Optimal. Leaf size=109 \[ -\frac {8 a^4 \sqrt {a \sin (e+f x)}}{45 b f \sqrt {b \tan (e+f x)}}-\frac {2 a^2 (a \sin (e+f x))^{5/2}}{45 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt {b \tan (e+f x)}} \]
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Rubi [A] time = 0.15, antiderivative size = 109, 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 = {2596, 2598, 2589} \[ -\frac {2 a^2 (a \sin (e+f x))^{5/2}}{45 b f \sqrt {b \tan (e+f x)}}-\frac {8 a^4 \sqrt {a \sin (e+f x)}}{45 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2589
Rule 2596
Rule 2598
Rubi steps
\begin {align*} \int \frac {(a \sin (e+f x))^{9/2}}{(b \tan (e+f x))^{3/2}} \, dx &=\frac {2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt {b \tan (e+f x)}}+\frac {a^2 \int (a \sin (e+f x))^{5/2} \sqrt {b \tan (e+f x)} \, dx}{9 b^2}\\ &=-\frac {2 a^2 (a \sin (e+f x))^{5/2}}{45 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt {b \tan (e+f x)}}+\frac {\left (4 a^4\right ) \int \sqrt {a \sin (e+f x)} \sqrt {b \tan (e+f x)} \, dx}{45 b^2}\\ &=-\frac {8 a^4 \sqrt {a \sin (e+f x)}}{45 b f \sqrt {b \tan (e+f x)}}-\frac {2 a^2 (a \sin (e+f x))^{5/2}}{45 b f \sqrt {b \tan (e+f x)}}+\frac {2 (a \sin (e+f x))^{9/2}}{9 b f \sqrt {b \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 57, normalized size = 0.52 \[ \frac {a^4 \cos ^2(e+f x) (5 \cos (2 (e+f x))-13) \sqrt {a \sin (e+f x)}}{45 b f \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 71, normalized size = 0.65 \[ \frac {2 \, {\left (5 \, a^{4} \cos \left (f x + e\right )^{5} - 9 \, a^{4} \cos \left (f x + e\right )^{3}\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}}}{45 \, b^{2} f \sin \left (f x + e\right )} \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 60, normalized size = 0.55 \[ \frac {2 \left (a \sin \left (f x +e \right )\right )^{\frac {9}{2}} \left (5 \left (\cos ^{2}\left (f x +e \right )\right )-9\right ) \cos \left (f x +e \right )}{45 f \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \sin \left (f x +e \right )^{3}} \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.48, size = 94, normalized size = 0.86 \[ \frac {a^4\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (47\,\sin \left (2\,e+2\,f\,x\right )+16\,\sin \left (4\,e+4\,f\,x\right )-5\,\sin \left (6\,e+6\,f\,x\right )\right )}{360\,b^2\,f\,\left (\cos \left (2\,e+2\,f\,x\right )-1\right )} \]
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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