Optimal. Leaf size=112 \[ -\frac {\sin (a+b x)}{6 b d \sqrt {d \tan (a+b x)}}+\frac {\sin ^3(a+b x)}{3 b d \sqrt {d \tan (a+b x)}}+\frac {\csc (a+b x) F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{12 b d^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2676, 2678,
2681, 2653, 2720} \begin {gather*} \frac {\sqrt {\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{12 b d^2}+\frac {\sin ^3(a+b x)}{3 b d \sqrt {d \tan (a+b x)}}-\frac {\sin (a+b x)}{6 b d \sqrt {d \tan (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2653
Rule 2676
Rule 2678
Rule 2681
Rule 2720
Rubi steps
\begin {align*} \int \frac {\sin ^3(a+b x)}{(d \tan (a+b x))^{3/2}} \, dx &=\frac {\sin ^3(a+b x)}{3 b d \sqrt {d \tan (a+b x)}}+\frac {\int \sin (a+b x) \sqrt {d \tan (a+b x)} \, dx}{6 d^2}\\ &=-\frac {\sin (a+b x)}{6 b d \sqrt {d \tan (a+b x)}}+\frac {\sin ^3(a+b x)}{3 b d \sqrt {d \tan (a+b x)}}+\frac {\int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx}{12 d^2}\\ &=-\frac {\sin (a+b x)}{6 b d \sqrt {d \tan (a+b x)}}+\frac {\sin ^3(a+b x)}{3 b d \sqrt {d \tan (a+b x)}}+\frac {\left (\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{12 d^2 \sqrt {\sin (a+b x)}}\\ &=-\frac {\sin (a+b x)}{6 b d \sqrt {d \tan (a+b x)}}+\frac {\sin ^3(a+b x)}{3 b d \sqrt {d \tan (a+b x)}}+\frac {\left (\csc (a+b x) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{12 d^2}\\ &=-\frac {\sin (a+b x)}{6 b d \sqrt {d \tan (a+b x)}}+\frac {\sin ^3(a+b x)}{3 b d \sqrt {d \tan (a+b x)}}+\frac {\csc (a+b x) F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{12 b d^2}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.43, size = 102, normalized size = 0.91 \begin {gather*} -\frac {\csc (a+b x) \left (\sqrt {\sec ^2(a+b x)} \sin (4 (a+b x))+4 \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right )\right |-1\right ) \sqrt {\tan (a+b x)}\right ) \sqrt {d \tan (a+b x)}}{24 b d^2 \sqrt {\sec ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 222, normalized size = 1.98
method | result | size |
default | \(-\frac {\left (-1+\cos \left (b x +a \right )\right ) \left (\sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+2 \left (\cos ^{4}\left (b x +a \right )\right ) \sqrt {2}-2 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}-\left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}+\cos \left (b x +a \right ) \sqrt {2}\right ) \left (\cos \left (b x +a \right )+1\right )^{2} \sqrt {2}}{12 b \sin \left (b x +a \right )^{2} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \cos \left (b x +a \right )^{2}}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\sin \left (a+b\,x\right )}^3}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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