [2]*Sqrt[d]) - Log[d + Sqrt[2]*d^(3/2)*Tan[e + f*x] + d^2*Tan[e + f*x]^2]/ (2*Sqrt[2]*Sqrt[d]))/2))/f)/(a*d^3)) - (18*a^2*d^2)/(f*Sqrt[d*Tan[e + f*x] ]))/(a*d^3))/(4*a^3*d)
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] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, 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[((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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
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_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Time = 1.40 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {2 d^{3} \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 d^{5} \left (d^{2}\right )^{\frac {1}{4}}}+\frac {\frac {\sqrt {d \tan \left (f x +e \right )}}{2 d \tan \left (f x +e \right )+2 d}+\frac {7 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{2 d^{5}}-\frac {1}{3 d^{4} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2}{d^{5} \sqrt {d \tan \left (f x +e \right )}}\right )}{f \,a^{2}}\) | \(227\) |
| default | \(\frac {2 d^{3} \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 d^{5} \left (d^{2}\right )^{\frac {1}{4}}}+\frac {\frac {\sqrt {d \tan \left (f x +e \right )}}{2 d \tan \left (f x +e \right )+2 d}+\frac {7 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{2 d^{5}}-\frac {1}{3 d^{4} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2}{d^{5} \sqrt {d \tan \left (f x +e \right )}}\right )}{f \,a^{2}}\) | \(227\) |
Input:
int(1/(d*tan(f*x+e))^(5/2)/(a+a*tan(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
2/f/a^2*d^3*(1/16/d^5/(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)-(d^2)^(1/4)*(d *tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f *x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x +e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1))+1/2/d ^5*(1/2*(d*tan(f*x+e))^(1/2)/(d*tan(f*x+e)+d)+7/2/d^(1/2)*arctan((d*tan(f* x+e))^(1/2)/d^(1/2)))-1/3/d^4/(d*tan(f*x+e))^(3/2)+2/d^5/(d*tan(f*x+e))^(1 /2))
Time = 0.10 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx=-\frac {84 \, {\left (\tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )^{2}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d} \tan \left (f x + e\right )}\right ) - \frac {6 \, \sqrt {2} {\left (d \tan \left (f x + e\right )^{3} + d \tan \left (f x + e\right )^{2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}} + 1\right )}{\sqrt {d}} - \frac {6 \, \sqrt {2} {\left (d \tan \left (f x + e\right )^{3} + d \tan \left (f x + e\right )^{2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}} - 1\right )}{\sqrt {d}} + \frac {3 \, \sqrt {2} {\left (d \tan \left (f x + e\right )^{3} + d \tan \left (f x + e\right )^{2}\right )} \log \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}} + \tan \left (f x + e\right ) + 1\right )}{\sqrt {d}} - \frac {3 \, \sqrt {2} {\left (d \tan \left (f x + e\right )^{3} + d \tan \left (f x + e\right )^{2}\right )} \log \left (-\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}} + \tan \left (f x + e\right ) + 1\right )}{\sqrt {d}} - 4 \, \sqrt {d \tan \left (f x + e\right )} {\left (27 \, \tan \left (f x + e\right )^{2} + 20 \, \tan \left (f x + e\right ) - 4\right )}}{24 \, {\left (a^{2} d^{3} f \tan \left (f x + e\right )^{3} + a^{2} d^{3} f \tan \left (f x + e\right )^{2}\right )}} \] Input:
