Optimal. Leaf size=95 \[ -\frac{2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}-\frac{4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}+\frac{8 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{15 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
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Rubi [A] time = 0.0998004, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2627, 3771, 2639} \[ -\frac{2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}-\frac{4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}+\frac{8 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{15 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sin ^4(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx &=-\frac{2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{2}{3} \int \frac{\sin ^2(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{4}{15} \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{4 \int \sqrt{\cos (e+f x)} \, dx}{15 \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 )}{15 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.33787, size = 63, normalized size = 0.66 \[ \frac{-68 \sin (2 (e+f x))+10 \sin (4 (e+f x))+\frac{192 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{\sqrt{\cos (e+f x)}}}{360 f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.15, size = 328, normalized size = 3.5 \begin{align*} -{\frac{2}{45\,f\sin \left ( fx+e \right ) b} \left ( 5\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+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}}}-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 ) \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}}}-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 ) -16\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+23\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-12\,\cos \left ( fx+e \right ) \right ) \sqrt{{\frac{b}{\cos \left ( fx+e \right ) }}}} \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}}{\sqrt{b \sec \left (f x + e\right )}}\,{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 \sec \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{4}{\left (e + f x \right )}}{\sqrt{b \sec{\left (e + f x \right )}}}\, dx \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}}{\sqrt{b \sec \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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