Integrand size = 25, antiderivative size = 377 \[ \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}+\frac {32 a^2 b}{35 d e^2 (e \cot (c+d x))^{5/2}}-\frac {2 a \left (a^2-3 b^2\right )}{3 d e^3 (e \cot (c+d x))^{3/2}}-\frac {2 b \left (3 a^2-b^2\right )}{d e^4 \sqrt {e \cot (c+d x)}}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d e^{9/2}} \]
32/35*a^2*b/d/e^2/(e*cot(d*x+c))^(5/2)-2/3*a*(a^2-3*b^2)/d/e^3/(e*cot(d*x+ c))^(3/2)+2/7*a^2*(a+b*cot(d*x+c))/d/e/(e*cot(d*x+c))^(7/2)+1/2*(a-b)*(a^2 +4*a*b+b^2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d/e^(9/2)*2^(1/ 2)-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2) )/d/e^(9/2)*2^(1/2)+1/4*(a+b)*(a^2-4*a*b+b^2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2 )-2^(1/2)*(e*cot(d*x+c))^(1/2))/d/e^(9/2)*2^(1/2)-1/4*(a+b)*(a^2-4*a*b+b^2 )*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/d/e^(9/2)*2^ (1/2)-2*b*(3*a^2-b^2)/d/e^4/(e*cot(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.80 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.31 \[ \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx=\frac {2 \sqrt {e \cot (c+d x)} \left (5 a \left (a^2-3 b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},1,-\frac {3}{4},-\cot ^2(c+d x)\right )+b \left (b (15 a+7 b \cot (c+d x))+7 \left (3 a^2-b^2\right ) \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},1,-\frac {1}{4},-\cot ^2(c+d x)\right )\right )\right ) \tan ^4(c+d x)}{35 d e^5} \]
(2*Sqrt[e*Cot[c + d*x]]*(5*a*(a^2 - 3*b^2)*Hypergeometric2F1[-7/4, 1, -3/4 , -Cot[c + d*x]^2] + b*(b*(15*a + 7*b*Cot[c + d*x]) + 7*(3*a^2 - b^2)*Cot[ c + d*x]*Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2]))*Tan[c + d*x]^ 4)/(35*d*e^5)
Time = 1.37 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.97, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.920, Rules used = {3042, 4048, 27, 3042, 4111, 27, 3042, 4012, 3042, 4012, 25, 3042, 4017, 25, 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.
| \(\displaystyle \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}-\frac {2 \int -\frac {-b \left (5 a^2-7 b^2\right ) \cot ^2(c+d x) e^2+16 a^2 b e^2-7 a \left (a^2-3 b^2\right ) \cot (c+d x) e^2}{2 (e \cot (c+d x))^{7/2}}dx}{7 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-b \left (5 a^2-7 b^2\right ) \cot ^2(c+d x) e^2+16 a^2 b e^2-7 a \left (a^2-3 b^2\right ) \cot (c+d x) e^2}{(e \cot (c+d x))^{7/2}}dx}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {-b \left (5 a^2-7 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^2+16 a^2 b e^2+7 a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e^2}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4111 |
| \(\displaystyle \frac {\frac {\int -\frac {7 \left (a \left (a^2-3 b^2\right ) e^3+b \left (3 a^2-b^2\right ) \cot (c+d x) e^3\right )}{(e \cot (c+d x))^{5/2}}dx}{e^2}+\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \int \frac {a \left (a^2-3 b^2\right ) e^3+b \left (3 a^2-b^2\right ) \cot (c+d x) e^3}{(e \cot (c+d x))^{5/2}}dx}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \int \frac {a \left (a^2-3 b^2\right ) e^3-b \left (3 a^2-b^2\right ) e^3 \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\int \frac {b \left (3 a^2-b^2\right ) e^4-a \left (a^2-3 b^2\right ) e^4 \cot (c+d x)}{(e \cot (c+d x))^{3/2}}dx}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\int \frac {b \left (3 a^2-b^2\right ) e^4+a \left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e^4}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {\int -\frac {a \left (a^2-3 b^2\right ) e^5+b \left (3 a^2-b^2\right ) \cot (c+d x) e^5}{\sqrt {e \cot (c+d x)}}dx}{e^2}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a \left (a^2-3 b^2\right ) e^5+b \left (3 a^2-b^2\right ) \cot (c+d x) e^5}{\sqrt {e \cot (c+d x)}}dx}{e^2}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a \left (a^2-3 b^2\right ) e^5-b \left (3 a^2-b^2\right ) e^5 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{e^2}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}-\frac {2 \int -\frac {e^5 \left (a \left (a^2-3 b^2\right ) e+b \left (3 a^2-b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d e^2}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 \int \frac {e^5 \left (a \left (a^2-3 b^2\right ) e+b \left (3 a^2-b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 e^3 \int \frac {a \left (a^2-3 b^2\right ) e+b \left (3 a^2-b^2\right ) \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 e^3 \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {32 a^2 b e}{5 d (e \cot (c+d x))^{5/2}}-\frac {7 \left (\frac {\frac {2 e^3 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}+\frac {2 b e^3 \left (3 a^2-b^2\right )}{d \sqrt {e \cot (c+d x)}}}{e^2}+\frac {2 a e^2 \left (a^2-3 b^2\right )}{3 d (e \cot (c+d x))^{3/2}}\right )}{e^2}}{7 e^3}+\frac {2 a^2 (a+b \cot (c+d x))}{7 d e (e \cot (c+d x))^{7/2}}\) |
(2*a^2*(a + b*Cot[c + d*x]))/(7*d*e*(e*Cot[c + d*x])^(7/2)) + ((32*a^2*b*e )/(5*d*(e*Cot[c + d*x])^(5/2)) - (7*((2*a*(a^2 - 3*b^2)*e^2)/(3*d*(e*Cot[c + d*x])^(3/2)) + ((2*b*(3*a^2 - b^2)*e^3)/(d*Sqrt[e*Cot[c + d*x]]) + (2*e ^3*(((a - b)*(a^2 + 4*a*b + b^2)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x ]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]] )/Sqrt[e]]/(Sqrt[2]*Sqrt[e])))/2 + ((a + b)*(a^2 - 4*a*b + b^2)*(-1/2*Log[ e + e*Cot[c + d*x] - Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(Sqrt[2]*Sqrt[e ]) + Log[e + e*Cot[c + d*x] + Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(2*Sqr t[2]*Sqrt[e])))/2))/d)/e^2))/e^2)/(7*e^3)
3.1.68.3.1 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[(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 = 0.04 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(-\frac {2 \left (\frac {\frac {\left (a^{3} e -3 a e \,b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}-\frac {a^{3} e}{7 \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {3 a^{2} b}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {a \left (a^{2}-3 b^{2}\right )}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {b \left (3 a^{2}-b^{2}\right )}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d \,e^{2}}\) | \(388\) |
| default | \(-\frac {2 \left (\frac {\frac {\left (a^{3} e -3 a e \,b^{2}\right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e^{2}}-\frac {a^{3} e}{7 \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}-\frac {3 a^{2} b}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {a \left (a^{2}-3 b^{2}\right )}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {b \left (3 a^{2}-b^{2}\right )}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d \,e^{2}}\) | \(388\) |
| parts | \(-\frac {2 a^{3} e \left (-\frac {1}{7 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {1}{3 e^{4} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{6}}\right )}{d}-\frac {2 b^{3} \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d \,e^{2}}-\frac {6 a \,b^{2} \left (-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{4}}-\frac {1}{3 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d e}+\frac {3 a^{2} b \left (\frac {2}{5 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {2}{e^{4} \sqrt {e \cot \left (d x +c \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e^{4} \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}\) | \(673\) |
-2/d/e^2*(1/e^2*(1/8*(a^3*e-3*a*b^2*e)*(e^2)^(1/4)/e^2*2^(1/2)*(ln((e*cot( d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c) -(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/( e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d* x+c))^(1/2)+1))+1/8*(3*a^2*b-b^3)/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e ^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1 /4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4 )*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/ 2)+1)))-1/7*a^3*e/(e*cot(d*x+c))^(7/2)-3/5*a^2*b/(e*cot(d*x+c))^(5/2)+1/3* a/e*(a^2-3*b^2)/(e*cot(d*x+c))^(3/2)+b*(3*a^2-b^2)/e^2/(e*cot(d*x+c))^(1/2 ))
Leaf count of result is larger than twice the leaf count of optimal. 1839 vs. \(2 (316) = 632\).
