Integrand size = 23, antiderivative size = 271 \[ \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {40 a^2 b}{63 d \tan ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)} \] 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)/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-40/63*a^2*b/d/tan(d*x+c)^(7/2)+2/5*a*(a^2-3*b^2)/d/tan(d*x+c)^(5/2) +2/3*b*(3*a^2-b^2)/d/tan(d*x+c)^(3/2)-2*a*(a^2-3*b^2)/d/tan(d*x+c)^(1/2)-2 /9*a^2*(a+b*tan(d*x+c))/d/tan(d*x+c)^(9/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.48 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.38 \[ \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 \left (7 a \left (a^2-3 b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},1,-\frac {5}{4},-\tan ^2(c+d x)\right )+3 b \left (3 \left (3 a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},1,-\frac {3}{4},-\tan ^2(c+d x)\right ) \tan (c+d x)+b (7 a+3 b \tan (c+d x))\right )\right )}{63 d \tan ^{\frac {9}{2}}(c+d x)} \] Input:
Integrate[(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(11/2),x]
Output:
(-2*(7*a*(a^2 - 3*b^2)*Hypergeometric2F1[-9/4, 1, -5/4, -Tan[c + d*x]^2] + 3*b*(3*(3*a^2 - b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Tan[c + d*x]^2]*Ta n[c + d*x] + b*(7*a + 3*b*Tan[c + d*x]))))/(63*d*Tan[c + d*x]^(9/2))
Time = 1.24 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.14, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4048, 27, 3042, 4111, 27, 3042, 4012, 3042, 4012, 25, 3042, 4012, 3042, 4017, 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.
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^3}{\tan (c+d x)^{11/2}}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {2}{9} \int \frac {20 b a^2-9 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (7 a^2-9 b^2\right ) \tan ^2(c+d x)}{2 \tan ^{\frac {9}{2}}(c+d x)}dx-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {20 b a^2-9 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (7 a^2-9 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac {9}{2}}(c+d x)}dx-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {20 b a^2-9 \left (a^2-3 b^2\right ) \tan (c+d x) a-b \left (7 a^2-9 b^2\right ) \tan (c+d x)^2}{\tan (c+d x)^{9/2}}dx-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4111 |
| \(\displaystyle \frac {1}{9} \left (\int -\frac {9 \left (a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)\right )}{\tan ^{\frac {7}{2}}(c+d x)}dx-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (-9 \int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^{\frac {7}{2}}(c+d x)}dx-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (-9 \int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)^{7/2}}dx-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (\int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)}dx-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (\int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan (c+d x)}{\tan (c+d x)^{5/2}}dx-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (\int -\frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)}dx-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)}dx-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\tan (c+d x)^{3/2}}dx-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\frac {2 \int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\frac {2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\frac {2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\frac {2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\frac {2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\frac {2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\frac {2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\frac {2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{9} \left (-9 \left (-\frac {2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 a \left (a^2-3 b^2\right )}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \left (3 a^2-b^2\right )}{3 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\tan (c+d x)}}\right )-\frac {40 a^2 b}{7 d \tan ^{\frac {7}{2}}(c+d x)}\right )-\frac {2 a^2 (a+b \tan (c+d x))}{9 d \tan ^{\frac {9}{2}}(c+d x)}\) |
