Optimal. Leaf size=126 \[ \frac{8 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{65 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}} \]
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Rubi [A] time = 0.13199, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2627, 3769, 3771, 2639} \[ \frac{8 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{65 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{6}{13} \int \frac{\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx\\ &=-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{4}{39} \int \frac{1}{(b \sec (e+f x))^{5/2}} \, dx\\ &=-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{4 \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx}{65 b^2}\\ &=-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{4 \int \sqrt{\cos (e+f x)} \, dx}{65 b^2 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=\frac{8 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{65 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.4978, size = 83, normalized size = 0.66 \[ \frac{192 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )+(-6 \sin (e+f x)-55 \sin (3 (e+f x))+15 \sin (5 (e+f x))) \cos ^{\frac{3}{2}}(e+f x)}{1560 f \cos ^{\frac{5}{2}}(e+f x) (b \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.158, size = 343, normalized size = 2.7 \begin{align*}{\frac{2}{195\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}\sin \left ( fx+e \right ) } \left ( -15\, \left ( \cos \left ( fx+e \right ) \right ) ^{8}+12\,i\cos \left ( fx+e \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) -12\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \cos \left ( fx+e \right ) \sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}+40\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+12\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) -12\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}-29\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+12\,\cos \left ( fx+e \right ) \right ) \left ({\frac{b}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (\cos \left (f x + e\right )^{4} - 2 \, \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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