Optimal. Leaf size=192 \[ \frac{d^{3/2} \log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} f}-\frac{d^{3/2} \log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} f} \]
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Rubi [A] time = 0.146697, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {16, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{d^{3/2} \log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} f}-\frac{d^{3/2} \log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} f}+\frac{d^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} f} \]
Antiderivative was successfully verified.
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Rule 16
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int (d \cot (e+f x))^{3/2} \tan ^2(e+f x) \, dx &=d^2 \int \frac{1}{\sqrt{d \cot (e+f x)}} \, dx\\ &=-\frac{d^3 \operatorname{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 ) \operatorname{Subst}\left (\int \frac{1}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}\\ &=-\frac{d^2 \operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}-\frac{d^2 \operatorname{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} \operatorname{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} \operatorname{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 \operatorname{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 \operatorname{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} \operatorname{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} \operatorname{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} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} f}-\frac{d^{3/2} \tan ^{-1}\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*}
Mathematica [A] time = 0.0235415, size = 134, normalized size = 0.7 \[ \frac{d^2 \sqrt{\cot (e+f x)} \left (\log \left (\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right )-\log \left (\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (e+f x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (e+f x)}+1\right )\right )}{2 \sqrt{2} f \sqrt{d \cot (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.152, size = 287, normalized size = 1.5 \begin{align*} -{\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( \cos \left ( fx+e \right ) -1 \right ) }{2\,f\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( i{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -i{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) \right ) \sqrt{{\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) -1+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}} \left ({\frac{d\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25654, size = 1258, normalized size = 6.55 \begin{align*} \sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{1}{4}} \arctan \left (-\frac{d^{6} + \sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{3}{4}} d f^{3} \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} - \sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{3}{4}} f^{3} \sqrt{\frac{\sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{1}{4}} d f \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \sin \left (f x + e\right ) + d^{3} \cos \left (f x + e\right ) + \sqrt{\frac{d^{6}}{f^{4}}} f^{2} \sin \left (f x + e\right )}{\sin \left (f x + e\right )}}}{d^{6}}\right ) + \sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{d^{6} - \sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{3}{4}} d f^{3} \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} + \sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{3}{4}} f^{3} \sqrt{-\frac{\sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{1}{4}} d f \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \sin \left (f x + e\right ) - d^{3} \cos \left (f x + e\right ) - \sqrt{\frac{d^{6}}{f^{4}}} f^{2} \sin \left (f x + e\right )}{\sin \left (f x + e\right )}}}{d^{6}}\right ) - \frac{1}{4} \, \sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{1}{4}} \log \left (\frac{\sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{1}{4}} d f \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \sin \left (f x + e\right ) + d^{3} \cos \left (f x + e\right ) + \sqrt{\frac{d^{6}}{f^{4}}} f^{2} \sin \left (f x + e\right )}{\sin \left (f x + e\right )}\right ) + \frac{1}{4} \, \sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{\sqrt{2} \left (\frac{d^{6}}{f^{4}}\right )^{\frac{1}{4}} d f \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \sin \left (f x + e\right ) - d^{3} \cos \left (f x + e\right ) - \sqrt{\frac{d^{6}}{f^{4}}} f^{2} \sin \left (f x + e\right )}{\sin \left (f x + e\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cot \left (f x + e\right )\right )^{\frac{3}{2}} \tan \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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