Integrand size = 23, antiderivative size = 273 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {2 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d} \] 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)/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)/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)/d-2*b*(3*a^2-b^2)*tan(d*x+c)^(1/2)/d+2/3*a*(a^2-3*b^2)*tan(d*x+c)^(3/ 2)/d+2/5*b*(3*a^2-b^2)*tan(d*x+c)^(5/2)/d+40/63*a*b^2*tan(d*x+c)^(7/2)/d+2 /9*b^2*tan(d*x+c)^(7/2)*(a+b*tan(d*x+c))/d
Result contains complex when optimal does not.
Time = 2.04 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.60 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {-315 (-1)^{3/4} (a-i b)^3 \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+315 (-1)^{3/4} (a+i b)^3 \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} \left (315 b \left (-3 a^2+b^2\right )+105 a \left (a^2-3 b^2\right ) \tan (c+d x)-63 b \left (-3 a^2+b^2\right ) \tan ^2(c+d x)+135 a b^2 \tan ^3(c+d x)+35 b^3 \tan ^4(c+d x)\right )}{315 d} \] Input:
Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3,x]
Output:
(-315*(-1)^(3/4)*(a - I*b)^3*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 315*( -1)^(3/4)*(a + I*b)^3*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[ c + d*x]]*(315*b*(-3*a^2 + b^2) + 105*a*(a^2 - 3*b^2)*Tan[c + d*x] - 63*b* (-3*a^2 + b^2)*Tan[c + d*x]^2 + 135*a*b^2*Tan[c + d*x]^3 + 35*b^3*Tan[c + d*x]^4))/(315*d)
Time = 1.24 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.12, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.957, Rules used = {3042, 4049, 27, 3042, 4113, 3042, 4011, 3042, 4011, 3042, 4011, 3042, 4017, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^{5/2} (a+b \tan (c+d x))^3dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{2} \tan ^{\frac {5}{2}}(c+d x) \left (20 a b^2 \tan ^2(c+d x)+9 b \left (3 a^2-b^2\right ) \tan (c+d x)+a \left (9 a^2-7 b^2\right )\right )dx+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \tan ^{\frac {5}{2}}(c+d x) \left (20 a b^2 \tan ^2(c+d x)+9 b \left (3 a^2-b^2\right ) \tan (c+d x)+a \left (9 a^2-7 b^2\right )\right )dx+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \tan (c+d x)^{5/2} \left (20 a b^2 \tan (c+d x)^2+9 b \left (3 a^2-b^2\right ) \tan (c+d x)+a \left (9 a^2-7 b^2\right )\right )dx+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {1}{9} \left (\int \tan ^{\frac {5}{2}}(c+d x) \left (9 a \left (a^2-3 b^2\right )+9 b \left (3 a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \tan (c+d x)^{5/2} \left (9 a \left (a^2-3 b^2\right )+9 b \left (3 a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{9} \left (\int \tan ^{\frac {3}{2}}(c+d x) \left (9 a \left (a^2-3 b^2\right ) \tan (c+d x)-9 b \left (3 a^2-b^2\right )\right )dx+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \tan (c+d x)^{3/2} \left (9 a \left (a^2-3 b^2\right ) \tan (c+d x)-9 b \left (3 a^2-b^2\right )\right )dx+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{9} \left (\int \sqrt {\tan (c+d x)} \left (-9 a \left (a^2-3 b^2\right )-9 b \left (3 a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \sqrt {\tan (c+d x)} \left (-9 a \left (a^2-3 b^2\right )-9 b \left (3 a^2-b^2\right ) \tan (c+d x)\right )dx+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {9 b \left (3 a^2-b^2\right )-9 a \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {9 b \left (3 a^2-b^2\right )-9 a \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 \int \frac {9 \left (b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 b \left (3 a^2-b^2\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 a \left (a^2-3 b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 b \left (3 a^2-b^2\right ) \sqrt {\tan (c+d x)}}{d}+\frac {40 a b^2 \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b^2 \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
Input:
Int[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^3,x]
Output:
(2*b^2*Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x]))/(9*d) + ((18*(-1/2*((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])) + ((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]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])))/2))/d - ( 18*b*(3*a^2 - b^2)*Sqrt[Tan[c + d*x]])/d + (6*a*(a^2 - 3*b^2)*Tan[c + d*x] ^(3/2))/d + (18*b*(3*a^2 - b^2)*Tan[c + d*x]^(5/2))/(5*d) + (40*a*b^2*Tan[ c + d*x]^(7/2))/(7*d))/9
