Integrand size = 23, antiderivative size = 382 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)}}+\frac {b^2}{2 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \] Output:
-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)/(a^ 2+b^2)^3/d-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*2^ (1/2)/(a^2+b^2)^3/d-1/4*b^(5/2)*(63*a^4+46*a^2*b^2+15*b^4)*arctan(b^(1/2)* tan(d*x+c)^(1/2)/a^(1/2))/a^(7/2)/(a^2+b^2)^3/d+1/2*(a+b)*(a^2-4*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)^3/d-1/4 *(8*a^4+31*a^2*b^2+15*b^4)/a^3/(a^2+b^2)^2/d/tan(d*x+c)^(1/2)+1/2*b^2/a/(a ^2+b^2)/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^2+1/4*b^2*(13*a^2+5*b^2)/a^2/( a^2+b^2)^2/d/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 2.70 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\frac {\frac {-8 a^{13/2}-39 a^{9/2} b^2-46 a^{5/2} b^4-15 \sqrt {a} b^6-4 (-1)^{3/4} a^{7/2} (a+i b)^3 \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}-63 a^4 b^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\tan (c+d x)}-46 a^2 b^{9/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\tan (c+d x)}-15 b^{13/2} \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {\tan (c+d x)}-4 \sqrt [4]{-1} a^{7/2} (i a+b)^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}}{a^{5/2} \left (a^2+b^2\right )^2}+\frac {2 b^2}{(a+b \tan (c+d x))^2}+\frac {13 a^2 b^2+5 b^4}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right ) d \sqrt {\tan (c+d x)}} \] Input:
Integrate[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]
Output:
((-8*a^(13/2) - 39*a^(9/2)*b^2 - 46*a^(5/2)*b^4 - 15*Sqrt[a]*b^6 - 4*(-1)^ (3/4)*a^(7/2)*(a + I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - 63*a^4*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqr t[Tan[c + d*x]] - 46*a^2*b^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[ a]]*Sqrt[Tan[c + d*x]] - 15*b^(13/2)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/S qrt[a]]*Sqrt[Tan[c + d*x]] - 4*(-1)^(1/4)*a^(7/2)*(I*a + b)^3*ArcTanh[(-1) ^(3/4)*Sqrt[Tan[c + d*x]]]*Sqrt[Tan[c + d*x]])/(a^(5/2)*(a^2 + b^2)^2) + ( 2*b^2)/(a + b*Tan[c + d*x])^2 + (13*a^2*b^2 + 5*b^4)/(a*(a^2 + b^2)*(a + b *Tan[c + d*x])))/(4*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]])
Time = 2.17 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.09, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.130, Rules used = {3042, 4052, 27, 3042, 4132, 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.
| \(\displaystyle \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \frac {\int \frac {4 a^2-4 b \tan (c+d x) a+5 b^2+5 b^2 \tan ^2(c+d x)}{2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 a^2-4 b \tan (c+d x) a+5 b^2+5 b^2 \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {4 a^2-4 b \tan (c+d x) a+5 b^2+5 b^2 \tan (c+d x)^2}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {\int \frac {8 a^4-16 b \tan (c+d x) a^3+31 b^2 a^2+15 b^4+3 b^2 \left (13 a^2+5 b^2\right ) \tan ^2(c+d x)}{2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {8 a^4-16 b \tan (c+d x) a^3+31 b^2 a^2+15 b^4+3 b^2 \left (13 a^2+5 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {8 a^4-16 b \tan (c+d x) a^3+31 b^2 a^2+15 b^4+3 b^2 \left (13 a^2+5 b^2\right ) \tan (c+d x)^2}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {-\frac {2 \int \frac {8 \left (a^2-b^2\right ) \tan (c+d x) a^3+b \left (8 a^4+31 b^2 a^2+15 b^4\right ) \tan ^2(c+d x)+b \left (24 a^4+31 b^2 a^2+15 b^4\right )}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {8 \left (a^2-b^2\right ) \tan (c+d x) a^3+b \left (8 a^4+31 b^2 a^2+15 b^4\right ) \tan ^2(c+d x)+b \left (24 a^4+31 b^2 a^2+15 b^4\right )}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {8 \left (a^2-b^2\right ) \tan (c+d x) a^3+b \left (8 a^4+31 b^2 a^2+15 b^4\right ) \tan (c+d x)^2+b \left (24 a^4+31 b^2 a^2+15 b^4\right )}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {\int \frac {8 \left (\left (a^2-3 b^2\right ) \tan (c+d x) a^4+b \left (3 a^2-b^2\right ) a^3\right )}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan ^2(c+d x)+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {8 \int \frac {\left (a^2-3 b^2\right ) \tan (c+d x) a^4+b \left (3 a^2-b^2\right ) a^3}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {8 \int \frac {\left (a^2-3 b^2\right ) \tan (c+d x) a^4+b \left (3 a^2-b^2\right ) a^3}{\sqrt {\tan (c+d x)}}dx}{a^2+b^2}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 \int \frac {a^3 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan (c+d x)^2+1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}dx}{a^2+b^2}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {-\frac {\frac {b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {1}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))}d\tan (c+d x)}{d \left (a^2+b^2\right )}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2 b^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {1}{a+b \tan (c+d x)}d\sqrt {\tan (c+d x)}}{d \left (a^2+b^2\right )}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}}{4 a \left (a^2+b^2\right )}+\frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b^2}{2 a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^2}+\frac {\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))}+\frac {-\frac {2 \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {\tan (c+d x)}}-\frac {\frac {2 b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}+\frac {16 a^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{a}}{2 a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}\) |
Input:
Int[1/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])^3),x]
Output:
b^2/(2*a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^2) + ((-((( 2*b^(5/2)*(63*a^4 + 46*a^2*b^2 + 15*b^4)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x] ])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)*d) + (16*a^3*(((a - b)*(a^2 + 4*a*b + b^ 2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2] *Sqrt[Tan[c + d*x]]]/Sqrt[2]))/2 - ((a + b)*(a^2 - 4*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]*S qrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/a) - (2*(8*a^4 + 31*a^2*b^2 + 15*b^4))/(a*d*Sqrt[Tan[c + d*x]]))/(2*a*(a^2 + b^ 2)) + (b^2*(13*a^2 + 5*b^2))/(a*(a^2 + b^2)*d*Sqrt[Tan[c + d*x]]*(a + b*Ta n[c + d*x])))/(4*a*(a^2 + b^2))
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 = 0.15 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.93
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2}{a^{3} \sqrt {\tan \left (d x +c \right )}}-\frac {2 b^{3} \left (\frac {\left (\frac {15}{8} b \,a^{4}+\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \tan \left (d x +c \right )^{\frac {3}{2}}+\frac {a \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \sqrt {\tan \left (d x +c \right )}}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(356\) |
| default | \(\frac {\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2}{a^{3} \sqrt {\tan \left (d x +c \right )}}-\frac {2 b^{3} \left (\frac {\left (\frac {15}{8} b \,a^{4}+\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \tan \left (d x +c \right )^{\frac {3}{2}}+\frac {a \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \sqrt {\tan \left (d x +c \right )}}{8}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {b \sqrt {\tan \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(356\) |
Input:
int(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(2/(a^2+b^2)^3*(1/8*(-3*a^2*b+b^3)*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^3+3*a* 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*arctan(-1+2 ^(1/2)*tan(d*x+c)^(1/2))))-2/a^3/tan(d*x+c)^(1/2)-2*b^3/a^3/(a^2+b^2)^3*(( (15/8*b*a^4+11/4*a^2*b^3+7/8*b^5)*tan(d*x+c)^(3/2)+1/8*a*(17*a^4+26*a^2*b^ 2+9*b^4)*tan(d*x+c)^(1/2))/(a+b*tan(d*x+c))^2+1/8*(63*a^4+46*a^2*b^2+15*b^ 4)/(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. 2483 vs. \(2 (341) = 682\).
