Integrand size = 33, antiderivative size = 330 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2 (A-B)-b^2 (A-B)-2 a b (A+B)\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (2 a b (A-B)+a^2 (A+B)-b^2 (A+B)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (9 A b+11 a B) \tan ^{\frac {7}{2}}(c+d x)}{63 d}+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d} \] Output:
-1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)) *2^(1/2)/d-1/2*(a^2*(A-B)-b^2*(A-B)-2*a*b*(A+B))*arctan(1+2^(1/2)*tan(d*x+ c)^(1/2))*2^(1/2)/d+1/2*(2*a*b*(A-B)+a^2*(A+B)-b^2*(A+B))*arctanh(2^(1/2)* tan(d*x+c)^(1/2)/(1+tan(d*x+c)))*2^(1/2)/d-2*(2*A*a*b+B*a^2-B*b^2)*tan(d*x +c)^(1/2)/d+2/3*(A*a^2-A*b^2-2*B*a*b)*tan(d*x+c)^(3/2)/d+2/5*(2*A*a*b+B*a^ 2-B*b^2)*tan(d*x+c)^(5/2)/d+2/63*b*(9*A*b+11*B*a)*tan(d*x+c)^(7/2)/d+2/9*b *B*tan(d*x+c)^(7/2)*(a+b*tan(d*x+c))/d
Result contains complex when optimal does not.
Time = 4.23 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.62 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {-315 \sqrt [4]{-1} (a-i b)^2 (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+315 (-1)^{3/4} (a+i b)^2 (A+i B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} \left (-315 \left (2 a A b+a^2 B-b^2 B\right )+105 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+63 \left (2 a A b+a^2 B-b^2 B\right ) \tan ^2(c+d x)+45 b (A b+2 a B) \tan ^3(c+d x)+35 b^2 B \tan ^4(c+d x)\right )}{315 d} \] Input:
Integrate[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x ]
Output:
(-315*(-1)^(1/4)*(a - I*b)^2*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x] ]] + 315*(-1)^(3/4)*(a + I*b)^2*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]]*(-315*(2*a*A*b + a^2*B - b^2*B) + 105*(a^2*A - A*b^2 - 2*a*b*B)*Tan[c + d*x] + 63*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d* x]^2 + 45*b*(A*b + 2*a*B)*Tan[c + d*x]^3 + 35*b^2*B*Tan[c + d*x]^4))/(315* d)
Time = 1.43 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.06, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4090, 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.
| \(\displaystyle \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^{5/2} (a+b \tan (c+d x))^2 (A+B \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4090 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{2} \tan ^{\frac {5}{2}}(c+d x) \left (b (9 A b+11 a B) \tan ^2(c+d x)+9 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a (9 a A-7 b B)\right )dx+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \tan ^{\frac {5}{2}}(c+d x) \left (b (9 A b+11 a B) \tan ^2(c+d x)+9 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a (9 a A-7 b B)\right )dx+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \tan (c+d x)^{5/2} \left (b (9 A b+11 a B) \tan (c+d x)^2+9 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a (9 a A-7 b B)\right )dx+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {1}{9} \left (\int \tan ^{\frac {5}{2}}(c+d x) \left (9 \left (A a^2-2 b B a-A b^2\right )+9 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)\right )dx+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \tan (c+d x)^{5/2} \left (9 \left (A a^2-2 b B a-A b^2\right )+9 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)\right )dx+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{9} \left (\int \tan ^{\frac {3}{2}}(c+d x) \left (9 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)-9 \left (B a^2+2 A b a-b^2 B\right )\right )dx+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \tan (c+d x)^{3/2} \left (9 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)-9 \left (B a^2+2 A b a-b^2 B\right )\right )dx+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{9} \left (\int \sqrt {\tan (c+d x)} \left (-9 \left (A a^2-2 b B a-A b^2\right )-9 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)\right )dx+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \sqrt {\tan (c+d x)} \left (-9 \left (A a^2-2 b B a-A b^2\right )-9 \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)\right )dx+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {9 \left (B a^2+2 A b a-b^2 B\right )-9 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \frac {9 \left (B a^2+2 A b a-b^2 B\right )-9 \left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 \int \frac {9 \left (B a^2+2 A b a-b^2 B-\left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)\right )}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \int \frac {B a^2+2 A b a-b^2 B-\left (A a^2-2 b B a-A b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{9} \left (\frac {18 \left (\frac {1}{2} \left (a^2 (A+B)+2 a b (A-B)-b^2 (A+B)\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} \left (a^2 (A-B)-2 a b (A+B)-b^2 (A-B)\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \tan ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {6 \left (a^2 A-2 a b B-A b^2\right ) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {18 \left (a^2 B+2 a A b-b^2 B\right ) \sqrt {\tan (c+d x)}}{d}+\frac {2 b (11 a B+9 A b) \tan ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b B \tan ^{\frac {7}{2}}(c+d x) (a+b \tan (c+d x))}{9 d}\) |
