Optimal. Leaf size=32 \[ -\frac {2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2589} \[ -\frac {2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2589
Rubi steps
\begin {align*} \int \frac {(a \sin (e+f x))^{5/2}}{(b \tan (e+f x))^{3/2}} \, dx &=-\frac {2 b (a \sin (e+f x))^{5/2}}{5 f (b \tan (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 45, normalized size = 1.41 \[ -\frac {2 a^2 \cos ^2(e+f x) \sqrt {a \sin (e+f x)}}{5 b f \sqrt {b \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 55, normalized size = 1.72 \[ -\frac {2 \, \sqrt {a \sin \left (f x + e\right )} a^{2} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{3}}{5 \, b^{2} f \sin \left (f x + e\right )} \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 48, normalized size = 1.50 \[ -\frac {2 \left (a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \cos \left (f x +e \right )}{5 f \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \sin \left (f x +e \right )} \]
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 \[ \int \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {5}{2}}}{\left (b \tan \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.01, size = 81, normalized size = 2.53 \[ \frac {a^2\,\sqrt {a\,\sin \left (e+f\,x\right )}\,\left (2\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )\right )\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}}{10\,b^2\,f\,\left (\cos \left (2\,e+2\,f\,x\right )-1\right )} \]
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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