Integrand size = 25, antiderivative size = 325 \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx=\frac {2 a^{5/2} e^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{b^{3/2} \left (a^2+b^2\right ) d}-\frac {(a+b) e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a+b) e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}+\frac {(a-b) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a-b) e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d} \]
2*a^(5/2)*e^(5/2)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2)/a^(1/2)/e^(1/2))/b^( 3/2)/(a^2+b^2)/d-1/2*(a+b)*e^(5/2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e ^(1/2))/(a^2+b^2)/d*2^(1/2)+1/2*(a+b)*e^(5/2)*arctan(1+2^(1/2)*(e*cot(d*x+ c))^(1/2)/e^(1/2))/(a^2+b^2)/d*2^(1/2)+1/4*(a-b)*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)/d*2^(1/2)-1/4*(a-b)*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)/d*2^(1/2)-2*e^2*(e*cot(d*x+c))^(1/2)/b/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.82 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.88 \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx=\frac {(e \cot (c+d x))^{5/2} \left (8 a b^{3/2} \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )-3 \left (2 \sqrt {2} b^{5/2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} b^{5/2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )-8 a^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {\cot (c+d x)}}{\sqrt {a}}\right )+8 a^2 \sqrt {b} \sqrt {\cot (c+d x)}+8 b^{5/2} \sqrt {\cot (c+d x)}+\sqrt {2} b^{5/2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} b^{5/2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right )}{12 b^{3/2} \left (a^2+b^2\right ) d \cot ^{\frac {5}{2}}(c+d x)} \]
((e*Cot[c + d*x])^(5/2)*(8*a*b^(3/2)*Cot[c + d*x]^(3/2)*Hypergeometric2F1[ 3/4, 1, 7/4, -Cot[c + d*x]^2] - 3*(2*Sqrt[2]*b^(5/2)*ArcTan[1 - Sqrt[2]*Sq rt[Cot[c + d*x]]] - 2*Sqrt[2]*b^(5/2)*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x] ]] - 8*a^(5/2)*ArcTan[(Sqrt[b]*Sqrt[Cot[c + d*x]])/Sqrt[a]] + 8*a^2*Sqrt[b ]*Sqrt[Cot[c + d*x]] + 8*b^(5/2)*Sqrt[Cot[c + d*x]] + Sqrt[2]*b^(5/2)*Log[ 1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*b^(5/2)*Log[1 + S qrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])))/(12*b^(3/2)*(a^2 + b^2)*d*Cot [c + d*x]^(5/2))
Time = 1.25 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.90, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4049, 27, 3042, 4136, 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.
\(\displaystyle \int \frac {(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{a-b \tan \left (c+d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4049 |
\(\displaystyle -\frac {2 \int \frac {a \cot ^2(c+d x) e^3+a e^3+b \cot (c+d x) e^3}{2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {a \cot ^2(c+d x) e^3+a e^3+b \cot (c+d x) e^3}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {a \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^3+a e^3-b \tan \left (c+d x+\frac {\pi }{2}\right ) e^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 4136 |
\(\displaystyle -\frac {\frac {\int \frac {b^2 e^3+a b \cot (c+d x) e^3}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}+\frac {a^3 e^3 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\int \frac {b^2 e^3-a b e^3 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}+\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 4017 |
\(\displaystyle -\frac {\frac {2 \int -\frac {b e^3 (b e+a \cot (c+d x) e)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 \int \frac {b e^3 (b e+a \cot (c+d x) e)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 b e^3 \int \frac {b e+a \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 1482 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )-\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle -\frac {\frac {a^3 e^3 \int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {-\frac {2 a^3 e^2 \int \frac {1}{\frac {b \cot ^2(c+d x)}{e}+a}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {\frac {2 a^{5/2} e^{5/2} \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {b} d \left (a^2+b^2\right )}-\frac {2 b e^3 \left (\frac {1}{2} (a+b) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{b}-\frac {2 e^2 \sqrt {e \cot (c+d x)}}{b d}\) |
(-2*e^2*Sqrt[e*Cot[c + d*x]])/(b*d) - ((2*a^(5/2)*e^(5/2)*ArcTan[(Sqrt[b]* Cot[c + d*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[b]*(a^2 + b^2)*d) - (2*b*e^3*(((a + b)*(-(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)*(-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]*Sq rt[e*Cot[c + d*x]]]/(2*Sqrt[2]*Sqrt[e])))/2))/((a^2 + b^2)*d))/b
3.1.69.3.1 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^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((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[(((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 = 0.11 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(-\frac {2 e^{2} \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{b}-\frac {a^{3} e \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{b \left (a^{2}+b^{2}\right ) \sqrt {a e b}}-\frac {e \left (\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{a^{2}+b^{2}}\right )}{d}\) | \(347\) |
default | \(-\frac {2 e^{2} \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{b}-\frac {a^{3} e \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{b \left (a^{2}+b^{2}\right ) \sqrt {a e b}}-\frac {e \left (\frac {b \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{a^{2}+b^{2}}\right )}{d}\) | \(347\) |
-2/d*e^2*((e*cot(d*x+c))^(1/2)/b-1/b*a^3*e/(a^2+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)*(1/8*b/e*(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))+1/8*a/(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. 1602 vs. \(2 (262) = 524\).
