Integrand size = 25, antiderivative size = 437 \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx=\frac {a^{5/2} \left (3 a^2+7 b^2\right ) e^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{b^{5/2} \left (a^2+b^2\right )^2 d}+\frac {\left (a^2-2 a b-b^2\right ) e^{7/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2-2 a b-b^2\right ) e^{7/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (3 a^2+2 b^2\right ) e^3 \sqrt {e \cot (c+d x)}}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b \left (a^2+b^2\right ) d (a+b \cot (c+d x))}+\frac {\left (a^2+2 a b-b^2\right ) e^{7/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d}-\frac {\left (a^2+2 a b-b^2\right ) e^{7/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^2 d} \]
a^(5/2)*(3*a^2+7*b^2)*e^(7/2)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/ e^(1/2))/b^(5/2)/(a^2+b^2)^2/d+a^2*e^2*(e*cot(d*x+c))^(3/2)/b/(a^2+b^2)/d/ (a+b*cot(d*x+c))+1/2*(a^2-2*a*b-b^2)*e^(7/2)*arctan(1-2^(1/2)*(e*cot(d*x+c ))^(1/2)/e^(1/2))/(a^2+b^2)^2/d*2^(1/2)-1/2*(a^2-2*a*b-b^2)*e^(7/2)*arctan (1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^2/d*2^(1/2)+1/4*(a^2+2* a*b-b^2)*e^(7/2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2 ))/(a^2+b^2)^2/d*2^(1/2)-1/4*(a^2+2*a*b-b^2)*e^(7/2)*ln(e^(1/2)+cot(d*x+c) *e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)^2/d*2^(1/2)-(3*a^2+2*b^2) *e^3*(e*cot(d*x+c))^(1/2)/b^2/(a^2+b^2)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.19 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.05 \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx=-\frac {(e \cot (c+d x))^{7/2} \left (\frac {4 a^{9/2} \arctan \left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{b^{5/2} \left (a^2+b^2\right )^2}-\frac {4 a^4 \sqrt {\cot (c+d x)}}{b^2 \left (a^2+b^2\right )^2}+\frac {4 a^3 \cot ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right )^2}-\frac {4 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 \left (a^2+b^2\right )^2}+\frac {4 a b \cot ^{\frac {7}{2}}(c+d x)}{7 \left (a^2+b^2\right )^2}+\frac {4 a b \left (7 \cot ^{\frac {3}{2}}(c+d x)-3 \cot ^{\frac {7}{2}}(c+d x)-7 \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )}{21 \left (a^2+b^2\right )^2}+\frac {2 b^2 \cot ^{\frac {9}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {9}{2},\frac {11}{2},-\frac {b \cot (c+d x)}{a}\right )}{9 a^2 \left (a^2+b^2\right )}-\frac {(a-b) (a+b) \left (10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+40 \sqrt {\cot (c+d x)}-8 \cot ^{\frac {5}{2}}(c+d x)+5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{20 \left (a^2+b^2\right )^2}\right )}{d \cot ^{\frac {7}{2}}(c+d x)} \]
-(((e*Cot[c + d*x])^(7/2)*((4*a^(9/2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/ Sqrt[a]])/(b^(5/2)*(a^2 + b^2)^2) - (4*a^4*Sqrt[Cot[c + d*x]])/(b^2*(a^2 + b^2)^2) + (4*a^3*Cot[c + d*x]^(3/2))/(3*b*(a^2 + b^2)^2) - (4*a^2*Cot[c + d*x]^(5/2))/(5*(a^2 + b^2)^2) + (4*a*b*Cot[c + d*x]^(7/2))/(7*(a^2 + b^2) ^2) + (4*a*b*(7*Cot[c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x] ^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^2 ) + (2*b^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[2, 9/2, 11/2, -((b*Cot[c + d*x])/a)])/(9*a^2*(a^2 + b^2)) - ((a - b)*(a + b)*(10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d *x]]] + 40*Sqrt[Cot[c + d*x]] - 8*Cot[c + d*x]^(5/2) + 5*Sqrt[2]*Log[1 - S qrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt [Cot[c + d*x]] + Cot[c + d*x]]))/(20*(a^2 + b^2)^2)))/(d*Cot[c + d*x]^(7/2 )))
Time = 2.03 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.91, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 4048, 27, 3042, 4130, 27, 3042, 4136, 27, 3042, 4017, 25, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}
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.
