Integrand size = 23, antiderivative size = 247 \[ \int \frac {(a+b \tan (c+d x))^3}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {32 a b^2}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2 (b+a \cot (c+d x))}{7 d \cot ^{\frac {7}{2}}(c+d x)} \] Output:
1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*cot(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)*cot(d*x+c)^(1/2))*2^(1/2)/d-1/2*( a+b)*(a^2-4*a*b+b^2)*arctanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1 /2)/d+32/35*a*b^2/d/cot(d*x+c)^(5/2)+2/3*b*(3*a^2-b^2)/d/cot(d*x+c)^(3/2)+ 2*a*(a^2-3*b^2)/d/cot(d*x+c)^(1/2)+2/7*b^2*(b+a*cot(d*x+c))/d/cot(d*x+c)^( 7/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.47 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.41 \[ \int \frac {(a+b \tan (c+d x))^3}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \left (5 b \left (-3 a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},1,-\frac {3}{4},-\cot ^2(c+d x)\right )+a \left (a (15 b+7 a \cot (c+d x))-7 \left (a^2-3 b^2\right ) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\cot ^2(c+d x)\right )\right )\right )}{35 d \cot ^{\frac {7}{2}}(c+d x)} \] Input:
Integrate[(a + b*Tan[c + d*x])^3/Cot[c + d*x]^(3/2),x]
Output:
(2*(5*b*(-3*a^2 + b^2)*Hypergeometric2F1[-7/4, 1, -3/4, -Cot[c + d*x]^2] + a*(a*(15*b + 7*a*Cot[c + d*x]) - 7*(a^2 - 3*b^2)*Cot[c + d*x]*Hypergeomet ric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2])))/(35*d*Cot[c + d*x]^(7/2))
Time = 1.17 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.14, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.043, Rules used = {3042, 4156, 3042, 4048, 27, 3042, 4111, 27, 3042, 4012, 3042, 4012, 25, 3042, 4017, 25, 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.
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^3}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (c+d x))^3}{\cot (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {(a \cot (c+d x)+b)^3}{\cot ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}-\frac {2}{7} \int -\frac {16 a b^2+7 \left (3 a^2-b^2\right ) \cot (c+d x) b+a \left (7 a^2-5 b^2\right ) \cot ^2(c+d x)}{2 \cot ^{\frac {7}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \frac {16 a b^2+7 \left (3 a^2-b^2\right ) \cot (c+d x) b+a \left (7 a^2-5 b^2\right ) \cot ^2(c+d x)}{\cot ^{\frac {7}{2}}(c+d x)}dx+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {16 a b^2-7 \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) b+a \left (7 a^2-5 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4111 |
| \(\displaystyle \frac {1}{7} \left (\int \frac {7 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\cot ^{\frac {5}{2}}(c+d x)}dx+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (7 \int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)}dx+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (7 \int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{7} \left (7 \left (\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)}dx+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (7 \left (\int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {1}{7} \left (7 \left (\int -\frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\frac {2 \int -\frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (7 \left (\frac {2 \int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\frac {2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\frac {2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\frac {2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\frac {2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\frac {2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\frac {2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\frac {2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{7} \left (7 \left (-\frac {2 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 b \left (3 a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (a^2-3 b^2\right )}{d \sqrt {\cot (c+d x)}}\right )+\frac {32 a b^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 b^2 (a \cot (c+d x)+b)}{7 d \cot ^{\frac {7}{2}}(c+d x)}\) |
Input:
Int[(a + b*Tan[c + d*x])^3/Cot[c + d*x]^(3/2),x]
Output:
(2*b^2*(b + a*Cot[c + d*x]))/(7*d*Cot[c + d*x]^(7/2)) + ((32*a*b^2)/(5*d*C ot[c + d*x]^(5/2)) + 7*((2*b*(3*a^2 - b^2))/(3*d*Cot[c + d*x]^(3/2)) + (2* a*(a^2 - 3*b^2))/(d*Sqrt[Cot[c + d*x]]) - (2*(-1/2*((a - b)*(a^2 + 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt [2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])) + ((a + b)*(a^2 - 4*a*b + b^2)*(-1/2*Log [1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]* Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/d))/7
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 ]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Leaf count of result is larger than twice the leaf count of optimal. \(515\) vs. \(2(215)=430\).
