Integrand size = 21, antiderivative size = 192 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]
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Time = 0.18 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {16, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} f}-\frac {d^{3/2} \log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} f} \]
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Rule 16
Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rubi steps \begin{align*} \text {integral}& = d^2 \int \frac {1}{\sqrt {d \cot (e+f x)}} \, dx \\ & = -\frac {d^3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )} \, dx,x,d \cot (e+f x)\right )}{f} \\ & = -\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {1}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f} \\ & = -\frac {d^2 \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f}-\frac {d^2 \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f} \\ & = \frac {d^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {d^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {d^2 \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 f}-\frac {d^2 \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 f} \\ & = \frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {d^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {d^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f} \\ & = \frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.70 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\frac {d^2 \sqrt {\cot (e+f x)} \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (e+f x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (e+f x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (e+f x)}+\cot (e+f x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (e+f x)}+\cot (e+f x)\right )\right )}{2 \sqrt {2} f \sqrt {d \cot (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(447\) vs. \(2(145)=290\).
Time = 2.05 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.33
method | result | size |
default | \(\frac {\left (\ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )+2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right )-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )-\ln \left (\frac {2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-\cot \left (f x +e \right ) \cos \left (f x +e \right )+\sin \left (f x +e \right )+2 \cos \left (f x +e \right )+2 \cot \left (f x +e \right )-\csc \left (f x +e \right )-2}{\cos \left (f x +e \right )-1}\right )-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )\right ) d \sqrt {\cot \left (f x +e \right ) d}\, \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right ) \sqrt {2}}{4 f \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(448\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.85 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=-\frac {1}{2} \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} \log \left (d \sqrt {\frac {d}{\tan \left (f x + e\right )}} + \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) - \frac {1}{2} i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} \log \left (d \sqrt {\frac {d}{\tan \left (f x + e\right )}} + i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) + \frac {1}{2} i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} \log \left (d \sqrt {\frac {d}{\tan \left (f x + e\right )}} - i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) + \frac {1}{2} \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} \log \left (d \sqrt {\frac {d}{\tan \left (f x + e\right )}} - \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f\right ) \]
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\[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )}\, dx \]
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none
Time = 0.34 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.87 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=-\frac {d^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}}\right )}}{4 \, f} \]
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\[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}} \tan \left (f x + e\right )^{2} \,d x } \]
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Time = 3.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.32 \[ \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx=\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{f}+\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{f} \]
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