Optimal. Leaf size=95 \[ -\frac{2 b \sin ^3(e+f x)}{7 f \sqrt{b \sec (e+f x)}}-\frac{4 b \sin (e+f x)}{7 f \sqrt{b \sec (e+f x)}}+\frac{8 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{7 f} \]
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Rubi [A] time = 0.0984107, 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, 2641} \[ -\frac{2 b \sin ^3(e+f x)}{7 f \sqrt{b \sec (e+f x)}}-\frac{4 b \sin (e+f x)}{7 f \sqrt{b \sec (e+f x)}}+\frac{8 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{7 f} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{b \sec (e+f x)} \sin ^4(e+f x) \, dx &=-\frac{2 b \sin ^3(e+f x)}{7 f \sqrt{b \sec (e+f x)}}+\frac{6}{7} \int \sqrt{b \sec (e+f x)} \sin ^2(e+f x) \, dx\\ &=-\frac{4 b \sin (e+f x)}{7 f \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^3(e+f x)}{7 f \sqrt{b \sec (e+f x)}}+\frac{4}{7} \int \sqrt{b \sec (e+f x)} \, dx\\ &=-\frac{4 b \sin (e+f x)}{7 f \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^3(e+f x)}{7 f \sqrt{b \sec (e+f x)}}+\frac{1}{7} \left (4 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx\\ &=\frac{8 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{7 f}-\frac{4 b \sin (e+f x)}{7 f \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^3(e+f x)}{7 f \sqrt{b \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.105073, size = 61, normalized size = 0.64 \[ \frac{\sqrt{b \sec (e+f x)} \left (-10 \sin (2 (e+f x))+\sin (4 (e+f x))+32 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{28 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.158, size = 143, normalized size = 1.5 \begin{align*}{\frac{ \left ( -2+2\,\cos \left ( fx+e \right ) \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{7\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}} \left ( -4\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) + \left ( \cos \left ( fx+e \right ) \right ) ^{4}- \left ( \cos \left ( fx+e \right ) \right ) ^{3}-3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+3\,\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 \sqrt{b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{4}\,{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 ({\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 )}, 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 \sqrt{b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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