Integrand size = 23, antiderivative size = 297 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {2 a^2+3 b^2}{a^2 \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}}+\frac {b^2}{a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \] Output:
-1/2*(a^2+2*a*b-b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/(a^2+b^2) ^2/d-1/2*(a^2+2*a*b-b^2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/(a^2+b ^2)^2/d-b^(5/2)*(7*a^2+3*b^2)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^( 5/2)/(a^2+b^2)^2/d+1/2*(a^2-2*a*b-b^2)*arctanh(2^(1/2)*tan(d*x+c)^(1/2)/(1 +tan(d*x+c)))*2^(1/2)/(a^2+b^2)^2/d-(2*a^2+3*b^2)/a^2/(a^2+b^2)/d/tan(d*x+ c)^(1/2)+b^2/a/(a^2+b^2)/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\frac {b^{5/2} \left (7 a^2+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right )}+\frac {(-1)^{3/4} a \left ((a+i b)^2 \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-(a-i b)^2 \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {2 a^2+3 b^2}{a \sqrt {\tan (c+d x)}}-\frac {b^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{a \left (a^2+b^2\right ) d} \] Input:
Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2),x]
Output:
-(((b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/ (a^(3/2)*(a^2 + b^2)) + ((-1)^(3/4)*a*((a + I*b)^2*ArcTan[(-1)^(3/4)*Sqrt[ Tan[c + d*x]]] - (a - I*b)^2*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]))/(a^2 + b^2) + (2*a^2 + 3*b^2)/(a*Sqrt[Tan[c + d*x]]) - b^2/(Sqrt[Tan[c + d*x]] *(a + b*Tan[c + d*x])))/(a*(a^2 + b^2)*d))
Time = 1.53 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.10, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}
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.
| \(\displaystyle \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \frac {\int \frac {2 a^2-2 b \tan (c+d x) a+3 b^2+3 b^2 \tan ^2(c+d x)}{2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 a^2-2 b \tan (c+d x) a+3 b^2+3 b^2 \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a^2-2 b \tan (c+d x) a+3 b^2+3 b^2 \tan (c+d x)^2}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {2 \int \frac {2 \tan (c+d x) a^3+b \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)+b \left (4 a^2+3 b^2\right )}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {2 \tan (c+d x) a^3+b \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)+b \left (4 a^2+3 b^2\right )}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {2 \tan (c+d x) a^3+b \left (2 a^2+3 b^2\right ) \tan (c+d x)^2+b \left (4 a^2+3 b^2\right )}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {-\frac {\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {\int \frac {2 \left (2 b a^3+\left (a^2-b^2\right ) \tan (c+d x) a^2\right )}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {2 \int \frac {2 b a^3+\left (a^2-b^2\right ) \tan (c+d x) a^2}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {2 \int \frac {2 b a^3+\left (a^2-b^2\right ) \tan (c+d x) a^2}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {-\frac {\frac {4 \int \frac {a^2 \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {4 a^2 \int \frac {2 a b+\left (a^2-b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-\frac {\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-\frac {\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {-\frac {\frac {b^3 \left (7 a^2+3 b^2\right ) \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}+\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {\frac {2 b^3 \left (7 a^2+3 b^2\right ) \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {-\frac {\frac {4 a^2 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 b^{5/2} \left (7 a^2+3 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{a}-\frac {2 \left (2 a^2+3 b^2\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}\) |
Input:
Int[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^2),x]
Output:
(-(((2*b^(5/2)*(7*a^2 + 3*b^2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a] ])/(Sqrt[a]*(a^2 + b^2)*d) + (4*a^2*(((a^2 + 2*a*b - b^2)*(-(ArcTan[1 - Sq rt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] ]/Sqrt[2]))/2 - ((a^2 - 2*a*b - b^2)*(-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[c + d* x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/a) - (2*(2*a^2 + 3*b^2))/(a*d*Sq rt[Tan[c + d*x]]))/(2*a*(a^2 + b^2)) + b^2/(a*(a^2 + b^2)*d*Sqrt[Tan[c + d *x]]*(a + b*Tan[c + d*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] :> 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[(-(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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && 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[((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[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
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 = 0.14 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {-\frac {2}{a^{2} \sqrt {\tan \left (d x +c \right )}}+\frac {-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{2}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 b^{3} \left (\frac {\left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \sqrt {\tan \left (d x +c \right )}}{a +b \tan \left (d x +c \right )}+\frac {\left (7 a^{2}+3 b^{2}\right ) \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(296\) |
| default | \(\frac {-\frac {2}{a^{2} \sqrt {\tan \left (d x +c \right )}}+\frac {-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{2}+\frac {\left (-a^{2}+b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 b^{3} \left (\frac {\left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \sqrt {\tan \left (d x +c \right )}}{a +b \tan \left (d x +c \right )}+\frac {\left (7 a^{2}+3 b^{2}\right ) \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(296\) |
Input:
int(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-2/a^2/tan(d*x+c)^(1/2)+2/(a^2+b^2)^2*(-1/4*a*b*2^(1/2)*(ln((tan(d*x+ c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(tan(d*x+c)-2^(1/2)*tan(d*x+c)^(1/2)+1))+2* arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+ 1/8*(-a^2+b^2)*2^(1/2)*(ln((tan(d*x+c)-2^(1/2)*tan(d*x+c)^(1/2)+1)/(tan(d* x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*a rctan(-1+2^(1/2)*tan(d*x+c)^(1/2))))-2*b^3/a^2/(a^2+b^2)^2*((1/2*a^2+1/2*b ^2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))+1/2*(7*a^2+3*b^2)/(a*b)^(1/2)*arctan (b*tan(d*x+c)^(1/2)/(a*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1595 vs. \(2 (266) = 532\).