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_)^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_) + (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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((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 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*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] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Time = 0.19 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b^{3} \tan \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {6 a \,b^{2} \tan \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} b \tan \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} \tan \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 a^{3} \tan \left (d x +c \right )^{\frac {3}{2}}}{3}-2 a \,b^{2} \tan \left (d x +c \right )^{\frac {3}{2}}-6 a^{2} b \sqrt {\tan \left (d x +c \right )}+2 b^{3} \sqrt {\tan \left (d x +c \right )}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(308\) |
| default | \(\frac {\frac {2 b^{3} \tan \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {6 a \,b^{2} \tan \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} b \tan \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {2 b^{3} \tan \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 a^{3} \tan \left (d x +c \right )^{\frac {3}{2}}}{3}-2 a \,b^{2} \tan \left (d x +c \right )^{\frac {3}{2}}-6 a^{2} b \sqrt {\tan \left (d x +c \right )}+2 b^{3} \sqrt {\tan \left (d x +c \right )}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(308\) |
| parts | \(\frac {a^{3} \left (\frac {2 \tan \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {\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}\right )}{d}+\frac {b^{3} \left (\frac {2 \tan \left (d x +c \right )^{\frac {9}{2}}}{9}-\frac {2 \tan \left (d x +c \right )^{\frac {5}{2}}}{5}+2 \sqrt {\tan \left (d x +c \right )}-\frac {\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}\right )}{d}+\frac {3 a^{2} b \left (\frac {2 \tan \left (d x +c \right )^{\frac {5}{2}}}{5}-2 \sqrt {\tan \left (d x +c \right )}+\frac {\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}\right )}{d}+\frac {3 a \,b^{2} \left (\frac {2 \tan \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {2 \tan \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {\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}\right )}{d}\) | \(458\) |
Input:
int(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(2/9*b^3*tan(d*x+c)^(9/2)+6/7*a*b^2*tan(d*x+c)^(7/2)+6/5*a^2*b*tan(d*x +c)^(5/2)-2/5*b^3*tan(d*x+c)^(5/2)+2/3*a^3*tan(d*x+c)^(3/2)-2*a*b^2*tan(d* x+c)^(3/2)-6*a^2*b*tan(d*x+c)^(1/2)+2*b^3*tan(d*x+c)^(1/2)+1/4*(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)))+1/4*(-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))))
Leaf count of result is larger than twice the leaf count of optimal. 976 vs. \(2 (238) = 476\).
Time = 0.09 (sec) , antiderivative size = 976, normalized size of antiderivative = 3.58 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx =\text {Too large to display} \] Input:
integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
-1/630*(630*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a ^2*b^4 - 6*a*b^5 + b^6)/d^2)*arctan((2*sqrt(1/2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^ 5 + b^6)/d^2)*sqrt(tan(d*x + c)) + d^2*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 2 0*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 - 6*a^5*b + 3*a^4*b^ 2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 - 15*a^4*b^2 + 15*a ^2*b^4 - b^6)) + 630*sqrt(1/2)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3* b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*arctan((2*sqrt(1/2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*d*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) - d^2*sqrt((a^6 + 6*a^5*b + 3*a^ 4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt((a^6 - 6*a^5*b + 3*a^4*b^2 + 20*a^3*b^3 + 3*a^2*b^4 - 6*a*b^5 + b^6)/d^2))/(a^6 - 15*a^4*b ^2 + 15*a^2*b^4 - b^6)) + 315*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*log(-a^3 - 3*a^2*b + 3*a*b^ 2 + b^3 + 2*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a ^2*b^4 + 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) - (a^3 + 3*a^2*b - 3*a*b^2 - b^3)*tan(d*x + c)) - 315*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2*b^4 + 6*a*b^5 + b^6)/d^2)*log(-a^3 - 3*a^2*b + 3*a*b^2 + b^3 - 2*sqrt(1/2)*d*sqrt((a^6 + 6*a^5*b + 3*a^4*b^2 - 20*a^3*b^3 + 3*a^2 *b^4 + 6*a*b^5 + b^6)/d^2)*sqrt(tan(d*x + c)) - (a^3 + 3*a^2*b - 3*a*b^...
