Integrand size = 25, antiderivative size = 138 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=-\frac {2 \sqrt {2} a^3 \sqrt {e} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e} \]
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Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3647, 3711, 3609, 3613, 211} \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=-\frac {2 \sqrt {2} a^3 \sqrt {e} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{3/2}}{5 d e} \]
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Rule 211
Rule 3609
Rule 3613
Rule 3647
Rule 3711
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac {2 \int \sqrt {e \cot (c+d x)} \left (-a^3 e-5 a^3 e \cot (c+d x)-6 a^3 e \cot ^2(c+d x)\right ) \, dx}{5 e} \\ & = -\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac {2 \int \sqrt {e \cot (c+d x)} \left (5 a^3 e-5 a^3 e \cot (c+d x)\right ) \, dx}{5 e} \\ & = -\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac {2 \int \frac {5 a^3 e^2+5 a^3 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{5 e} \\ & = -\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}+\frac {\left (20 a^6 e^3\right ) \text {Subst}\left (\int \frac {1}{-50 a^6 e^4-e x^2} \, dx,x,\frac {5 a^3 e^2-5 a^3 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d} \\ & = -\frac {2 \sqrt {2} a^3 \sqrt {e} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(360\) vs. \(2(138)=276\).
Time = 3.43 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.61 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=-\frac {a^3 \sqrt {e \cot (c+d x)} (1+\cot (c+d x))^3 \left (-20 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)} \sin ^3(c+d x)+20 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)} \sin ^3(c+d x)+\sqrt [4]{\cot (c+d x)} \sin (c+d x) \left (4 \cos ^2(c+d x) \sqrt {\cot (c+d x)}+10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sin ^2(c+d x)-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sin ^2(c+d x)+40 \sqrt {\cot (c+d x)} \sin ^2(c+d x)+5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)-5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)+10 \sqrt {\cot (c+d x)} \sin (2 (c+d x))\right )\right )}{10 d \cot ^{\frac {3}{4}}(c+d x) (\cos (c+d x)+\sin (c+d x))^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(322\) vs. \(2(117)=234\).
Time = 0.05 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.34
method | result | size |
derivativedivides | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+2 \sqrt {e \cot \left (d x +c \right )}\, e^{2}-2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(323\) |
default | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+2 \sqrt {e \cot \left (d x +c \right )}\, e^{2}-2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(323\) |
parts | \(-\frac {a^{3} e \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 d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\sqrt {e \cot \left (d x +c \right )}\, e^{2}+\frac {e^{2} \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}\right )}{d \,e^{2}}+\frac {3 a^{3} \left (-2 \sqrt {e \cot \left (d x +c \right )}+\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{d}-\frac {6 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \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 )}{d e}\) | \(623\) |
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Time = 0.27 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.65 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=\left [\frac {5 \, \sqrt {2} {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt {-e} \log \left (\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) - 2 \, {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - 11 \, a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{5 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}, -\frac {2 \, {\left (5 \, \sqrt {2} {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt {e} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) + {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - 11 \, a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{5 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}\right ] \]
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\[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=a^{3} \left (\int \sqrt {e \cot {\left (c + d x \right )}}\, dx + \int 3 \sqrt {e \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )}\, dx + \int 3 \sqrt {e \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )}\, dx + \int \sqrt {e \cot {\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \]
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Exception generated. \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=\text {Exception raised: ValueError} \]
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\[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=\int { {\left (a \cot \left (d x + c\right ) + a\right )}^{3} \sqrt {e \cot \left (d x + c\right )} \,d x } \]
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Time = 13.52 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=\frac {\sqrt {2}\,a^3\,\sqrt {e}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{d}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{d\,e}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d\,e^2}-\frac {4\,a^3\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d} \]
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