Integrand size = 25, antiderivative size = 470 \[ \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx=-\frac {\sqrt {a} \left (a^4+18 a^2 b^2-15 b^4\right ) e^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (a+b \cot (c+d x))^2}-\frac {a \left (a^2+9 b^2\right ) e^2 \sqrt {e \cot (c+d x)}}{4 b \left (a^2+b^2\right )^2 d (a+b \cot (c+d x))}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]
-1/2*(a-b)*(a^2+4*a*b+b^2)*e^(5/2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e ^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/2*(a-b)*(a^2+4*a*b+b^2)*e^(5/2)*arctan(1+2 ^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/4*(a+b)*(a^2- 4*a*b+b^2)*e^(5/2)*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)^3/d*2^(1/2)-1/4*(a+b)*(a^2-4*a*b+b^2)*e^(5/2)*ln(e^(1/2)+co t(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*( a^4+18*a^2*b^2-15*b^4)*e^(5/2)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2) /e^(1/2))*a^(1/2)/b^(3/2)/(a^2+b^2)^3/d+1/2*a^2*e^2*(e*cot(d*x+c))^(1/2)/b /(a^2+b^2)/d/(a+b*cot(d*x+c))^2-1/4*a*(a^2+9*b^2)*e^2*(e*cot(d*x+c))^(1/2) /b/(a^2+b^2)^2/d/(a+b*cot(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.23 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.11 \[ \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx=-\frac {(e \cot (c+d x))^{5/2} \left (-\frac {2 a^{5/2} \left (3 a^2-b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )}{b^{3/2} \left (a^2+b^2\right )^3}+\frac {2 a^2 \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)}}{b \left (a^2+b^2\right )^3}-\frac {2 a \left (3 a^2-b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 \left (a^2+b^2\right )^3}+\frac {2 b \left (3 a^2-b^2\right ) \cot ^{\frac {5}{2}}(c+d x)}{5 \left (a^2+b^2\right )^3}+\frac {2 a \left (a^2-3 b^2\right ) \left (\cot ^{\frac {3}{2}}(c+d x)-\cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )}{3 \left (a^2+b^2\right )^3}+\frac {4 b^2 \cot ^{\frac {7}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {7}{2},\frac {9}{2},-\frac {b \cot (c+d x)}{a}\right )}{7 a \left (a^2+b^2\right )^2}+\frac {2 b^2 \cot ^{\frac {7}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (3,\frac {7}{2},\frac {9}{2},-\frac {b \cot (c+d x)}{a}\right )}{7 a^3 \left (a^2+b^2\right )}+\frac {b \left (3 a^2-b^2\right ) \left (10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+40 \sqrt {\cot (c+d x)}-8 \cot ^{\frac {5}{2}}(c+d x)+5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{20 \left (a^2+b^2\right )^3}\right )}{d \cot ^{\frac {5}{2}}(c+d x)} \]
-(((e*Cot[c + d*x])^(5/2)*((-2*a^(5/2)*(3*a^2 - b^2)*ArcTan[(Sqrt[b]*Sqrt[ Cot[c + d*x]])/Sqrt[a]])/(b^(3/2)*(a^2 + b^2)^3) + (2*a^2*(3*a^2 - b^2)*Sq rt[Cot[c + d*x]])/(b*(a^2 + b^2)^3) - (2*a*(3*a^2 - b^2)*Cot[c + d*x]^(3/2 ))/(3*(a^2 + b^2)^3) + (2*b*(3*a^2 - b^2)*Cot[c + d*x]^(5/2))/(5*(a^2 + b^ 2)^3) + (2*a*(a^2 - 3*b^2)*(Cot[c + d*x]^(3/2) - Cot[c + d*x]^(3/2)*Hyperg eometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(3*(a^2 + b^2)^3) + (4*b^2*Cot [c + d*x]^(7/2)*Hypergeometric2F1[2, 7/2, 9/2, -((b*Cot[c + d*x])/a)])/(7* a*(a^2 + b^2)^2) + (2*b^2*Cot[c + d*x]^(7/2)*Hypergeometric2F1[3, 7/2, 9/2 , -((b*Cot[c + d*x])/a)])/(7*a^3*(a^2 + b^2)) + (b*(3*a^2 - b^2)*(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*Sqr t[2]*Log[1 - Sqrt[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)^3)))/(d*Cot [c + d*x]^(5/2)))
Time = 2.13 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.94, 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, 4132, 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.