Input:
Int[(a + b*Tan[c + d*x])^3/Tan[c + d*x]^(11/2),x]
Output:
(-9*((-2*(-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]] + Ta n[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/( 2*Sqrt[2])))/2))/d - (2*a*(a^2 - 3*b^2))/(5*d*Tan[c + d*x]^(5/2)) - (2*b*( 3*a^2 - b^2))/(3*d*Tan[c + d*x]^(3/2)) + (2*a*(a^2 - 3*b^2))/(d*Sqrt[Tan[c + d*x]])) - (40*a^2*b)/(7*d*Tan[c + d*x]^(7/2)))/9 - (2*a^2*(a + b*Tan[c + d*x]))/(9*d*Tan[c + d*x]^(9/2))
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_)^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[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
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*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x ] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B , C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0 ]
Time = 0.14 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.07
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a^{3}}{9 \tan \left (d x +c \right )^{\frac {9}{2}}}-\frac {2 a \left (a^{2}-3 b^{2}\right )}{\sqrt {\tan \left (d x +c \right )}}+\frac {2 a \left (a^{2}-3 b^{2}\right )}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}-\frac {6 a^{2} b}{7 \tan \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 b \left (3 a^{2}-b^{2}\right )}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(289\) |
| default | \(\frac {-\frac {2 a^{3}}{9 \tan \left (d x +c \right )^{\frac {9}{2}}}-\frac {2 a \left (a^{2}-3 b^{2}\right )}{\sqrt {\tan \left (d x +c \right )}}+\frac {2 a \left (a^{2}-3 b^{2}\right )}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}-\frac {6 a^{2} b}{7 \tan \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 b \left (3 a^{2}-b^{2}\right )}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right )\right )}{4}}{d}\) | \(289\) |
| parts | \(\frac {a^{3} \left (-\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}-\frac {2}{9 \tan \left (d x +c \right )^{\frac {9}{2}}}-\frac {2}{\sqrt {\tan \left (d x +c \right )}}+\frac {2}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}\right )}{d}+\frac {b^{3} \left (-\frac {2}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}-\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 {\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 {2}{5 \tan \left (d x +c \right )^{\frac {5}{2}}}+\frac {2}{\sqrt {\tan \left (d x +c \right )}}\right )}{d}+\frac {3 a^{2} b \left (\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}-\frac {2}{7 \tan \left (d x +c \right )^{\frac {7}{2}}}+\frac {2}{3 \tan \left (d x +c \right )^{\frac {3}{2}}}\right )}{d}\) | \(458\) |
Input:
int((a+b*tan(d*x+c))^3/tan(d*x+c)^(11/2),x,method=_RETURNVERBOSE)
Output:
1/d*(-2/9*a^3/tan(d*x+c)^(9/2)-2*a*(a^2-3*b^2)/tan(d*x+c)^(1/2)+2/5*a*(a^2 -3*b^2)/tan(d*x+c)^(5/2)-6/7*a^2*b/tan(d*x+c)^(7/2)+2/3*b*(3*a^2-b^2)/tan( d*x+c)^(3/2)+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)*t an(d*x+c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1017 vs. \(2 (236) = 472\).
Time = 0.10 (sec) , antiderivative size = 1017, normalized size of antiderivative = 3.75 \[ \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((a+b*tan(d*x+c))^3/tan(d*x+c)^(11/2),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))*tan(d*x + c)^5 + 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))*tan(d*x + c)^5 + 315*sqrt(1/2)*d*sq rt((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))*tan(d*x + c)^5 - 315*s qrt(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)...
