Integrand size = 23, antiderivative size = 299 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}+\frac {2 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{35 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (b+a \cot (c+d x))}{7 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d} \]
2/3*a*(a^2-3*b^2)*cot(d*x+c)^(3/2)/d-32/35*a^2*b*cot(d*x+c)^(5/2)/d-2/7*a^ 2*cot(d*x+c)^(5/2)*(b+a*cot(d*x+c))/d-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+ 2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1+2^( 1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)-1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1+cot(d*x+c) -2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)+1/4*(a+b)*(a^2-4*a*b+b^2)*ln(1+cot(d* x+c)+2^(1/2)*cot(d*x+c)^(1/2))/d*2^(1/2)+2*b*(3*a^2-b^2)*cot(d*x+c)^(1/2)/ d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.32 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.77 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {\frac {6}{5} a^2 b \cot ^{\frac {5}{2}}(c+d x)+\frac {2}{7} a^3 \cot ^{\frac {7}{2}}(c+d x)+\frac {2}{3} a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )+\frac {1}{4} b \left (-3 a^2+b^2\right ) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{d} \]
-(((6*a^2*b*Cot[c + d*x]^(5/2))/5 + (2*a^3*Cot[c + d*x]^(7/2))/7 + (2*a*(a ^2 - 3*b^2)*Cot[c + d*x]^(3/2)*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/3 + (b*(-3*a^2 + b^2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[ c + d*x]]] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot [c + d*x]] + Sqrt[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]]))/4)/d)
Time = 1.14 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.93, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.957, Rules used = {3042, 4156, 3042, 4049, 27, 3042, 4113, 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.
| \(\displaystyle \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cot (c+d x)^{9/2} (a+b \tan (c+d x))^3dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) (a \cot (c+d x)+b)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle -\frac {2}{7} \int \frac {1}{2} \cot ^{\frac {3}{2}}(c+d x) \left (-16 a^2 b \cot ^2(c+d x)+7 a \left (a^2-3 b^2\right ) \cot (c+d x)+b \left (5 a^2-7 b^2\right )\right )dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{7} \int \cot ^{\frac {3}{2}}(c+d x) \left (-16 a^2 b \cot ^2(c+d x)+7 a \left (a^2-3 b^2\right ) \cot (c+d x)+b \left (5 a^2-7 b^2\right )\right )dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{7} \int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-16 a^2 b \tan \left (c+d x+\frac {\pi }{2}\right )^2-7 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )+b \left (5 a^2-7 b^2\right )\right )dx-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {1}{7} \left (-\int \cot ^{\frac {3}{2}}(c+d x) \left (7 b \left (3 a^2-b^2\right )+7 a \left (a^2-3 b^2\right ) \cot (c+d x)\right )dx-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (-\int \left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (7 b \left (3 a^2-b^2\right )-7 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{7} \left (-\int \sqrt {\cot (c+d x)} \left (7 b \left (3 a^2-b^2\right ) \cot (c+d x)-7 a \left (a^2-3 b^2\right )\right )dx+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (-\int \sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (-7 a \left (a^2-3 b^2\right )-7 b \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {1}{7} \left (-\int \frac {-7 b \left (3 a^2-b^2\right )-7 a \left (a^2-3 b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (-\int \frac {7 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )-7 b \left (3 a^2-b^2\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {1}{7} \left (-\frac {2 \int \frac {7 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \int \frac {b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )\right )}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{7} \left (-\frac {14 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {14 a \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {14 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{d}-\frac {32 a^2 b \cot ^{\frac {5}{2}}(c+d x)}{5 d}\right )-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x) (a \cot (c+d x)+b)}{7 d}\) |
(-2*a^2*Cot[c + d*x]^(5/2)*(b + a*Cot[c + d*x]))/(7*d) + ((14*b*(3*a^2 - b ^2)*Sqrt[Cot[c + d*x]])/d + (14*a*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2))/(3*d) - (32*a^2*b*Cot[c + d*x]^(5/2))/(5*d) - (14*(((a - b)*(a^2 + 4*a*b + b^2)* (-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sq rt[Cot[c + d*x]]]/Sqrt[2]))/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
3.9.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
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. \(599\) vs. \(2(257)=514\).