| \(\displaystyle \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}-\frac {\int -\frac {a^2 e^3+\left (a^2+4 b^2\right ) \cot ^2(c+d x) e^3-4 a b \cot (c+d x) e^3}{2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}dx}{2 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a^2 e^3+\left (a^2+4 b^2\right ) \cot ^2(c+d x) e^3-4 a b \cot (c+d x) e^3}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}dx}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a^2 e^3+\left (a^2+4 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^3+4 a b \tan \left (c+d x+\frac {\pi }{2}\right ) e^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {\int -\frac {a^2 \left (a^2+9 b^2\right ) \cot ^2(c+d x) e^4+a^2 \left (a^2-7 b^2\right ) e^4-8 a b \left (a^2-b^2\right ) \cot (c+d x) e^4}{2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 \left (a^2+9 b^2\right ) \cot ^2(c+d x) e^4+a^2 \left (a^2-7 b^2\right ) e^4-8 a b \left (a^2-b^2\right ) \cot (c+d x) e^4}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {a^2 \left (a^2+9 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^4+a^2 \left (a^2-7 b^2\right ) e^4+8 a b \left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e^4}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {8 \left (a b^2 \left (3 a^2-b^2\right ) e^4+a^2 b \left (a^2-3 b^2\right ) \cot (c+d x) e^4\right )}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}-\frac {8 \int \frac {a b^2 \left (3 a^2-b^2\right ) e^4+a^2 b \left (a^2-3 b^2\right ) \cot (c+d x) e^4}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {8 \int \frac {a b^2 \left (3 a^2-b^2\right ) e^4-a^2 b \left (a^2-3 b^2\right ) e^4 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int -\frac {a b e^4 \left (b \left (3 a^2-b^2\right ) e+a \left (a^2-3 b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {16 \int \frac {a b e^4 \left (b \left (3 a^2-b^2\right ) e+a \left (a^2-3 b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {16 a b e^4 \int \frac {b \left (3 a^2-b^2\right ) e+a \left (a^2-3 b^2\right ) \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {\frac {a^2 e^4 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}+\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 a^2 e^3 \left (a^4+18 a^2 b^2-15 b^4\right ) \int \frac {1}{\frac {b \cot ^2(c+d x)}{e}+a}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}+\frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a^2 e^2 \sqrt {e \cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a+b \cot (c+d x))^2}+\frac {\frac {\frac {16 a b e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 a^{3/2} e^{7/2} \left (a^4+18 a^2 b^2-15 b^4\right ) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}}{2 a e \left (a^2+b^2\right )}-\frac {a e^2 \left (a^2+9 b^2\right ) \sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}}{4 b \left (a^2+b^2\right )}\) |
(a^2*e^2*Sqrt[e*Cot[c + d*x]])/(2*b*(a^2 + b^2)*d*(a + b*Cot[c + d*x])^2) + (-((a*(a^2 + 9*b^2)*e^2*Sqrt[e*Cot[c + d*x]])/((a^2 + b^2)*d*(a + b*Cot[ c + d*x]))) + ((2*a^(3/2)*(a^4 + 18*a^2*b^2 - 15*b^4)*e^(7/2)*ArcTan[(Sqrt [b]*Cot[c + d*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[b]*(a^2 + b^2)*d) + (16*a*b*e^ 4*(((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*Sqrt [2]*Sqrt[e])))/2))/((a^2 + b^2)*d))/(2*a*(a^2 + b^2)*e))/(4*b*(a^2 + b^2))
3.1.83.3.1 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[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[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] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !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 = 1.15 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.97
| method | result | size |
| derivativedivides | \(-\frac {2 e^{4} \left (\frac {a \left (\frac {\left (\frac {1}{8} a^{4}+\frac {5}{4} a^{2} b^{2}+\frac {9}{8} b^{4}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}-\frac {a e \left (a^{4}-6 a^{2} b^{2}-7 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8 b}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (a^{4}+18 a^{2} b^{2}-15 b^{4}\right ) \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{8 b \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{3} e}+\frac {\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e \left (a^{2}+b^{2}\right )^{3}}\right )}{d}\) | \(457\) |
| default | \(-\frac {2 e^{4} \left (\frac {a \left (\frac {\left (\frac {1}{8} a^{4}+\frac {5}{4} a^{2} b^{2}+\frac {9}{8} b^{4}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}-\frac {a e \left (a^{4}-6 a^{2} b^{2}-7 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8 b}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (a^{4}+18 a^{2} b^{2}-15 b^{4}\right ) \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{8 b \sqrt {a e b}}\right )}{\left (a^{2}+b^{2}\right )^{3} e}+\frac {\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{e \left (a^{2}+b^{2}\right )^{3}}\right )}{d}\) | \(457\) |
-2/d*e^4*(a/(a^2+b^2)^3/e*(((1/8*a^4+5/4*a^2*b^2+9/8*b^4)*(e*cot(d*x+c))^( 3/2)-1/8*a*e*(a^4-6*a^2*b^2-7*b^4)/b*(e*cot(d*x+c))^(1/2))/(e*cot(d*x+c)*b +a*e)^2+1/8*(a^4+18*a^2*b^2-15*b^4)/b/(a*e*b)^(1/2)*arctan((e*cot(d*x+c))^ (1/2)*b/(a*e*b)^(1/2)))+1/e/(a^2+b^2)^3*(1/8*(-3*a^2*b*e+b^3*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*(-a^3+3*a*b^2)/(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. 4446 vs. \(2 (399) = 798\).