\[ \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{3}}{\tan ^{\frac {11}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*tan(d*x+c))**3/tan(d*x+c)**(11/2),x)
Output:
Integral((a + b*tan(c + d*x))**3/tan(c + d*x)**(11/2), x)
Time = 0.12 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {630 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 630 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - 315 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 315 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \frac {8 \, {\left (315 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{4} + 135 \, a^{2} b \tan \left (d x + c\right ) - 105 \, {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} + 35 \, a^{3} - 63 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}\right )}}{\tan \left (d x + c\right )^{\frac {9}{2}}}}{1260 \, d} \] Input:
integrate((a+b*tan(d*x+c))^3/tan(d*x+c)^(11/2),x, algorithm="maxima")
Output:
-1/1260*(630*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(s qrt(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) + 8*(315*(a^3 - 3*a*b^2)*tan(d*x + c)^4 + 1 35*a^2*b*tan(d*x + c) - 105*(3*a^2*b - b^3)*tan(d*x + c)^3 + 35*a^3 - 63*( a^3 - 3*a*b^2)*tan(d*x + c)^2)/tan(d*x + c)^(9/2))/d
\[ \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{3}}{\tan \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*tan(d*x+c))^3/tan(d*x+c)^(11/2),x, algorithm="giac")
Output:
undef
Time = 7.70 (sec) , antiderivative size = 1821, normalized size of antiderivative = 6.72 \[ \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\text {Too large to display} \] Input:
int((a + b*tan(c + d*x))^3/tan(c + d*x)^(11/2),x)
Output:
2*atanh((32*a^6*d^3*tan(c + d*x)^(1/2)*((b^6*1i)/(4*d^2) - (a^6*1i)/(4*d^2 ) + (3*a*b^5)/(2*d^2) + (3*a^5*b)/(2*d^2) - (a^2*b^4*15i)/(4*d^2) - (5*a^3 *b^3)/d^2 + (a^4*b^2*15i)/(4*d^2))^(1/2))/(16*a^9*d^2 + b^9*d^2*16i + 48*a *b^8*d^2 + a^8*b*d^2*48i - a^2*b^7*d^2*288i - 736*a^3*b^6*d^2 + a^4*b^5*d^ 2*960i + 960*a^5*b^4*d^2 - a^6*b^3*d^2*736i - 288*a^7*b^2*d^2) - (32*b^6*d ^3*tan(c + d*x)^(1/2)*((b^6*1i)/(4*d^2) - (a^6*1i)/(4*d^2) + (3*a*b^5)/(2* d^2) + (3*a^5*b)/(2*d^2) - (a^2*b^4*15i)/(4*d^2) - (5*a^3*b^3)/d^2 + (a^4* b^2*15i)/(4*d^2))^(1/2))/(16*a^9*d^2 + b^9*d^2*16i + 48*a*b^8*d^2 + a^8*b* d^2*48i - a^2*b^7*d^2*288i - 736*a^3*b^6*d^2 + a^4*b^5*d^2*960i + 960*a^5* b^4*d^2 - a^6*b^3*d^2*736i - 288*a^7*b^2*d^2) + (480*a^2*b^4*d^3*tan(c + d *x)^(1/2)*((b^6*1i)/(4*d^2) - (a^6*1i)/(4*d^2) + (3*a*b^5)/(2*d^2) + (3*a^ 5*b)/(2*d^2) - (a^2*b^4*15i)/(4*d^2) - (5*a^3*b^3)/d^2 + (a^4*b^2*15i)/(4* d^2))^(1/2))/(16*a^9*d^2 + b^9*d^2*16i + 48*a*b^8*d^2 + a^8*b*d^2*48i - a^ 2*b^7*d^2*288i - 736*a^3*b^6*d^2 + a^4*b^5*d^2*960i + 960*a^5*b^4*d^2 - a^ 6*b^3*d^2*736i - 288*a^7*b^2*d^2) - (480*a^4*b^2*d^3*tan(c + d*x)^(1/2)*(( b^6*1i)/(4*d^2) - (a^6*1i)/(4*d^2) + (3*a*b^5)/(2*d^2) + (3*a^5*b)/(2*d^2) - (a^2*b^4*15i)/(4*d^2) - (5*a^3*b^3)/d^2 + (a^4*b^2*15i)/(4*d^2))^(1/2)) /(16*a^9*d^2 + b^9*d^2*16i + 48*a*b^8*d^2 + a^8*b*d^2*48i - a^2*b^7*d^2*28 8i - 736*a^3*b^6*d^2 + a^4*b^5*d^2*960i + 960*a^5*b^4*d^2 - a^6*b^3*d^2*73 6i - 288*a^7*b^2*d^2))*((6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*...
\[ \int \frac {(a+b \tan (c+d x))^3}{\tan ^{\frac {11}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )^{6}}d x \right ) a^{3}+3 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )^{5}}d x \right ) a^{2} b +3 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )^{4}}d x \right ) a \,b^{2}+\left (\int \frac {\sqrt {\tan \left (d x +c \right )}}{\tan \left (d x +c \right )^{3}}d x \right ) b^{3} \] Input:
int((a+b*tan(d*x+c))^3/tan(d*x+c)^(11/2),x)
Output:
int(sqrt(tan(c + d*x))/tan(c + d*x)**6,x)*a**3 + 3*int(sqrt(tan(c + d*x))/ tan(c + d*x)**5,x)*a**2*b + 3*int(sqrt(tan(c + d*x))/tan(c + d*x)**4,x)*a* b**2 + int(sqrt(tan(c + d*x))/tan(c + d*x)**3,x)*b**3