integrate(1/(d*tan(f*x+e))^(5/2)/(a+a*tan(f*x+e))^2,x, algorithm="fricas")
Output:
-1/24*(84*(tan(f*x + e)^3 + tan(f*x + e)^2)*sqrt(d)*arctan(sqrt(d*tan(f*x + e))/(sqrt(d)*tan(f*x + e))) - 6*sqrt(2)*(d*tan(f*x + e)^3 + d*tan(f*x + e)^2)*arctan(sqrt(2)*sqrt(d*tan(f*x + e))/sqrt(d) + 1)/sqrt(d) - 6*sqrt(2) *(d*tan(f*x + e)^3 + d*tan(f*x + e)^2)*arctan(sqrt(2)*sqrt(d*tan(f*x + e)) /sqrt(d) - 1)/sqrt(d) + 3*sqrt(2)*(d*tan(f*x + e)^3 + d*tan(f*x + e)^2)*lo g(sqrt(2)*sqrt(d*tan(f*x + e))/sqrt(d) + tan(f*x + e) + 1)/sqrt(d) - 3*sqr t(2)*(d*tan(f*x + e)^3 + d*tan(f*x + e)^2)*log(-sqrt(2)*sqrt(d*tan(f*x + e ))/sqrt(d) + tan(f*x + e) + 1)/sqrt(d) - 4*sqrt(d*tan(f*x + e))*(27*tan(f* x + e)^2 + 20*tan(f*x + e) - 4))/(a^2*d^3*f*tan(f*x + e)^3 + a^2*d^3*f*tan (f*x + e)^2)
\[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx=\frac {\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan ^{2}{\left (e + f x \right )} + 2 \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan {\left (e + f x \right )} + \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx}{a^{2}} \] Input:
integrate(1/(d*tan(f*x+e))**(5/2)/(a+a*tan(f*x+e))**2,x)
Output:
Integral(1/((d*tan(e + f*x))**(5/2)*tan(e + f*x)**2 + 2*(d*tan(e + f*x))** (5/2)*tan(e + f*x) + (d*tan(e + f*x))**(5/2)), x)/a**2
Time = 0.12 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx=\frac {\frac {4 \, {\left (27 \, d^{2} \tan \left (f x + e\right )^{2} + 20 \, d^{2} \tan \left (f x + e\right ) - 4 \, d^{2}\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} a^{2} d + \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{2} d^{2}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )}}{a^{2} d} + \frac {84 \, \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{2} d^{\frac {3}{2}}}}{24 \, d f} \] Input:
integrate(1/(d*tan(f*x+e))^(5/2)/(a+a*tan(f*x+e))^2,x, algorithm="maxima")
Output:
1/24*(4*(27*d^2*tan(f*x + e)^2 + 20*d^2*tan(f*x + e) - 4*d^2)/((d*tan(f*x + e))^(5/2)*a^2*d + (d*tan(f*x + e))^(3/2)*a^2*d^2) + 3*(2*sqrt(2)*arctan( 1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2*sqrt(d*tan(f*x + e)))/s qrt(d))/sqrt(d) - sqrt(2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e) )*sqrt(d) + d)/sqrt(d) + sqrt(2)*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f *x + e))*sqrt(d) + d)/sqrt(d))/(a^2*d) + 84*arctan(sqrt(d*tan(f*x + e))/sq rt(d))/(a^2*d^(3/2)))/(d*f)
Timed out. \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx=\text {Timed out} \] Input:
integrate(1/(d*tan(f*x+e))^(5/2)/(a+a*tan(f*x+e))^2,x, algorithm="giac")
Output:
Timed out