Time = 0.21 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.09
| method | result | size |
| derivativedivides | \(\frac {120 b^{3} \tan \left (d x +c \right )^{\frac {7}{2}}+504 a \,b^{2} \tan \left (d x +c \right )^{\frac {5}{2}}+840 a^{2} b \tan \left (d x +c \right )^{\frac {3}{2}}-280 b^{3} \tan \left (d x +c \right )^{\frac {3}{2}}-105 \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{3}+315 \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a \,b^{2}-210 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}-630 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +630 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}+210 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}-210 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}-630 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +630 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}+210 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}-315 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a^{2} b +105 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) b^{3}+840 a^{3} \sqrt {\tan \left (d x +c \right )}-2520 a \,b^{2} \sqrt {\tan \left (d x +c \right )}}{420 d \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right )^{\frac {3}{2}}}\) | \(516\) |
| default | \(\frac {120 b^{3} \tan \left (d x +c \right )^{\frac {7}{2}}+504 a \,b^{2} \tan \left (d x +c \right )^{\frac {5}{2}}+840 a^{2} b \tan \left (d x +c \right )^{\frac {3}{2}}-280 b^{3} \tan \left (d x +c \right )^{\frac {3}{2}}-105 \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a^{3}+315 \sqrt {2}\, \ln \left (-\frac {\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}{\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}\right ) a \,b^{2}-210 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}-630 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +630 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}+210 \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}-210 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{3}-630 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a^{2} b +630 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) a \,b^{2}+210 \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}\right ) b^{3}-315 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) a^{2} b +105 \sqrt {2}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}-\tan \left (d x +c \right )-1}{\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {\tan \left (d x +c \right )}+1}\right ) b^{3}+840 a^{3} \sqrt {\tan \left (d x +c \right )}-2520 a \,b^{2} \sqrt {\tan \left (d x +c \right )}}{420 d \left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {3}{2}} \tan \left (d x +c \right )^{\frac {3}{2}}}\) | \(516\) |
Input:
int((a+b*tan(d*x+c))^3/cot(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/420/d*(120*b^3*tan(d*x+c)^(7/2)+504*a*b^2*tan(d*x+c)^(5/2)+840*a^2*b*tan (d*x+c)^(3/2)-280*b^3*tan(d*x+c)^(3/2)-105*2^(1/2)*ln(-(tan(d*x+c)+2^(1/2) *tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*a^3+315*2^(1 /2)*ln(-(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1)/(2^(1/2)*tan(d*x+c)^(1/2)- tan(d*x+c)-1))*a*b^2-210*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^3-63 0*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+630*2^(1/2)*arctan(1+2^ (1/2)*tan(d*x+c)^(1/2))*a*b^2+210*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2 ))*b^3-210*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^3-630*2^(1/2)*arc tan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^2*b+630*2^(1/2)*arctan(-1+2^(1/2)*tan(d *x+c)^(1/2))*a*b^2+210*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b^3-315 *2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(tan(d*x+c)+2^(1/2)*t an(d*x+c)^(1/2)+1))*a^2*b+105*2^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d* x+c)-1)/(tan(d*x+c)+2^(1/2)*tan(d*x+c)^(1/2)+1))*b^3+840*a^3*tan(d*x+c)^(1 /2)-2520*a*b^2*tan(d*x+c)^(1/2))/(1/tan(d*x+c))^(3/2)/tan(d*x+c)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 953 vs. \(2 (215) = 430\).
Time = 0.09 (sec) , antiderivative size = 953, normalized size of antiderivative = 3.86 \[ \int \frac {(a+b \tan (c+d x))^3}{\cot ^{\frac {3}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((a+b*tan(d*x+c))^3/cot(d*x+c)^(3/2),x, algorithm="fricas")
Output:
1/210*(210*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)) + 210*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)) - 105*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)) + 105*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...
\[ \int \frac {(a+b \tan (c+d x))^3}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{3}}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*tan(d*x+c))**3/cot(d*x+c)**(3/2),x)
Output:
Integral((a + b*tan(c + d*x))**3/cot(c + d*x)**(3/2), x)
Time = 0.21 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \tan (c+d x))^3}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {8 \, {\left (15 \, b^{3} + \frac {63 \, a b^{2}}{\tan \left (d x + c\right )} + \frac {35 \, {\left (3 \, a^{2} b - b^{3}\right )}}{\tan \left (d x + c\right )^{2}} + \frac {105 \, {\left (a^{3} - 3 \, a b^{2}\right )}}{\tan \left (d x + c\right )^{3}}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} + 210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{420 \, d} \] Input:
integrate((a+b*tan(d*x+c))^3/cot(d*x+c)^(3/2),x, algorithm="maxima")
Output:
1/420*(8*(15*b^3 + 63*a*b^2/tan(d*x + c) + 35*(3*a^2*b - b^3)/tan(d*x + c) ^2 + 105*(a^3 - 3*a*b^2)/tan(d*x + c)^3)*tan(d*x + c)^(7/2) + 210*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)))) + 210*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)))) - 105*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^ 2 + b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 105*sqrt(2 )*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan( d*x + c) + 1))/d
\[ \int \frac {(a+b \tan (c+d x))^3}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{3}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*tan(d*x+c))^3/cot(d*x+c)^(3/2),x, algorithm="giac")
Output:
integrate((b*tan(d*x + c) + a)^3/cot(d*x + c)^(3/2), x)
Timed out. \[ \int \frac {(a+b \tan (c+d x))^3}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int((a + b*tan(c + d*x))^3/cot(c + d*x)^(3/2),x)
Output:
int((a + b*tan(c + d*x))^3/cot(c + d*x)^(3/2), x)
\[ \int \frac {(a+b \tan (c+d x))^3}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )^{2}}d x \right ) a^{3}+\left (\int \frac {\sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )^{3}}{\cot \left (d x +c \right )^{2}}d x \right ) b^{3}+3 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )^{2}}{\cot \left (d x +c \right )^{2}}d x \right ) a \,b^{2}+3 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}\, \tan \left (d x +c \right )}{\cot \left (d x +c \right )^{2}}d x \right ) a^{2} b \] Input:
int((a+b*tan(d*x+c))^3/cot(d*x+c)^(3/2),x)
Output:
int(sqrt(cot(c + d*x))/cot(c + d*x)**2,x)*a**3 + int((sqrt(cot(c + d*x))*t an(c + d*x)**3)/cot(c + d*x)**2,x)*b**3 + 3*int((sqrt(cot(c + d*x))*tan(c + d*x)**2)/cot(c + d*x)**2,x)*a*b**2 + 3*int((sqrt(cot(c + d*x))*tan(c + d *x))/cot(c + d*x)**2,x)*a**2*b