| \(\displaystyle \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}{\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}-\frac {\int -\frac {\sqrt {e \cot (c+d x)} \left (3 a^2 e^3+\left (3 a^2+2 b^2\right ) \cot ^2(c+d x) e^3-2 a b \cot (c+d x) e^3\right )}{2 (a+b \cot (c+d x))}dx}{b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {e \cot (c+d x)} \left (3 a^2 e^3+\left (3 a^2+2 b^2\right ) \cot ^2(c+d x) e^3-2 a b \cot (c+d x) e^3\right )}{a+b \cot (c+d x)}dx}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (3 a^2 e^3+\left (3 a^2+2 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^3+2 a b \tan \left (c+d x+\frac {\pi }{2}\right ) e^3\right )}{a-b \tan \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {-\frac {2 \int \frac {a \left (3 a^2+4 b^2\right ) \cot ^2(c+d x) e^4+a \left (3 a^2+2 b^2\right ) e^4+2 b^3 \cot (c+d x) e^4}{2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {a \left (3 a^2+4 b^2\right ) \cot ^2(c+d x) e^4+a \left (3 a^2+2 b^2\right ) e^4+2 b^3 \cot (c+d x) e^4}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {a \left (3 a^2+4 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^4+a \left (3 a^2+2 b^2\right ) e^4-2 b^3 \tan \left (c+d x+\frac {\pi }{2}\right ) e^4}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {2 \left (b^2 \left (a^2-b^2\right ) e^4-2 a b^3 e^4 \cot (c+d x)\right )}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}-\frac {2 \int \frac {b^2 \left (a^2-b^2\right ) e^4-2 a b^3 e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 \int \frac {b^2 \left (a^2-b^2\right ) e^4+2 a b^3 \tan \left (c+d x+\frac {\pi }{2}\right ) e^4}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {-\frac {\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {4 \int -\frac {b^2 e^4 \left (\left (a^2-b^2\right ) e-2 a b e \cot (c+d x)\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {4 \int \frac {b^2 e^4 \left (\left (a^2-b^2\right ) e-2 a b e \cot (c+d x)\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 e^4 \int \frac {\left (a^2-b^2\right ) e-2 a b e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \left (a^2-2 a b-b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {-\frac {\frac {a^3 e^4 \left (3 a^2+7 b^2\right ) \int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}+\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 a^3 e^3 \left (3 a^2+7 b^2\right ) \int \frac {1}{\frac {b \cot ^2(c+d x)}{e}+a}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}}{2 b \left (a^2+b^2\right )}+\frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a^2 e^2 (e \cot (c+d x))^{3/2}}{b d \left (a^2+b^2\right ) (a+b \cot (c+d x))}+\frac {-\frac {2 e^3 \left (3 a^2+2 b^2\right ) \sqrt {e \cot (c+d x)}}{b d}-\frac {\frac {4 b^2 e^4 \left (\frac {1}{2} \left (a^2-2 a b-b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (a^2+2 a b-b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 a^{5/2} e^{7/2} \left (3 a^2+7 b^2\right ) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}}{b}}{2 b \left (a^2+b^2\right )}\) |
(a^2*e^2*(e*Cot[c + d*x])^(3/2))/(b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])) + ((-2*(3*a^2 + 2*b^2)*e^3*Sqrt[e*Cot[c + d*x]])/(b*d) - ((2*a^(5/2)*(3*a^2 + 7*b^2)*e^(7/2)*ArcTan[(Sqrt[b]*Cot[c + d*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[b ]*(a^2 + b^2)*d) + (4*b^2*e^4*(((a^2 - 2*a*b - b^2)*(-(ArcTan[1 - (Sqrt[2] *Sqrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*S qrt[e*Cot[c + d*x]])/Sqrt[e]]/(Sqrt[2]*Sqrt[e])))/2 + ((a^2 + 2*a*b - b^2) *(-1/2*Log[e + e*Cot[c + d*x] - Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(Sqr t[2]*Sqrt[e]) + Log[e + e*Cot[c + d*x] + Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d* x]]]/(2*Sqrt[2]*Sqrt[e])))/2))/((a^2 + b^2)*d))/b)/(2*b*(a^2 + b^2))
3.1.75.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((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_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((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 - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Time = 0.33 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.94
| method | result | size |
| derivativedivides | \(-\frac {2 e^{3} \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{b^{2}}-\frac {a^{3} e \left (\frac {\left (-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) \sqrt {e \cot \left (d x +c \right )}}{e \cot \left (d x +c \right ) b +a e}+\frac {\left (3 a^{2}+7 b^{2}\right ) \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{b^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {e \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{\left (a^{2}+b^{2}\right )^{2}}\right )}{d}\) | \(409\) |
| default | \(-\frac {2 e^{3} \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{b^{2}}-\frac {a^{3} e \left (\frac {\left (-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) \sqrt {e \cot \left (d x +c \right )}}{e \cot \left (d x +c \right ) b +a e}+\frac {\left (3 a^{2}+7 b^{2}\right ) \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}\right )}{b^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {e \left (\frac {\left (a^{2} e -b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}-\frac {a b \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{\left (a^{2}+b^{2}\right )^{2}}\right )}{d}\) | \(409\) |
-2/d*e^3*((e*cot(d*x+c))^(1/2)/b^2-a^3*e/b^2/(a^2+b^2)^2*((-1/2*a^2-1/2*b^ 2)*(e*cot(d*x+c))^(1/2)/(e*cot(d*x+c)*b+a*e)+1/2*(3*a^2+7*b^2)/(a*e*b)^(1/ 2)*arctan((e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2)))+e/(a^2+b^2)^2*(1/8*(a^2*e -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))-1/4*a*b/(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))))
Leaf count of result is larger than twice the leaf count of optimal. 3170 vs. \(2 (372) = 744\).
