Optimal. Leaf size=232 \[ -\frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} d^{3/2} f}+\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}} \]
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Rubi [A] time = 0.21, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {16, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} d^{3/2} f}+\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3474
Rule 3476
Rubi steps
\begin {align*} \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx &=d^2 \int \frac {1}{(d \cot (e+f x))^{7/2}} \, dx\\ &=\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\int \frac {1}{(d \cot (e+f x))^{3/2}} \, dx\\ &=\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}+\frac {\int \sqrt {d \cot (e+f x)} \, dx}{d^2}\\ &=\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{d f}\\ &=\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f}\\ &=\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}+\frac {\operatorname {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f}-\frac {\operatorname {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f}\\ &=\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\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} d^{3/2} f}-\frac {\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} d^{3/2} f}-\frac {\operatorname {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 d f}-\frac {\operatorname {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 d f}\\ &=\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\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} d^{3/2} f}+\frac {\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} d^{3/2} f}\\ &=\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 38, normalized size = 0.16 \[ \frac {2 d \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\cot ^2(e+f x)\right )}{5 f (d \cot (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 607, normalized size = 2.62 \[ \frac {20 \, \sqrt {2} d^{2} f \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \arctan \left (-\sqrt {2} d f \sqrt {\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} + \sqrt {2} d f \sqrt {\frac {\sqrt {2} d^{5} f^{3} \sqrt {\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} \sin \left (f x + e\right ) + d^{4} f^{2} \sqrt {\frac {1}{d^{6} f^{4}}} \sin \left (f x + e\right ) + d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} - 1\right ) \cos \left (f x + e\right )^{3} + 20 \, \sqrt {2} d^{2} f \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \arctan \left (-\sqrt {2} d f \sqrt {\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} + \sqrt {2} d f \sqrt {-\frac {\sqrt {2} d^{5} f^{3} \sqrt {\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} \sin \left (f x + e\right ) - d^{4} f^{2} \sqrt {\frac {1}{d^{6} f^{4}}} \sin \left (f x + e\right ) - d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} + 1\right ) \cos \left (f x + e\right )^{3} + 5 \, \sqrt {2} d^{2} f \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \cos \left (f x + e\right )^{3} \log \left (\frac {\sqrt {2} d^{5} f^{3} \sqrt {\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} \sin \left (f x + e\right ) + d^{4} f^{2} \sqrt {\frac {1}{d^{6} f^{4}}} \sin \left (f x + e\right ) + d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right ) - 5 \, \sqrt {2} d^{2} f \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \cos \left (f x + e\right )^{3} \log \left (-\frac {\sqrt {2} d^{5} f^{3} \sqrt {\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} \sin \left (f x + e\right ) - d^{4} f^{2} \sqrt {\frac {1}{d^{6} f^{4}}} \sin \left (f x + e\right ) - d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right ) - 8 \, {\left (6 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \sin \left (f x + e\right )}{20 \, d^{2} f \cos \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (f x + e\right )^{2}}{\left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.64, size = 728, normalized size = 3.14 \[ \frac {\left (-1+\cos \left (f x +e \right )\right ) \left (5 i \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \EllipticPi \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}-5 i \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \EllipticPi \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}-5 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \EllipticPi \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}+10 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \EllipticF \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}-5 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \EllipticPi \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}-12 \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}+12 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}+2 \cos \left (f x +e \right ) \sqrt {2}-2 \sqrt {2}\right ) \left (1+\cos \left (f x +e \right )\right )^{2} \sqrt {2}}{10 f \left (\frac {d \cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )^{\frac {3}{2}} \sin \left (f x +e \right )^{5} \cos \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 207, normalized size = 0.89 \[ -\frac {d^{3} {\left (\frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}\right )}}{d^{4}} - \frac {8 \, {\left (d^{2} - \frac {5 \, d^{2}}{\tan \left (f x + e\right )^{2}}\right )}}{d^{4} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {5}{2}}}\right )}}{20 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.58, size = 93, normalized size = 0.40 \[ \frac {\frac {2\,d}{5}-\frac {2\,d}{{\mathrm {tan}\left (e+f\,x\right )}^2}}{f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{d^{3/2}\,f}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{d^{3/2}\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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