Optimal. Leaf size=135 \[ -\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {a^2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{12 b^2 f \sqrt {a \sin (e+f x)}} \]
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Rubi [A]
time = 0.15, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2662, 2663,
2665, 2653, 2720} \begin {gather*} \frac {a^2 \sqrt {\sin (2 e+2 f x)} F\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {b \sec (e+f x)}}{12 b^2 f \sqrt {a \sin (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}-\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2653
Rule 2662
Rule 2663
Rule 2665
Rule 2720
Rubi steps
\begin {align*} \int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx &=\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2} \, dx}{6 b^2}\\ &=-\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {a^2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx}{12 b^2}\\ &=-\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}} \, dx}{12 b^2}\\ &=-\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (a^2 \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{12 b^2 \sqrt {a \sin (e+f x)}}\\ &=-\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {a^2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{12 b^2 f \sqrt {a \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.31, size = 87, normalized size = 0.64 \begin {gather*} \frac {a \sqrt {a \sin (e+f x)} \left (-2 \cos (2 (e+f x))+\csc ^2(e+f x) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{3/4}\right )}{12 b f \sqrt {b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 218, normalized size = 1.61
method | result | size |
default | \(-\frac {\left (2 \left (\cos ^{4}\left (f x +e \right )\right ) \sqrt {2}+\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (f x +e \right )-2 \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}-\left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}+\cos \left (f x +e \right ) \sqrt {2}\right ) \left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {2}}{12 f \left (-1+\cos \left (f x +e \right )\right ) \sin \left (f x +e \right ) \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \cos \left (f x +e \right )^{2}}\) | \(218\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {{\left (a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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