Time = 39.25 (sec) , antiderivative size = 8955, normalized size of antiderivative = 19.05 \[ \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx=\int { \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (b \cot \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Time = 19.08 (sec) , antiderivative size = 19256, normalized size of antiderivative = 40.97 \[ \int \frac {(e \cot (c+d x))^{5/2}}{(a+b \cot (c+d x))^3} \, dx=\text {Too large to display} \]
atan(((((10*a^16*b*d^2*e^18 - 2398*a^2*b^15*d^2*e^18 + 5238*a^4*b^13*d^2*e ^18 + 7386*a^6*b^11*d^2*e^18 - 8322*a^8*b^9*d^2*e^18 - 5498*a^10*b^7*d^2*e ^18 + 2946*a^12*b^5*d^2*e^18 + 382*a^14*b^3*d^2*e^18)/(b^17*d^5 + a^16*b*d ^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a^14*b^3*d^5) - (((832*a*b^22*d^4*e ^13 + 5952*a^3*b^20*d^4*e^13 + 17664*a^5*b^18*d^4*e^13 + 26880*a^7*b^16*d^ 4*e^13 + 18816*a^9*b^14*d^4*e^13 - 2688*a^11*b^12*d^4*e^13 - 16128*a^13*b^ 10*d^4*e^13 - 13056*a^15*b^8*d^4*e^13 - 4800*a^17*b^6*d^4*e^13 - 704*a^19* b^4*d^4*e^13)/(b^17*d^5 + a^16*b*d^5 + 8*a^2*b^15*d^5 + 28*a^4*b^13*d^5 + 56*a^6*b^11*d^5 + 70*a^8*b^9*d^5 + 56*a^10*b^7*d^5 + 28*a^12*b^5*d^5 + 8*a ^14*b^3*d^5) + ((e*cot(c + d*x))^(1/2)*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^ 2*d^2)))^(1/2)*(512*b^26*d^4*e^10 + 4608*a^2*b^24*d^4*e^10 + 17920*a^4*b^2 2*d^4*e^10 + 38400*a^6*b^20*d^4*e^10 + 46080*a^8*b^18*d^4*e^10 + 21504*a^1 0*b^16*d^4*e^10 - 21504*a^12*b^14*d^4*e^10 - 46080*a^14*b^12*d^4*e^10 - 38 400*a^16*b^10*d^4*e^10 - 17920*a^18*b^8*d^4*e^10 - 4608*a^20*b^6*d^4*e^10 - 512*a^22*b^4*d^4*e^10))/(b^17*d^4 + a^16*b*d^4 + 8*a^2*b^15*d^4 + 28*a^4 *b^13*d^4 + 56*a^6*b^11*d^4 + 70*a^8*b^9*d^4 + 56*a^10*b^7*d^4 + 28*a^12*b ^5*d^4 + 8*a^14*b^3*d^4))*(-(e^5*1i)/(4*(b^6*d^2 - a^6*d^2 + a*b^5*d^2*6i + a^5*b*d^2*6i - 15*a^2*b^4*d^2 - a^3*b^3*d^2*20i + 15*a^4*b^2*d^2)))^(...