Time = 2.05 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.54 \[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx=\frac {\frac {9\,{\mathrm {tan}\left (e+f\,x\right )}^2}{2}+\frac {10\,\mathrm {tan}\left (e+f\,x\right )}{3}-\frac {2}{3}}{a^2\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}+a^2\,d\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}+\frac {\mathrm {atan}\left (\frac {2048\,a^{10}\,d^{18}\,f^5\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {1}{a^8\,d^{10}\,f^4}\right )}^{1/4}}{2048\,a^8\,d^{16}\,f^4+100352\,a^{12}\,d^{21}\,f^6\,\sqrt {-\frac {1}{a^8\,d^{10}\,f^4}}}+\frac {100352\,a^{14}\,d^{23}\,f^7\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {1}{a^8\,d^{10}\,f^4}\right )}^{3/4}}{2048\,a^8\,d^{16}\,f^4+100352\,a^{12}\,d^{21}\,f^6\,\sqrt {-\frac {1}{a^8\,d^{10}\,f^4}}}\right )\,{\left (-\frac {1}{a^8\,d^{10}\,f^4}\right )}^{1/4}}{2}+\mathrm {atan}\left (\frac {a^{10}\,d^{18}\,f^5\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {1}{256\,a^8\,d^{10}\,f^4}\right )}^{1/4}\,8192{}\mathrm {i}}{2048\,a^8\,d^{16}\,f^4-1605632\,a^{12}\,d^{21}\,f^6\,\sqrt {-\frac {1}{256\,a^8\,d^{10}\,f^4}}}-\frac {a^{14}\,d^{23}\,f^7\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {1}{256\,a^8\,d^{10}\,f^4}\right )}^{3/4}\,6422528{}\mathrm {i}}{2048\,a^8\,d^{16}\,f^4-1605632\,a^{12}\,d^{21}\,f^6\,\sqrt {-\frac {1}{256\,a^8\,d^{10}\,f^4}}}\right )\,{\left (-\frac {1}{256\,a^8\,d^{10}\,f^4}\right )}^{1/4}\,2{}\mathrm {i}+\frac {\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-d^5}\,1{}\mathrm {i}}{d^3}\right )\,\sqrt {-d^5}\,7{}\mathrm {i}}{2\,a^2\,d^5\,f} \] Input:
int(1/((d*tan(e + f*x))^(5/2)*(a + a*tan(e + f*x))^2),x)
Output:
((10*tan(e + f*x))/3 + (9*tan(e + f*x)^2)/2 - 2/3)/(a^2*f*(d*tan(e + f*x)) ^(5/2) + a^2*d*f*(d*tan(e + f*x))^(3/2)) + (atan((2048*a^10*d^18*f^5*(d*ta n(e + f*x))^(1/2)*(-1/(a^8*d^10*f^4))^(1/4))/(2048*a^8*d^16*f^4 + 100352*a ^12*d^21*f^6*(-1/(a^8*d^10*f^4))^(1/2)) + (100352*a^14*d^23*f^7*(d*tan(e + f*x))^(1/2)*(-1/(a^8*d^10*f^4))^(3/4))/(2048*a^8*d^16*f^4 + 100352*a^12*d ^21*f^6*(-1/(a^8*d^10*f^4))^(1/2)))*(-1/(a^8*d^10*f^4))^(1/4))/2 + atan((a ^10*d^18*f^5*(d*tan(e + f*x))^(1/2)*(-1/(256*a^8*d^10*f^4))^(1/4)*8192i)/( 2048*a^8*d^16*f^4 - 1605632*a^12*d^21*f^6*(-1/(256*a^8*d^10*f^4))^(1/2)) - (a^14*d^23*f^7*(d*tan(e + f*x))^(1/2)*(-1/(256*a^8*d^10*f^4))^(3/4)*64225 28i)/(2048*a^8*d^16*f^4 - 1605632*a^12*d^21*f^6*(-1/(256*a^8*d^10*f^4))^(1 /2)))*(-1/(256*a^8*d^10*f^4))^(1/4)*2i + (atan(((d*tan(e + f*x))^(1/2)*(-d ^5)^(1/2)*1i)/d^3)*(-d^5)^(1/2)*7i)/(2*a^2*d^5*f)
\[ \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx=\frac {\sqrt {d}\, \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )^{5}+2 \tan \left (f x +e \right )^{4}+\tan \left (f x +e \right )^{3}}d x \right )}{a^{2} d^{3}} \] Input:
int(1/(d*tan(f*x+e))^(5/2)/(a+a*tan(f*x+e))^2,x)
Output:
(sqrt(d)*int(sqrt(tan(e + f*x))/(tan(e + f*x)**5 + 2*tan(e + f*x)**4 + tan (e + f*x)**3),x))/(a**2*d**3)