Optimal. Leaf size=214 \[ -\frac {d^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {d^{3/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {2 d^3}{3 f (d \cot (e+f x))^{3/2}}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]
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Rubi [A]
time = 0.13, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {16, 3555,
3557, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {d^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {d^{3/2} \text {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}+\frac {2 d^3}{3 f (d \cot (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rubi steps
\begin {align*} \int (d \cot (e+f x))^{3/2} \tan ^4(e+f x) \, dx &=d^4 \int \frac {1}{(d \cot (e+f x))^{5/2}} \, dx\\ &=\frac {2 d^3}{3 f (d \cot (e+f x))^{3/2}}-d^2 \int \frac {1}{\sqrt {d \cot (e+f x)}} \, dx\\ &=\frac {2 d^3}{3 f (d \cot (e+f x))^{3/2}}+\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 {2 d^3}{3 f (d \cot (e+f x))^{3/2}}+\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 {2 d^3}{3 f (d \cot (e+f x))^{3/2}}+\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 {2 d^3}{3 f (d \cot (e+f x))^{3/2}}-\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 {2 d^3}{3 f (d \cot (e+f x))^{3/2}}-\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} \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 {2 d^3}{3 f (d \cot (e+f x))^{3/2}}-\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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.05, size = 45, normalized size = 0.21 \begin {gather*} \frac {2 (d \cot (e+f x))^{3/2} \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\cot ^2(e+f x)\right ) \tan ^3(e+f x)}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.39, size = 548, normalized size = 2.56
method | result | size |
default | \(-\frac {\left (\cos \left (f x +e \right )-1\right ) \left (3 i \sqrt {\frac {\cos \left (f x +e \right )-1}{\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 {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-3 i \sqrt {\frac {\cos \left (f x +e \right )-1}{\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 {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+3 \sqrt {\frac {\cos \left (f x +e \right )-1}{\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 {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+3 \sqrt {\frac {\cos \left (f x +e \right )-1}{\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 {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-2 \cos \left (f x +e \right ) \sqrt {2}+2 \sqrt {2}\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \left (\frac {d \cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )^{\frac {3}{2}} \sqrt {2}}{6 f \sin \left (f x +e \right ) \cos \left (f x +e \right )^{3}}\) | \(548\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 197, normalized size = 0.92 \begin {gather*} \frac {d^{5} {\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 623 vs.
\(2 (170) = 340\).
time = 0.38, size = 623, normalized size = 2.91 \begin {gather*} -\frac {12 \, \sqrt {2} \left (\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \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 ) \cos \left (f x + e\right )^{2} + 12 \, \sqrt {2} \left (\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \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 ) \cos \left (f x + e\right )^{2} - 3 \, \sqrt {2} \left (\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \cos \left (f x + e\right )^{2} \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 ) + 3 \, \sqrt {2} \left (\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \cos \left (f x + e\right )^{2} \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 ) + 8 \, {\left (d \cos \left (f x + e\right )^{2} - d\right )} \sqrt {\frac {d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}}}{12 \, f \cos \left (f x + e\right )^{2}} \end {gather*}
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 \begin {gather*} \int \left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{4}{\left (e + f x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.52, size = 83, normalized size = 0.39 \begin {gather*} \frac {2\,d^3}{3\,f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{3/2}}-\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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