\[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \tan ^{\frac {5}{2}}{\left (c + d x \right )}\, dx \] Input:
integrate(tan(d*x+c)**(5/2)*(a+b*tan(d*x+c))**3,x)
Output:
Integral((a + b*tan(c + d*x))**3*tan(c + d*x)**(5/2), x)
Time = 0.12 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.03 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {280 \, b^{3} \tan \left (d x + c\right )^{\frac {9}{2}} + 1080 \, a b^{2} \tan \left (d x + c\right )^{\frac {7}{2}} + 504 \, {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} - 630 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 630 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 315 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 315 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 840 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} - 2520 \, {\left (3 \, a^{2} b - b^{3}\right )} \sqrt {\tan \left (d x + c\right )}}{1260 \, d} \] Input:
integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
1/1260*(280*b^3*tan(d*x + c)^(9/2) + 1080*a*b^2*tan(d*x + c)^(7/2) + 504*( 3*a^2*b - b^3)*tan(d*x + c)^(5/2) - 630*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)))) - 630*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)))) + 315*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) - 315*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) + 840*(a^3 - 3*a *b^2)*tan(d*x + c)^(3/2) - 2520*(3*a^2*b - b^3)*sqrt(tan(d*x + c)))/d
Exception generated. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 7.75 (sec) , antiderivative size = 1796, normalized size of antiderivative = 6.58 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Too large to display} \] Input:
int(tan(c + d*x)^(5/2)*(a + b*tan(c + d*x))^3,x)
Output:
tan(c + d*x)^(3/2)*((2*a^3)/(3*d) - (2*a*b^2)/d) + tan(c + d*x)^(1/2)*((2* b^3)/d - (6*a^2*b)/d) - tan(c + d*x)^(5/2)*((2*b^3)/(5*d) - (6*a^2*b)/(5*d )) + atan((((8*(4*b^3*d^2 - 12*a^2*b*d^2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b ^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2))/d^3 - (16* tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^ 2))^(1/2)*1i - ((8*(4*b^3*d^2 - 12*a^2*b*d^2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2))/d^3 + (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*((6*a*b ^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/( 4*d^2))^(1/2)*1i)/(((8*(4*b^3*d^2 - 12*a^2*b*d^2)*((6*a*b^5 + 6*a^5*b - a^ 6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2))/d^ 3 - (16*tan(c + d*x)^(1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*((6 *a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15 i)/(4*d^2))^(1/2) - (16*(3*a*b^8 - a^9 + 8*a^3*b^6 + 6*a^5*b^4))/d^3 + ((8 *(4*b^3*d^2 - 12*a^2*b*d^2)*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^ 4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2))/d^3 + (16*tan(c + d*x)^( 1/2)*(a^6 - b^6 + 15*a^2*b^4 - 15*a^4*b^2))/d^2)*((6*a*b^5 + 6*a^5*b - a^6 *1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)/(4*d^2))^(1/2)))*(( 6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^...
\[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {10 \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{4} b^{3}+54 \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{2} a^{2} b -18 \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{2} b^{3}-270 \sqrt {\tan \left (d x +c \right )}\, a^{2} b +90 \sqrt {\tan \left (d x +c \right )}\, b^{3}+135 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )}d x \right ) a^{2} b d -45 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )}d x \right ) b^{3} d +135 \left (\int \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{4}d x \right ) a \,b^{2} d +45 \left (\int \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{2}d x \right ) a^{3} d}{45 d} \] Input:
int(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^3,x)
Output:
(10*sqrt(tan(c + d*x))*tan(c + d*x)**4*b**3 + 54*sqrt(tan(c + d*x))*tan(c + d*x)**2*a**2*b - 18*sqrt(tan(c + d*x))*tan(c + d*x)**2*b**3 - 270*sqrt(t an(c + d*x))*a**2*b + 90*sqrt(tan(c + d*x))*b**3 + 135*int(sqrt(tan(c + d* x))/tan(c + d*x),x)*a**2*b*d - 45*int(sqrt(tan(c + d*x))/tan(c + d*x),x)*b **3*d + 135*int(sqrt(tan(c + d*x))*tan(c + d*x)**4,x)*a*b**2*d + 45*int(sq rt(tan(c + d*x))*tan(c + d*x)**2,x)*a**3*d)/(45*d)