Time = 0.34 (sec) , antiderivative size = 1839, normalized size of antiderivative = 4.88 \[ \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx=\text {Too large to display} \]
1/210*(105*(d*e^5*cos(2*d*x + 2*c)^2 + 2*d*e^5*cos(2*d*x + 2*c) + d*e^5)*s qrt(-(d^2*e^9*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*e^18)) + 6*a^5*b - 20*a^3*b^3 + 6*a*b^5 )/(d^2*e^9))*log(-(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((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((3*a^2*b - b^3)*d^3*e^14*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*e^18)) + (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d*e^5)*sqrt(-(d^2*e^9*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*e^18)) + 6*a^5*b - 20*a^3*b^3 + 6*a*b^5)/(d^2*e^9))) - 105*(d*e^5*cos(2*d*x + 2* c)^2 + 2*d*e^5*cos(2*d*x + 2*c) + d*e^5)*sqrt(-(d^2*e^9*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*e^18)) + 6*a^5*b - 20*a^3*b^3 + 6*a*b^5)/(d^2*e^9))*log(-(a^12 - 12*a^1 0*b^2 - 27*a^8*b^4 + 27*a^4*b^8 + 12*a^2*b^10 - b^12)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - ((3*a^2*b - b^3)*d^3*e^14*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*e^18)) + (a^9 - 18*a^7*b^2 + 60*a^5*b^4 - 46*a^3*b^6 + 3*a*b^8)*d*e^5)* sqrt(-(d^2*e^9*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*e^18)) + 6*a^5*b - 20*a^3*b^3 + 6*a*b^ 5)/(d^2*e^9))) - 105*(d*e^5*cos(2*d*x + 2*c)^2 + 2*d*e^5*cos(2*d*x + 2*...
\[ \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx=\int \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{3}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {9}{2}}}\, dx \]
Exception generated. \[ \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Timed out. \[ \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx=\text {Timed out} \]
Time = 18.02 (sec) , antiderivative size = 1992, normalized size of antiderivative = 5.28 \[ \int \frac {(a+b \cot (c+d x))^3}{(e \cot (c+d x))^{9/2}} \, dx=\text {Too large to display} \]
atan((((e*cot(c + d*x))^(1/2)*(16*a^6*d^3*e^14 - 16*b^6*d^3*e^14 + 240*a^2 *b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14) + (32*a^3*d^4*e^19 - 96*a*b^2*d^4*e^ 19)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4 *b^2)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2* b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2)*1i + ((e*cot(c + d* x))^(1/2)*(16*a^6*d^3*e^14 - 16*b^6*d^3*e^14 + 240*a^2*b^4*d^3*e^14 - 240* a^4*b^2*d^3*e^14) - (32*a^3*d^4*e^19 - 96*a*b^2*d^4*e^19)*(((a*b^5*6i + a^ 5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9 ))^(1/2))*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2)*1i)/(((e*cot(c + d*x))^(1/2)*(16*a^6*d^ 3*e^14 - 16*b^6*d^3*e^14 + 240*a^2*b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14) + (32*a^3*d^4*e^19 - 96*a*b^2*d^4*e^19)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6 i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d ^2*e^9))^(1/2) - ((e*cot(c + d*x))^(1/2)*(16*a^6*d^3*e^14 - 16*b^6*d^3*e^1 4 + 240*a^2*b^4*d^3*e^14 - 240*a^4*b^2*d^3*e^14) - (32*a^3*d^4*e^19 - 96*a *b^2*d^4*e^19)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*2 0i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2))*(((a*b^5*6i + a^5*b*6i - a^6 + b^ 6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e^9))^(1/2) - 16*b^9 *d^2*e^10 + 48*a^8*b*d^2*e^10 + 96*a^4*b^5*d^2*e^10 + 128*a^6*b^3*d^2*e...