Time = 0.46 (sec) , antiderivative size = 3219, normalized size of antiderivative = 10.84 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{2} \tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate(1/tan(d*x+c)**(3/2)/(a+b*tan(d*x+c))**2,x)
Output:
Integral(1/((a + b*tan(c + d*x))**2*tan(c + d*x)**(3/2)), x)
Time = 0.12 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (7 \, a^{2} b^{3} + 3 \, b^{5}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {4 \, {\left (2 \, a^{3} + 2 \, a b^{2} + {\left (2 \, a^{2} b + 3 \, b^{3}\right )} \tan \left (d x + c\right )\right )}}{{\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (a^{5} + a^{3} b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \] Input:
integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
Output:
-1/4*(4*(7*a^2*b^3 + 3*b^5)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^6 + 2*a^4*b^2 + a^2*b^4)*sqrt(a*b)) + (2*sqrt(2)*(a^2 + 2*a*b - b^2)*arctan(1 /2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^2 + 2*a*b - b^ 2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*(a^2 - 2*a*b - b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)* (a^2 - 2*a*b - b^2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/( a^4 + 2*a^2*b^2 + b^4) + 4*(2*a^3 + 2*a*b^2 + (2*a^2*b + 3*b^3)*tan(d*x + c))/((a^4*b + a^2*b^3)*tan(d*x + c)^(3/2) + (a^5 + a^3*b^2)*sqrt(tan(d*x + c))))/d
Exception generated. \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Time = 4.91 (sec) , antiderivative size = 12063, normalized size of antiderivative = 40.62 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \] Input:
int(1/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^2),x)
Output:
atan(((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2* d^2)))^(1/2)*(tan(c + d*x)^(1/2)*(144*a^14*b^23*d^5 + 1248*a^16*b^21*d^5 + 4224*a^18*b^19*d^5 + 6720*a^20*b^17*d^5 + 3872*a^22*b^15*d^5 - 2816*a^24* b^13*d^5 - 5632*a^26*b^11*d^5 - 3136*a^28*b^9*d^5 - 560*a^30*b^7*d^5 + 32* a^32*b^5*d^5) + (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(26496*a^25*b^14*d^6 - 1152*a^15*b^24*d^6 - 8448*a^ 17*b^22*d^6 - 23776*a^19*b^20*d^6 - 29664*a^21*b^18*d^6 - 6528*a^23*b^16*d ^6 - (1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a^2*b^2*d ^2)))^(1/2)*((1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d^2*4i - 6*a ^2*b^2*d^2)))^(1/2)*(768*a^16*b^27*d^8 + 8704*a^18*b^25*d^8 + 44288*a^20*b ^23*d^8 + 133120*a^22*b^21*d^8 + 261120*a^24*b^19*d^8 + 347136*a^26*b^17*d ^8 + 311808*a^28*b^15*d^8 + 178176*a^30*b^13*d^8 + 49920*a^32*b^11*d^8 - 7 680*a^34*b^9*d^8 - 12032*a^36*b^7*d^8 - 4096*a^38*b^5*d^8 - 512*a^40*b^3*d ^8 - tan(c + d*x)^(1/2)*(1i/(4*(a^4*d^2 + b^4*d^2 + a*b^3*d^2*4i - a^3*b*d ^2*4i - 6*a^2*b^2*d^2)))^(1/2)*(512*a^18*b^27*d^9 + 5120*a^20*b^25*d^9 + 2 2528*a^22*b^23*d^9 + 56320*a^24*b^21*d^9 + 84480*a^26*b^19*d^9 + 67584*a^2 8*b^17*d^9 - 67584*a^32*b^13*d^9 - 84480*a^34*b^11*d^9 - 56320*a^36*b^9*d^ 9 - 22528*a^38*b^7*d^9 - 5120*a^40*b^5*d^9 - 512*a^42*b^3*d^9)) + tan(c + d*x)^(1/2)*(1152*a^15*b^26*d^7 + 13440*a^17*b^24*d^7 + 69056*a^19*b^22*d^7 + 202752*a^21*b^20*d^7 + 372800*a^23*b^18*d^7 + 443136*a^25*b^16*d^7 +...
\[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx=\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )^{4} b^{2}+2 \tan \left (d x +c \right )^{3} a b +\tan \left (d x +c \right )^{2} a^{2}}d x \] Input:
int(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^2,x)
Output:
int(sqrt(tan(c + d*x))/(tan(c + d*x)**4*b**2 + 2*tan(c + d*x)**3*a*b + tan (c + d*x)**2*a**2),x)