Optimal. Leaf size=100 \[ \frac {4 b^3 \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}-\frac {8 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}+\frac {2 b \sin ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f} \]
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Rubi [A] time = 0.10, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2624, 2627, 3771, 2641} \[ \frac {4 b^3 \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}-\frac {8 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}+\frac {2 b \sin ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f} \]
Antiderivative was successfully verified.
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Rule 2624
Rule 2627
Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int (b \sec (e+f x))^{5/2} \sin ^4(e+f x) \, dx &=\frac {2 b (b \sec (e+f x))^{3/2} \sin ^3(e+f x)}{3 f}-\left (2 b^2\right ) \int \sqrt {b \sec (e+f x)} \sin ^2(e+f x) \, dx\\ &=\frac {4 b^3 \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {2 b (b \sec (e+f x))^{3/2} \sin ^3(e+f x)}{3 f}-\frac {1}{3} \left (4 b^2\right ) \int \sqrt {b \sec (e+f x)} \, dx\\ &=\frac {4 b^3 \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {2 b (b \sec (e+f x))^{3/2} \sin ^3(e+f x)}{3 f}-\frac {1}{3} \left (4 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=-\frac {8 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{3 f}+\frac {4 b^3 \sin (e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {2 b (b \sec (e+f x))^{3/2} \sin ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 64, normalized size = 0.64 \[ -\frac {b^2 \sqrt {b \sec (e+f x)} \left (-\sin (2 (e+f x))-2 \tan (e+f x)+8 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right )}{3 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \cos \left (f x + e\right )^{4} - 2 \, b^{2} \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt {b \sec \left (f x + e\right )} \sec \left (f x + e\right )^{2}, x\right ) \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}} \sin \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 144, normalized size = 1.44 \[ \frac {2 \left (-1+\cos \left (f x +e \right )\right ) \left (4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )+\cos ^{3}\left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right )+\cos \left (f x +e \right )-1\right ) \cos \left (f x +e \right ) \left (\cos \left (f x +e \right )+1\right )^{2} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}{3 f \sin \left (f x +e \right )^{3}} \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}} \sin \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int {\sin \left (e+f\,x\right )}^4\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2} \,d x \]
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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