Time = 1.83 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.01
| method | result | size |
| derivativedivides | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {9}{2}} \tan \left (d x +c \right ) \left (105 \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{3}-315 \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a \,b^{2}+210 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{3}+630 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{2} b -630 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a \,b^{2}-210 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) b^{3}+210 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{3}+630 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{2} b -630 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a \,b^{2}-210 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) b^{3}+315 \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{2} b -105 \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) b^{3}+2520 \left (\tan ^{3}\left (d x +c \right )\right ) a^{2} b -840 \left (\tan ^{3}\left (d x +c \right )\right ) b^{3}+280 \left (\tan ^{2}\left (d x +c \right )\right ) a^{3}-840 \left (\tan ^{2}\left (d x +c \right )\right ) a \,b^{2}-504 \tan \left (d x +c \right ) a^{2} b -120 a^{3}\right )}{420 d}\) | \(600\) |
| default | \(\frac {\left (\frac {1}{\tan \left (d x +c \right )}\right )^{\frac {9}{2}} \tan \left (d x +c \right ) \left (105 \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{3}-315 \ln \left (-\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a \,b^{2}+210 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{3}+630 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{2} b -630 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a \,b^{2}-210 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) b^{3}+210 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{3}+630 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{2} b -630 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a \,b^{2}-210 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) b^{3}+315 \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) a^{2} b -105 \ln \left (-\frac {\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )-\tan \left (d x +c \right )-1}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) \sqrt {2}\, \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right ) b^{3}+2520 \left (\tan ^{3}\left (d x +c \right )\right ) a^{2} b -840 \left (\tan ^{3}\left (d x +c \right )\right ) b^{3}+280 \left (\tan ^{2}\left (d x +c \right )\right ) a^{3}-840 \left (\tan ^{2}\left (d x +c \right )\right ) a \,b^{2}-504 \tan \left (d x +c \right ) a^{2} b -120 a^{3}\right )}{420 d}\) | \(600\) |
1/420/d*(1/tan(d*x+c))^(9/2)*tan(d*x+c)*(105*ln(-(1+2^(1/2)*tan(d*x+c)^(1/ 2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c) ^(7/2)*a^3-315*ln(-(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d* x+c)^(1/2)-tan(d*x+c)-1))*2^(1/2)*tan(d*x+c)^(7/2)*a*b^2+210*arctan(1+2^(1 /2)*tan(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^3+630*arctan(1+2^(1/2)*ta n(d*x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b-630*arctan(1+2^(1/2)*tan(d* x+c)^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*a*b^2-210*arctan(1+2^(1/2)*tan(d*x+c) ^(1/2))*2^(1/2)*tan(d*x+c)^(7/2)*b^3+210*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2 ))*2^(1/2)*tan(d*x+c)^(7/2)*a^3+630*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^ (1/2)*tan(d*x+c)^(7/2)*a^2*b-630*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/ 2)*tan(d*x+c)^(7/2)*a*b^2-210*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*2^(1/2)* tan(d*x+c)^(7/2)*b^3+315*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^ (1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*2^(1/2)*tan(d*x+c)^(7/2)*a^2*b-105*ln( -(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d *x+c)))*2^(1/2)*tan(d*x+c)^(7/2)*b^3+2520*tan(d*x+c)^3*a^2*b-840*tan(d*x+c )^3*b^3+280*tan(d*x+c)^2*a^3-840*tan(d*x+c)^2*a*b^2-504*tan(d*x+c)*a^2*b-1 20*a^3)
Leaf count of result is larger than twice the leaf count of optimal. 1468 vs. \(2 (257) = 514\).
Time = 0.37 (sec) , antiderivative size = 1468, normalized size of antiderivative = 4.91 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Too large to display} \]
-1/210*(105*d*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30 *a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/ d^4))/d^2)*log(((3*a^2*b - b^3)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^ 4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (a^9 - 18*a^7*b ^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6 *a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a ^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c)))*tan(d*x + c)^3 - 1 05*d*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2 )*log(-((3*a^2*b - b^3)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452* a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) + (a^9 - 18*a^7*b^2 + 60* a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d)*sqrt(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 + d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2) - (a^12 - 12*a^10*b^2 - 27*a^8*b^4 + 27*a^ 4*b^8 + 12*a^2*b^10 - b^12)*sqrt(tan(d*x + c)))*tan(d*x + c)^3 - 105*d*sqr t(-(6*a^5*b - 20*a^3*b^3 + 6*a*b^5 - d^2*sqrt(-(a^12 - 30*a^10*b^2 + 255*a ^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4))/d^2)*log((( 3*a^2*b - b^3)*d^3*sqrt(-(a^12 - 30*a^10*b^2 + 255*a^8*b^4 - 452*a^6*b^6 + 255*a^4*b^8 - 30*a^2*b^10 + b^12)/d^4) - (a^9 - 18*a^7*b^2 + 60*a^5*b^...
Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.33 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.88 \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {504 \, a^{2} b}{\tan \left (d x + c\right )^{\frac {5}{2}}} - \frac {840 \, {\left (3 \, a^{2} b - b^{3}\right )}}{\sqrt {\tan \left (d x + c\right )}} + \frac {120 \, a^{3}}{\tan \left (d x + c\right )^{\frac {7}{2}}} - \frac {280 \, {\left (a^{3} - 3 \, a b^{2}\right )}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{420 \, d} \]
-1/420*(210*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sq rt(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(ta n(d*x + c)) + 1/tan(d*x + c) + 1) + 504*a^2*b/tan(d*x + c)^(5/2) - 840*(3* a^2*b - b^3)/sqrt(tan(d*x + c)) + 120*a^3/tan(d*x + c)^(7/2) - 280*(a^3 - 3*a*b^2)/tan(d*x + c)^(3/2))/d
\[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
Timed out. \[ \int \cot ^{\frac {9}{2}}(c+d x) (a+b \tan (c+d x))^3 \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{9/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]