Optimal. Leaf size=98 \[ \frac {4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 98, 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 = {2627, 3769, 3771, 2639} \[ \frac {4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}+\frac {4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 2639
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}+\frac {2}{9} \int \frac {1}{(b \sec (e+f x))^{5/2}} \, dx\\ &=-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}+\frac {4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}+\frac {2 \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx}{15 b^2}\\ &=-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}+\frac {4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}+\frac {2 \int \sqrt {\cos (e+f x)} \, dx}{15 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ &=\frac {4 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}+\frac {4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 66, normalized size = 0.67 \[ \frac {-4 \sin (2 (e+f x))-10 \sin (4 (e+f x))+\frac {96 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\sqrt {\cos (e+f x)}}}{360 b^2 f \sqrt {b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt {b \sec \left (f x + e\right )}}{b^{3} \sec \left (f x + e\right )^{3}}, x\right ) \]
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 \[ \int \frac {\sin \left (f x + e\right )^{2}}{\left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 333, normalized size = 3.40 \[ \frac {\frac {4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )}{15}-\frac {4 i \cos \left (f x +e \right ) \sin \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )}{15}+\frac {2 \left (\cos ^{6}\left (f x +e \right )\right )}{9}+\frac {4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )}{15}-\frac {4 i \sin \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )}{15}-\frac {14 \left (\cos ^{4}\left (f x +e \right )\right )}{45}-\frac {8 \left (\cos ^{2}\left (f x +e \right )\right )}{45}+\frac {4 \cos \left (f x +e \right )}{15}}{f \cos \left (f x +e \right )^{3} \sin \left (f x +e \right ) \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {\sin \left (f x + e\right )^{2}}{\left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\sin \left (e+f\,x\right )}^2}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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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