Input:
Int[Tan[c + d*x]^(5/2)*(a + b*Tan[c + d*x])^2*(A + B*Tan[c + d*x]),x]
Output:
(2*b*B*Tan[c + d*x]^(7/2)*(a + b*Tan[c + d*x]))/(9*d) + ((18*(-1/2*((a^2*( A - B) - b^2*(A - B) - 2*a*b*(A + B))*(-(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 - B) + a^2*(A + B) - b^2*(A + B))*(-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*(2*a*A*b + a^2*B - b^2*B)*Sqrt[Tan[c + d*x]])/d + (6*(a^2*A - A*b^2 - 2*a*b*B)*Tan[c + d*x]^(3/2))/d + (18*(2*a*A *b + a^2*B - b^2*B)*Tan[c + d*x]^(5/2))/(5*d) + (2*b*(9*A*b + 11*a*B)*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_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[b*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Ta n[e + f*x])^n*Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b *(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2 , 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 1] && ( !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 = 0.13 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {\frac {2 B \,b^{2} \tan \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 A \,b^{2} \tan \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 B a b \tan \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 A a b \tan \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 B \,a^{2} \tan \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {2 B \,b^{2} \tan \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 A \,a^{2} \tan \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {2 A \,b^{2} \tan \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {4 B a b \tan \left (d x +c \right )^{\frac {3}{2}}}{3}-4 A a b \sqrt {\tan \left (d x +c \right )}-2 B \,a^{2} \sqrt {\tan \left (d x +c \right )}+2 \sqrt {\tan \left (d x +c \right )}\, B \,b^{2}+\frac {\left (2 A a b +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-A \,a^{2}+A \,b^{2}+2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(374\) |
| default | \(\frac {\frac {2 B \,b^{2} \tan \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 A \,b^{2} \tan \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 B a b \tan \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 A a b \tan \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 B \,a^{2} \tan \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {2 B \,b^{2} \tan \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 A \,a^{2} \tan \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {2 A \,b^{2} \tan \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {4 B a b \tan \left (d x +c \right )^{\frac {3}{2}}}{3}-4 A a b \sqrt {\tan \left (d x +c \right )}-2 B \,a^{2} \sqrt {\tan \left (d x +c \right )}+2 \sqrt {\tan \left (d x +c \right )}\, B \,b^{2}+\frac {\left (2 A a b +B \,a^{2}-B \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-A \,a^{2}+A \,b^{2}+2 B a b \right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(374\) |
| parts | \(\frac {\left (A \,b^{2}+2 B a b \right ) \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}+\frac {\left (2 A a b +B \,a^{2}\right ) \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 {A \,a^{2} \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 \,b^{2} \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}\) | \(472\) |
Input:
int(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x,method=_RETURNV ERBOSE)
Output:
1/d*(2/9*B*b^2*tan(d*x+c)^(9/2)+2/7*A*b^2*tan(d*x+c)^(7/2)+4/7*B*a*b*tan(d *x+c)^(7/2)+4/5*A*a*b*tan(d*x+c)^(5/2)+2/5*B*a^2*tan(d*x+c)^(5/2)-2/5*B*b^ 2*tan(d*x+c)^(5/2)+2/3*A*a^2*tan(d*x+c)^(3/2)-2/3*A*b^2*tan(d*x+c)^(3/2)-4 /3*B*a*b*tan(d*x+c)^(3/2)-4*A*a*b*tan(d*x+c)^(1/2)-2*B*a^2*tan(d*x+c)^(1/2 )+2*tan(d*x+c)^(1/2)*B*b^2+1/4*(2*A*a*b+B*a^2-B*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)))+ 1/4*(-A*a^2+A*b^2+2*B*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))))
Leaf count of result is larger than twice the leaf count of optimal. 1546 vs. \(2 (295) = 590\).
