Integrand size = 25, antiderivative size = 87 \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx=-\frac {e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d}+\frac {e^{3/2} \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d} \]
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Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3654, 3613, 214, 3715, 65, 211} \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx=\frac {e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d}-\frac {e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d} \]
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Rule 65
Rule 211
Rule 214
Rule 3613
Rule 3654
Rule 3715
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {-a e^2+a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{2 a^2}+\frac {1}{2} e^2 \int \frac {1+\cot ^2(c+d x)}{\sqrt {e \cot (c+d x)} (a+a \cot (c+d x))} \, dx \\ & = \frac {e^2 \text {Subst}\left (\int \frac {1}{\sqrt {-e x} (a-a x)} \, dx,x,-\cot (c+d x)\right )}{2 d}-\frac {e^4 \text {Subst}\left (\int \frac {1}{2 a^2 e^4-e x^2} \, dx,x,\frac {-a e^2-a e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d} \\ & = \frac {e^{3/2} \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d}-\frac {e \text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{e}} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d} \\ & = -\frac {e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{a d}+\frac {e^{3/2} \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{\sqrt {2} a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(280\) vs. \(2(87)=174\).
Time = 0.55 (sec) , antiderivative size = 280, normalized size of antiderivative = 3.22 \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx=-\frac {8 e^{3/2} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-4 \left (-e^2\right )^{3/4} \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )+2 \sqrt {2} e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )-2 \sqrt {2} e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )+4 \left (-e^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt [4]{-e^2}}\right )+\sqrt {2} e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )-\sqrt {2} e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{8 a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(71)=142\).
Time = 0.05 (sec) , antiderivative size = 298, normalized size of antiderivative = 3.43
method | result | size |
derivativedivides | \(-\frac {2 e^{2} \left (-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}\right )}{d a}\) | \(298\) |
default | \(-\frac {2 e^{2} \left (-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 \left (e^{2}\right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{2 \sqrt {e}}\right )}{d a}\) | \(298\) |
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Time = 0.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.83 \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx=\left [-\frac {\sqrt {2} \sqrt {-e} e \arctan \left (\frac {{\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) + \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) + \sqrt {2}\right )} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) - \sqrt {-e} e \log \left (\frac {e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) - 2 \, \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right )}{2 \, a d}, \frac {\sqrt {2} e^{\frac {3}{2}} \log \left (-{\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) - \sqrt {2}\right )} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) - 4 \, e^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}}\right )}{4 \, a d}\right ] \]
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\[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx=\frac {\int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\cot {\left (c + d x \right )} + 1}\, dx}{a} \]
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Exception generated. \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx=\int { \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}}{a \cot \left (d x + c\right ) + a} \,d x } \]
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Time = 12.78 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.91 \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \cot (c+d x)} \, dx=\frac {\sqrt {2}\,e^{3/2}\,\mathrm {atanh}\left (\frac {12\,\sqrt {2}\,e^{25/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{12\,e^{13}\,\mathrm {cot}\left (c+d\,x\right )+12\,e^{13}}\right )}{2\,a\,d}-\frac {e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{a\,d} \]
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