Optimal. Leaf size=94 \[ \frac {\sqrt {a \sin (e+f x)}}{a b f \sqrt {b \sec (e+f x)}}+\frac {F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{2 b^2 f \sqrt {a \sin (e+f x)}} \]
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Rubi [A]
time = 0.11, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2662, 2665,
2653, 2720} \begin {gather*} \frac {\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)}}{2 b^2 f \sqrt {a \sin (e+f x)}}+\frac {\sqrt {a \sin (e+f x)}}{a b f \sqrt {b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2653
Rule 2662
Rule 2665
Rule 2720
Rubi steps
\begin {align*} \int \frac {1}{(b \sec (e+f x))^{3/2} \sqrt {a \sin (e+f x)}} \, dx &=\frac {\sqrt {a \sin (e+f x)}}{a b f \sqrt {b \sec (e+f x)}}+\frac {\int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx}{2 b^2}\\ &=\frac {\sqrt {a \sin (e+f x)}}{a b f \sqrt {b \sec (e+f x)}}+\frac {\left (\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}{2 b^2}\\ &=\frac {\sqrt {a \sin (e+f x)}}{a b f \sqrt {b \sec (e+f x)}}+\frac {\left (\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}{2 b^2 \sqrt {a \sin (e+f x)}}\\ &=\frac {\sqrt {a \sin (e+f x)}}{a b f \sqrt {b \sec (e+f x)}}+\frac {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)}}{2 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 = 10.41, size = 84, normalized size = 0.89 \begin {gather*} -\frac {\cot (e+f x) \sqrt {b \sec (e+f x)} \left (-1+\cos (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 )}{2 b^2 f \sqrt {a \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 190, normalized size = 2.02
method | result | size |
default | \(-\frac {\left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sin \left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}+\cos \left (f x +e \right ) \sqrt {2}\right ) \sin \left (f x +e \right ) \sqrt {2}}{2 f \left (-1+\cos \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x +e \right )}}\) | \(190\) |
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 {1}{\sqrt {a\,\sin \left (e+f\,x\right )}\,{\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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