Time = 0.11 (sec) , antiderivative size = 1546, normalized size of antiderivative = 4.68 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:
integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorith m="fricas")
Output:
-1/630*(630*sqrt(1/2)*d*sqrt(((A^2 - 2*A*B + B^2)*a^4 - 4*(A^2 - B^2)*a^3* b + 2*(A^2 + 6*A*B + B^2)*a^2*b^2 + 4*(A^2 - B^2)*a*b^3 + (A^2 - 2*A*B + B ^2)*b^4)/d^2)*arctan(-(2*sqrt(1/2)*((A + B)*a^2 + 2*(A - B)*a*b - (A + B)* b^2)*d*sqrt(((A^2 - 2*A*B + B^2)*a^4 - 4*(A^2 - B^2)*a^3*b + 2*(A^2 + 6*A* B + B^2)*a^2*b^2 + 4*(A^2 - B^2)*a*b^3 + (A^2 - 2*A*B + B^2)*b^4)/d^2)*sqr t(tan(d*x + c)) + d^2*sqrt(((A^2 + 2*A*B + B^2)*a^4 + 4*(A^2 - B^2)*a^3*b + 2*(A^2 - 6*A*B + B^2)*a^2*b^2 - 4*(A^2 - B^2)*a*b^3 + (A^2 + 2*A*B + B^2 )*b^4)/d^2)*sqrt(((A^2 - 2*A*B + B^2)*a^4 - 4*(A^2 - B^2)*a^3*b + 2*(A^2 + 6*A*B + B^2)*a^2*b^2 + 4*(A^2 - B^2)*a*b^3 + (A^2 - 2*A*B + B^2)*b^4)/d^2 ))/(8*A*B*a^3*b - 8*A*B*a*b^3 - (A^2 - B^2)*a^4 + 6*(A^2 - B^2)*a^2*b^2 - (A^2 - B^2)*b^4)) + 630*sqrt(1/2)*d*sqrt(((A^2 - 2*A*B + B^2)*a^4 - 4*(A^2 - B^2)*a^3*b + 2*(A^2 + 6*A*B + B^2)*a^2*b^2 + 4*(A^2 - B^2)*a*b^3 + (A^2 - 2*A*B + B^2)*b^4)/d^2)*arctan(-(2*sqrt(1/2)*((A + B)*a^2 + 2*(A - B)*a* b - (A + B)*b^2)*d*sqrt(((A^2 - 2*A*B + B^2)*a^4 - 4*(A^2 - B^2)*a^3*b + 2 *(A^2 + 6*A*B + B^2)*a^2*b^2 + 4*(A^2 - B^2)*a*b^3 + (A^2 - 2*A*B + B^2)*b ^4)/d^2)*sqrt(tan(d*x + c)) - d^2*sqrt(((A^2 + 2*A*B + B^2)*a^4 + 4*(A^2 - B^2)*a^3*b + 2*(A^2 - 6*A*B + B^2)*a^2*b^2 - 4*(A^2 - B^2)*a*b^3 + (A^2 + 2*A*B + B^2)*b^4)/d^2)*sqrt(((A^2 - 2*A*B + B^2)*a^4 - 4*(A^2 - B^2)*a^3* b + 2*(A^2 + 6*A*B + B^2)*a^2*b^2 + 4*(A^2 - B^2)*a*b^3 + (A^2 - 2*A*B + B ^2)*b^4)/d^2))/(8*A*B*a^3*b - 8*A*B*a*b^3 - (A^2 - B^2)*a^4 + 6*(A^2 - ...