Time = 0.71 (sec) , antiderivative size = 6403, normalized size of antiderivative = 14.65 \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx=\int { \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (b \cot \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Time = 16.41 (sec) , antiderivative size = 13244, normalized size of antiderivative = 30.31 \[ \int \frac {(e \cot (c+d x))^{7/2}}{(a+b \cot (c+d x))^2} \, dx=\text {Too large to display} \]
(atan(((((16*(e*cot(c + d*x))^(1/2)*(9*a^12*e^24 + 2*b^12*e^24 + 4*a^2*b^1 0*e^24 + 2*a^4*b^8*e^24 - 49*a^6*b^6*e^24 + 7*a^8*b^4*e^24 + 33*a^10*b^2*e ^24))/(b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3* d^4) + (((16*(30*a^6*b^8*d^2*e^21 - 224*a^4*b^10*d^2*e^21 - 18*a^14*d^2*e^ 21 + 600*a^8*b^6*d^2*e^21 + 388*a^10*b^4*d^2*e^21 + 24*a^12*b^2*d^2*e^21)) /(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 + 4*a^6*b^5*d^5 + a^8*b^3*d^5) - (((16*(e*cot(c + d*x))^(1/2)*(72*a^15*b*d^2*e^17 - 60*a*b^15*d^2*e^17 - 52*a^3*b^13*d^2*e^17 + 72*a^5*b^11*d^2*e^17 + 448*a^7*b^9*d^2*e^17 + 1108* a^9*b^7*d^2*e^17 + 1132*a^11*b^5*d^2*e^17 + 480*a^13*b^3*d^2*e^17))/(b^11* d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3*d^4) + (((16 *(8*a*b^17*d^4*e^14 + 96*a^3*b^15*d^4*e^14 + 360*a^5*b^13*d^4*e^14 + 640*a ^7*b^11*d^4*e^14 + 600*a^9*b^9*d^4*e^14 + 288*a^11*b^7*d^4*e^14 + 56*a^13* b^5*d^4*e^14))/(b^11*d^5 + 4*a^2*b^9*d^5 + 6*a^4*b^7*d^5 + 4*a^6*b^5*d^5 + a^8*b^3*d^5) - (8*(e*cot(c + d*x))^(1/2)*(3*a^2 + 7*b^2)*(-a^5*b^5*e^7)^( 1/2)*(32*b^20*d^4*e^10 + 160*a^2*b^18*d^4*e^10 + 288*a^4*b^16*d^4*e^10 + 1 60*a^6*b^14*d^4*e^10 - 160*a^8*b^12*d^4*e^10 - 288*a^10*b^10*d^4*e^10 - 16 0*a^12*b^8*d^4*e^10 - 32*a^14*b^6*d^4*e^10))/((b^9*d + 2*a^2*b^7*d + a^4*b ^5*d)*(b^11*d^4 + 4*a^2*b^9*d^4 + 6*a^4*b^7*d^4 + 4*a^6*b^5*d^4 + a^8*b^3* d^4)))*(3*a^2 + 7*b^2)*(-a^5*b^5*e^7)^(1/2))/(2*(b^9*d + 2*a^2*b^7*d + a^4 *b^5*d)))*(3*a^2 + 7*b^2)*(-a^5*b^5*e^7)^(1/2))/(2*(b^9*d + 2*a^2*b^7*d...