Time = 1.88 (sec) , antiderivative size = 4995, normalized size of antiderivative = 13.08 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{3} \tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate(1/tan(d*x+c)**(3/2)/(a+b*tan(d*x+c))**3,x)
Output:
Integral(1/((a + b*tan(c + d*x))**3*tan(c + d*x)**(3/2)), x)
Time = 0.12 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=-\frac {\frac {{\left (63 \, a^{4} b^{3} + 46 \, a^{2} b^{5} + 15 \, b^{7}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} \sqrt {a b}} + \frac {8 \, a^{6} + 16 \, a^{4} b^{2} + 8 \, a^{2} b^{4} + {\left (8 \, a^{4} b^{2} + 31 \, a^{2} b^{4} + 15 \, b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (16 \, a^{5} b + 49 \, a^{3} b^{3} + 25 \, a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \sqrt {\tan \left (d x + c\right )}} + \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}}}{4 \, d} \] Input:
integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
-1/4*((63*a^4*b^3 + 46*a^2*b^5 + 15*b^7)*arctan(b*sqrt(tan(d*x + c))/sqrt( a*b))/((a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*sqrt(a*b)) + (8*a^6 + 16*a^ 4*b^2 + 8*a^2*b^4 + (8*a^4*b^2 + 31*a^2*b^4 + 15*b^6)*tan(d*x + c)^2 + (16 *a^5*b + 49*a^3*b^3 + 25*a*b^5)*tan(d*x + c))/((a^7*b^2 + 2*a^5*b^4 + a^3* b^6)*tan(d*x + c)^(5/2) + 2*(a^8*b + 2*a^6*b^3 + a^4*b^5)*tan(d*x + c)^(3/ 2) + (a^9 + 2*a^7*b^2 + a^5*b^4)*sqrt(tan(d*x + c))) + (2*sqrt(2)*(a^3 + 3 *a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)) )) + 2*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2 ) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(s qrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*(a^3 - 3*a^2*b - 3 *a*b^2 + b^3)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6))/d
\[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \tan \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
undef
Time = 11.54 (sec) , antiderivative size = 16740, normalized size of antiderivative = 43.82 \[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int(1/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))^3),x)
Output:
(log(29491200*a^22*b^35*d^4 - ((((-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^ 2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/ 2)*(((((-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4* d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2*d^2*15i))^(1/2)*(251658240*a^24*b^45*d^ 8 + 5049942016*a^26*b^43*d^8 + 48368713728*a^28*b^41*d^8 + 293819383808*a^ 30*b^39*d^8 + 1268458192896*a^32*b^37*d^8 + 4132731617280*a^34*b^35*d^8 + 10531192700928*a^36*b^33*d^8 + 21462823993344*a^38*b^31*d^8 + 354696183152 64*a^40*b^29*d^8 + 47896904859648*a^42*b^27*d^8 + 52983958077440*a^44*b^25 *d^8 + 47896904859648*a^46*b^23*d^8 + 35090285461504*a^48*b^21*d^8 + 20487 396655104*a^50*b^19*d^8 + 9230622916608*a^52*b^17*d^8 + 2994733056000*a^54 *b^15*d^8 + 565576728576*a^56*b^13*d^8 - 18572378112*a^58*b^11*d^8 - 50281 316352*a^60*b^9*d^8 - 16089350144*a^62*b^7*d^8 - 2516582400*a^64*b^5*d^8 - 167772160*a^66*b^3*d^8 + (tan(c + d*x)^(1/2)*(-1/(b^6*d^2*1i - a^6*d^2*1i + 6*a*b^5*d^2 + 6*a^5*b*d^2 - a^2*b^4*d^2*15i - 20*a^3*b^3*d^2 + a^4*b^2* d^2*15i))^(1/2)*(134217728*a^27*b^45*d^9 + 2550136832*a^29*b^43*d^9 + 2281 7013760*a^31*b^41*d^9 + 127506841600*a^33*b^39*d^9 + 497276682240*a^35*b^3 7*d^9 + 1430626762752*a^37*b^35*d^9 + 3121367482368*a^39*b^33*d^9 + 520227 9137280*a^41*b^31*d^9 + 6502848921600*a^43*b^29*d^9 + 5635802398720*a^45*b ^27*d^9 + 2254320959488*a^47*b^25*d^9 - 2254320959488*a^49*b^23*d^9 - 5635 802398720*a^51*b^21*d^9 - 6502848921600*a^53*b^19*d^9 - 5202279137280*a...
\[ \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )^{5} b^{3}+3 \tan \left (d x +c \right )^{4} a \,b^{2}+3 \tan \left (d x +c \right )^{3} a^{2} b +\tan \left (d x +c \right )^{2} a^{3}}d x \] Input:
int(1/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c))^3,x)
Output:
int(sqrt(tan(c + d*x))/(tan(c + d*x)**5*b**3 + 3*tan(c + d*x)**4*a*b**2 + 3*tan(c + d*x)**3*a**2*b + tan(c + d*x)**2*a**3),x)