Integrand size = 25, antiderivative size = 114 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx=\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{3/2}}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{d e \sqrt {e \cot (c+d x)}} \]
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Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3646, 3711, 3613, 211} \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx=\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{3/2}}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{d e \sqrt {e \cot (c+d x)}} \]
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Rule 211
Rule 3613
Rule 3646
Rule 3711
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \int \frac {-2 a^3 e^2-a^3 e^2 \cot (c+d x)-a^3 e^2 \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{e^3} \\ & = -\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{d e \sqrt {e \cot (c+d x)}}-\frac {2 \int \frac {-a^3 e^2-a^3 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{e^3} \\ & = -\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{d e \sqrt {e \cot (c+d x)}}+\frac {\left (4 a^6 e\right ) \text {Subst}\left (\int \frac {1}{-2 a^6 e^4-e x^2} \, dx,x,\frac {-a^3 e^2+a^3 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d} \\ & = \frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{3/2}}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d e^2}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{d e \sqrt {e \cot (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(357\) vs. \(2(114)=228\).
Time = 3.12 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.13 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx=\frac {a^3 (1+\cot (c+d x))^3 \sin (c+d x) \left (-4 \cos ^2(c+d x)+4 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) (-\cot (c+d x))^{5/4} \sqrt [4]{\cot (c+d x)} \sin ^2(c+d x)+4 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)} \cot ^{\frac {5}{4}}(c+d x) \sin ^2(c+d x)+2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x) \sin ^2(c+d x)-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x) \sin ^2(c+d x)+\sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)-\sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)+2 \sin (2 (c+d x))\right )}{2 d (e \cot (c+d x))^{3/2} (\cos (c+d x)+\sin (c+d x))^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(99)=198\).
Time = 0.04 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.68
method | result | size |
derivativedivides | \(-\frac {2 a^{3} \left (\sqrt {e \cot \left (d x +c \right )}+2 e \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )-\frac {e}{\sqrt {e \cot \left (d x +c \right )}}\right )}{d \,e^{2}}\) | \(305\) |
default | \(-\frac {2 a^{3} \left (\sqrt {e \cot \left (d x +c \right )}+2 e \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )-\frac {e}{\sqrt {e \cot \left (d x +c \right )}}\right )}{d \,e^{2}}\) | \(305\) |
parts | \(-\frac {2 a^{3} e \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}-\frac {2 a^{3} \left (\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 )}{8}\right )}{d \,e^{2}}-\frac {3 a^{3} \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 d \,e^{2}}-\frac {3 a^{3} \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 e \left (e^{2}\right )^{\frac {1}{4}}}\) | \(594\) |
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Time = 0.28 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.26 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx=\left [\frac {\sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \sqrt {-\frac {1}{e}} \log \left (-\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt {-\frac {1}{e}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) - 2 \, {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3} \sin \left (2 \, d x + 2 \, c\right ) + a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}}, \frac {2 \, {\left (\frac {\sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right )}{\sqrt {e}} - {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3} \sin \left (2 \, d x + 2 \, c\right ) + a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}}\right ] \]
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\[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx=a^{3} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {3 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {3 \cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
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Exception generated. \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx=\int { \frac {{\left (a \cot \left (d x + c\right ) + a\right )}^{3}}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Time = 12.45 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx=\frac {2\,a^3}{d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}-\frac {2\,a^3\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d\,e^2}-\frac {\sqrt {2}\,a^3\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{d\,e^{3/2}} \]
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