Timed out. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(tan(d*x+c)**(5/2)*(a+b*tan(d*x+c))**2*(A+B*tan(d*x+c)),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {280 \, B b^{2} \tan \left (d x + c\right )^{\frac {9}{2}} + 360 \, {\left (2 \, B a b + A b^{2}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} + 504 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} - 630 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 630 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - 2 \, {\left (A + B\right )} a b - {\left (A - B\right )} b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 315 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 315 \, \sqrt {2} {\left ({\left (A + B\right )} a^{2} + 2 \, {\left (A - B\right )} a b - {\left (A + B\right )} b^{2}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 840 \, {\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \tan \left (d x + c\right )^{\frac {3}{2}} - 2520 \, {\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \sqrt {\tan \left (d x + c\right )}}{1260 \, d} \] Input:
integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorith m="maxima")
Output:
1/1260*(280*B*b^2*tan(d*x + c)^(9/2) + 360*(2*B*a*b + A*b^2)*tan(d*x + c)^ (7/2) + 504*(B*a^2 + 2*A*a*b - B*b^2)*tan(d*x + c)^(5/2) - 630*sqrt(2)*((A - B)*a^2 - 2*(A + B)*a*b - (A - B)*b^2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*s qrt(tan(d*x + c)))) - 630*sqrt(2)*((A - B)*a^2 - 2*(A + B)*a*b - (A - B)*b ^2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + 315*sqrt(2)*(( A + B)*a^2 + 2*(A - B)*a*b - (A + B)*b^2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - 315*sqrt(2)*((A + B)*a^2 + 2*(A - B)*a*b - (A + B)*b^ 2)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + 840*(A*a^2 - 2*B* a*b - A*b^2)*tan(d*x + c)^(3/2) - 2520*(B*a^2 + 2*A*a*b - B*b^2)*sqrt(tan( d*x + c)))/d
Exception generated. \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),x, algorith m="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 = 23.80 (sec) , antiderivative size = 3914, normalized size of antiderivative = 11.86 \[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \] Input:
int(tan(c + d*x)^(5/2)*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^2,x)
Output:
atan((B^2*a^4*tan(c + d*x)^(1/2)*((B^2*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2) /(4*d^4) - (B^2*a^3*b)/d^2)^(1/2)*32i)/((16*B*a^2*(12*B^4*a^2*b^6*d^4 - B^ 4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/ d^3 - (192*B^3*a^3*b^3)/d - (16*B*b^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 + (32*B^ 3*a*b^5)/d + (32*B^3*a^5*b)/d) + (B^2*b^4*tan(c + d*x)^(1/2)*((B^2*a*b^3)/ d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (B^2*a^3*b)/d^2)^(1/2)*32i)/((16*B* a^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*B^3*a^3*b^3)/d - (16*B*b^2*(12*B^4* a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6* b^2*d^4)^(1/2))/d^3 + (32*B^3*a*b^5)/d + (32*B^3*a^5*b)/d) - (B^2*a^2*b^2* tan(c + d*x)^(1/2)*((B^2*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2)/(4*d^4) - (B^ 2*a^3*b)/d^2)^(1/2)*192i)/((16*B*a^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B ^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 - (192*B^ 3*a^3*b^3)/d - (16*B*b^2*(12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4 - B^4*a^8*d^4 - 38*B^4*a^4*b^4*d^4 + 12*B^4*a^6*b^2*d^4)^(1/2))/d^3 + (32*B^3*a*b^5)/d + (32*B^3*a^5*b)/d))*((B^2*a*b^3)/d^2 - (12*B^4*a^2*b^6*d^4 - B^4*b^8*d^4...
\[ \int \tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {10 \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{4} b^{3}+54 \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{2} a^{2} b -18 \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{2} b^{3}-270 \sqrt {\tan \left (d x +c \right )}\, a^{2} b +90 \sqrt {\tan \left (d x +c \right )}\, b^{3}+135 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )}d x \right ) a^{2} b d -45 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )}d x \right ) b^{3} d +135 \left (\int \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{4}d x \right ) a \,b^{2} d +45 \left (\int \sqrt {\tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{2}d x \right ) a^{3} d}{45 d} \] Input:
int(tan(d*x+c)^(5/2)*(a+b*tan(d*x+c))^2*(A+B*tan(d*x+c)),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)