Optimal. Leaf size=211 \[ -\frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} d^{3/2} f}+\frac {2}{3 f (d \cot (e+f x))^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {16, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} d^{3/2} f}+\frac {2}{3 f (d \cot (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 3474
Rule 3476
Rubi steps
\begin {align*} \int \frac {\tan (e+f x)}{(d \cot (e+f x))^{3/2}} \, dx &=d \int \frac {1}{(d \cot (e+f x))^{5/2}} \, dx\\ &=\frac {2}{3 f (d \cot (e+f x))^{3/2}}-\frac {\int \frac {1}{\sqrt {d \cot (e+f x)}} \, dx}{d}\\ &=\frac {2}{3 f (d \cot (e+f x))^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )} \, dx,x,d \cot (e+f x)\right )}{f}\\ &=\frac {2}{3 f (d \cot (e+f x))^{3/2}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f}\\ &=\frac {2}{3 f (d \cot (e+f x))^{3/2}}+\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}{3 f (d \cot (e+f x))^{3/2}}-\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}{3 f (d \cot (e+f x))^{3/2}}-\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}{3 f (d \cot (e+f x))^{3/2}}-\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 = 0.08, size = 37, normalized size = 0.18 \[ \frac {2 \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\cot ^2(e+f x)\right )}{3 f (d \cot (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 603, normalized size = 2.86 \[ -\frac {12 \, \sqrt {2} d^{2} f \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \arctan \left (-\sqrt {2} d^{4} 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}} + \sqrt {2} d^{4} f^{3} \sqrt {\frac {d^{4} f^{2} \sqrt {\frac {1}{d^{6} f^{4}}} \sin \left (f x + e\right ) + \sqrt {2} d^{2} 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}} \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 {3}{4}} - 1\right ) \cos \left (f x + e\right )^{2} + 12 \, \sqrt {2} d^{2} f \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \arctan \left (-\sqrt {2} d^{4} 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}} + \sqrt {2} d^{4} f^{3} \sqrt {\frac {d^{4} f^{2} \sqrt {\frac {1}{d^{6} f^{4}}} \sin \left (f x + e\right ) - \sqrt {2} d^{2} 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}} \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 {3}{4}} + 1\right ) \cos \left (f x + e\right )^{2} - 3 \, \sqrt {2} d^{2} f \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \cos \left (f x + e\right )^{2} \log \left (\frac {d^{4} f^{2} \sqrt {\frac {1}{d^{6} f^{4}}} \sin \left (f x + e\right ) + \sqrt {2} d^{2} 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}} \sin \left (f x + e\right ) + d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right ) + 3 \, \sqrt {2} d^{2} f \left (\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \cos \left (f x + e\right )^{2} \log \left (\frac {d^{4} f^{2} \sqrt {\frac {1}{d^{6} f^{4}}} \sin \left (f x + e\right ) - \sqrt {2} d^{2} 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}} \sin \left (f x + e\right ) + d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right ) + 8 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}}}{12 \, d^{2} f \cos \left (f x + e\right )^{2}} \]
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 \[ \int \frac {\tan \left (f x + e\right )}{\left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.59, size = 540, normalized size = 2.56 \[ \frac {\left (-1+\cos \left (f x +e \right )\right ) \left (3 i \cos \left (f x +e \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 )}}\, \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 )-3 i \cos \left (f x +e \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 )}}\, \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 )-3 \cos \left (f x +e \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 )}}\, \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 )-3 \cos \left (f x +e \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 )}}\, \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 )+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}}{6 f \sin \left (f x +e \right )^{4} \left (\frac {d \cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 190, normalized size = 0.90 \[ \frac {d^{2} {\left (\frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}}\right )}}{d^{2}} + \frac {8}{d^{2} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {3}{2}}}\right )}}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 80, normalized size = 0.38 \[ \frac {2}{3\,f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{d^{3/2}\,f}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{d^{3/2}\,f} \]
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 \[ \int \frac {\tan {\left (e + f x \right )}}{\left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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