Time = 0.37 (sec) , antiderivative size = 3267, normalized size of antiderivative = 10.05 \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx=\text {Too large to display} \]
[1/2*(2*a^2*sqrt(-a*e/b)*e^2*log((b*e*cos(2*d*x + 2*c) - a*e*sin(2*d*x + 2 *c) + 2*b*sqrt(-a*e/b)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sin (2*d*x + 2*c) + b*e)/(b*cos(2*d*x + 2*c) + a*sin(2*d*x + 2*c) + b)) - 4*(a ^2 + b^2)*e^2*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) - (a^2*b + b ^3)*d*sqrt(-(2*a*b*e^5 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^10/((a^8 + 4*a^6* b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(a^4 + 2*a^2*b^2 + b^4)*d^2)/((a^ 4 + 2*a^2*b^2 + b^4)*d^2))*log(-(a^2 - b^2)*e^7*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) + ((a^2*b - b^3)*d*e^5 - sqrt(-(a^4 - 2*a^2*b^2 + b^ 4)*e^10/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(a^5 + 2*a^ 3*b^2 + a*b^4)*d^3)*sqrt(-(2*a*b*e^5 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^10/ ((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(a^4 + 2*a^2*b^2 + b^4)*d^2)/((a^4 + 2*a^2*b^2 + b^4)*d^2))) + (a^2*b + b^3)*d*sqrt(-(2*a*b*e ^5 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^10/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4* a^2*b^6 + b^8)*d^4))*(a^4 + 2*a^2*b^2 + b^4)*d^2)/((a^4 + 2*a^2*b^2 + b^4) *d^2))*log(-(a^2 - b^2)*e^7*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c) ) - ((a^2*b - b^3)*d*e^5 - sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^10/((a^8 + 4*a^ 6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(a^5 + 2*a^3*b^2 + a*b^4)*d^3)* sqrt(-(2*a*b*e^5 + sqrt(-(a^4 - 2*a^2*b^2 + b^4)*e^10/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d^4))*(a^4 + 2*a^2*b^2 + b^4)*d^2)/((a^4 + 2* a^2*b^2 + b^4)*d^2))) - (a^2*b + b^3)*d*sqrt(-(2*a*b*e^5 - sqrt(-(a^4 -...
\[ \int \frac {(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx=\int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}{a + b \cot {\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx=\int { \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}}}{b \cot \left (d x + c\right ) + a} \,d x } \]
Time = 14.68 (sec) , antiderivative size = 5579, normalized size of antiderivative = 17.17 \[ \int \frac {(e \cot (c+d x))^{5/2}}{a+b \cot (c+d x)} \, dx=\text {Too large to display} \]
(atan(((((32*(e*cot(c + d*x))^(1/2)*(2*a^6*e^20 - b^6*e^20))/(b*d^4) + ((( 32*(12*a^6*b*d^2*e^18 + a^2*b^5*d^2*e^18 - 15*a^4*b^3*d^2*e^18))/(b*d^5) + (((32*(e*cot(c + d*x))^(1/2)*(16*a^7*b*d^2*e^15 - 14*a*b^7*d^2*e^15 + 4*a ^3*b^5*d^2*e^15 + 2*a^5*b^3*d^2*e^15))/(b*d^4) - (((32*(4*a*b^8*d^4*e^13 + 8*a^3*b^6*d^4*e^13 + 4*a^5*b^4*d^4*e^13))/(b*d^5) + (32*(e*cot(c + d*x))^ (1/2)*(-a^5*b^3*e^5)^(1/2)*(16*b^10*d^4*e^10 + 16*a^2*b^8*d^4*e^10 - 16*a^ 4*b^6*d^4*e^10 - 16*a^6*b^4*d^4*e^10))/(b^4*d^5*(a^2 + b^2)))*(-a^5*b^3*e^ 5)^(1/2))/(b^3*d*(a^2 + b^2)))*(-a^5*b^3*e^5)^(1/2))/(b^3*d*(a^2 + b^2)))* (-a^5*b^3*e^5)^(1/2))/(b^3*d*(a^2 + b^2)))*(-a^5*b^3*e^5)^(1/2)*1i)/(b^3*d *(a^2 + b^2)) + (((32*(e*cot(c + d*x))^(1/2)*(2*a^6*e^20 - b^6*e^20))/(b*d ^4) - (((32*(12*a^6*b*d^2*e^18 + a^2*b^5*d^2*e^18 - 15*a^4*b^3*d^2*e^18))/ (b*d^5) - (((32*(e*cot(c + d*x))^(1/2)*(16*a^7*b*d^2*e^15 - 14*a*b^7*d^2*e ^15 + 4*a^3*b^5*d^2*e^15 + 2*a^5*b^3*d^2*e^15))/(b*d^4) + (((32*(4*a*b^8*d ^4*e^13 + 8*a^3*b^6*d^4*e^13 + 4*a^5*b^4*d^4*e^13))/(b*d^5) - (32*(e*cot(c + d*x))^(1/2)*(-a^5*b^3*e^5)^(1/2)*(16*b^10*d^4*e^10 + 16*a^2*b^8*d^4*e^1 0 - 16*a^4*b^6*d^4*e^10 - 16*a^6*b^4*d^4*e^10))/(b^4*d^5*(a^2 + b^2)))*(-a ^5*b^3*e^5)^(1/2))/(b^3*d*(a^2 + b^2)))*(-a^5*b^3*e^5)^(1/2))/(b^3*d*(a^2 + b^2)))*(-a^5*b^3*e^5)^(1/2))/(b^3*d*(a^2 + b^2)))*(-a^5*b^3*e^5)^(1/2)*1 i)/(b^3*d*(a^2 + b^2)))/((64*(a^5*e^23 - a^3*b^2*e^23))/(b*d^5) + (((32*(e *cot(c + d*x))^(1/2)*(2*a^6*e^20 - b^6*e^20))/(b*d^4) + (((32*(12*a^6*b...