Optimal. Leaf size=93 \[ \frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}}+\frac {2 a^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{3 b^2 f \sqrt {a \sin (e+f x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2676, 2681,
2720} \begin {gather*} \frac {2 a^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{3 b^2 f \sqrt {a \sin (e+f x)}}+\frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2676
Rule 2681
Rule 2720
Rubi steps
\begin {align*} \int \frac {(a \sin (e+f x))^{3/2}}{(b \tan (e+f x))^{3/2}} \, dx &=\frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}}+\frac {a^2 \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx}{3 b^2}\\ &=\frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}}+\frac {\left (a^2 \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{3 b^2 \sqrt {a \sin (e+f x)}}\\ &=\frac {2 (a \sin (e+f x))^{3/2}}{3 b f \sqrt {b \tan (e+f x)}}+\frac {2 a^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \tan (e+f x)}}{3 b^2 f \sqrt {a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 80, normalized size = 0.86 \begin {gather*} \frac {2 a \sqrt {a \sin (e+f x)} \left (F\left (\left .\frac {1}{2} \text {ArcSin}(\sin (e+f x))\right |2\right )+\sqrt [4]{\cos ^2(e+f x)} \sin (e+f x)\right )}{3 b f \sqrt [4]{\cos ^2(e+f x)} \sqrt {b \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.33, size = 137, normalized size = 1.47
method | result | size |
default | \(-\frac {2 \sin \left (f x +e \right ) \left (i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right )+\cos \left (f x +e \right )\right ) \left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{3 f \left (\cos \left (f x +e \right )-1\right ) \cos \left (f x +e \right )^{2} \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}\) | \(137\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 112, normalized size = 1.20 \begin {gather*} \frac {2 \, \sqrt {a \sin \left (f x + e\right )} a \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) + \sqrt {2} \sqrt {-a b} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2} \sqrt {-a b} a {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}{3 \, b^{2} f} \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 (b\,\mathrm {tan}\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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