Optimal. Leaf size=123 \[ \frac {8 a^4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {a \sin (e+f x)}}{15 f \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}-\frac {4 a^2 b (a \sin (e+f x))^{5/2}}{15 f (b \tan (e+f x))^{3/2}}-\frac {2 b (a \sin (e+f x))^{9/2}}{9 f (b \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2598, 2601, 2639} \[ -\frac {4 a^2 b (a \sin (e+f x))^{5/2}}{15 f (b \tan (e+f x))^{3/2}}+\frac {8 a^4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {a \sin (e+f x)}}{15 f \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}-\frac {2 b (a \sin (e+f x))^{9/2}}{9 f (b \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2598
Rule 2601
Rule 2639
Rubi steps
\begin {align*} \int \frac {(a \sin (e+f x))^{9/2}}{\sqrt {b \tan (e+f x)}} \, dx &=-\frac {2 b (a \sin (e+f x))^{9/2}}{9 f (b \tan (e+f x))^{3/2}}+\frac {1}{3} \left (2 a^2\right ) \int \frac {(a \sin (e+f x))^{5/2}}{\sqrt {b \tan (e+f x)}} \, dx\\ &=-\frac {4 a^2 b (a \sin (e+f x))^{5/2}}{15 f (b \tan (e+f x))^{3/2}}-\frac {2 b (a \sin (e+f x))^{9/2}}{9 f (b \tan (e+f x))^{3/2}}+\frac {1}{15} \left (4 a^4\right ) \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \tan (e+f x)}} \, dx\\ &=-\frac {4 a^2 b (a \sin (e+f x))^{5/2}}{15 f (b \tan (e+f x))^{3/2}}-\frac {2 b (a \sin (e+f x))^{9/2}}{9 f (b \tan (e+f x))^{3/2}}+\frac {\left (4 a^4 \sqrt {a \sin (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{15 \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}\\ &=-\frac {4 a^2 b (a \sin (e+f x))^{5/2}}{15 f (b \tan (e+f x))^{3/2}}-\frac {2 b (a \sin (e+f x))^{9/2}}{9 f (b \tan (e+f x))^{3/2}}+\frac {8 a^4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {a \sin (e+f x)}}{15 f \sqrt {\cos (e+f x)} \sqrt {b \tan (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.52, size = 100, normalized size = 0.81 \[ \frac {a^4 \sin (2 (e+f x)) \sqrt {a \sin (e+f x)} \left (12 \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};\sin ^2(e+f x)\right )+\cos ^2(e+f x)^{3/4} (5 \cos (2 (e+f x))-17)\right )}{90 f \cos ^2(e+f x)^{3/4} \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} \cos \left (f x + e\right )^{4} - 2 \, a^{4} \cos \left (f x + e\right )^{2} + a^{4}\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )}}{b \tan \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}}}{\sqrt {b \tan \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.58, size = 349, normalized size = 2.84 \[ \frac {2 \left (12 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-12 i \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-5 \left (\cos ^{6}\left (f x +e \right )\right )+12 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-12 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+16 \left (\cos ^{4}\left (f x +e \right )\right )-23 \left (\cos ^{2}\left (f x +e \right )\right )+12 \cos \left (f x +e \right )\right ) \left (a \sin \left (f x +e \right )\right )^{\frac {9}{2}}}{45 f \cos \left (f x +e \right ) \sin \left (f x +e \right )^{5} \sqrt {\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}} \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}}}{\sqrt {b \tan \left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^{9/2}}{\sqrt {b\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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