Optimal. Leaf size=123 \[ -\frac{2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}-\frac{20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac{8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}+\frac{16 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{39 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
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Rubi [A] time = 0.143405, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2627, 3771, 2639} \[ -\frac{2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}-\frac{20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac{8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}+\frac{16 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{39 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sin ^6(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx &=-\frac{2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}+\frac{10}{13} \int \frac{\sin ^4(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}+\frac{20}{39} \int \frac{\sin ^2(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}-\frac{20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}+\frac{8}{39} \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}-\frac{20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}+\frac{8 \int \sqrt{\cos (e+f x)} \, dx}{39 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=\frac{16 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{39 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}-\frac{20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.480945, size = 73, normalized size = 0.59 \[ \frac{-317 \sin (2 (e+f x))+76 \sin (4 (e+f x))-9 \sin (6 (e+f x))+\frac{768 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{\sqrt{\cos (e+f x)}}}{1872 f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.185, size = 338, normalized size = 2.8 \begin{align*} -{\frac{2}{117\,f\sin \left ( fx+e \right ) b} \left ( -9\, \left ( \cos \left ( fx+e \right ) \right ) ^{8}+24\,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}}}-24\,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 ) +37\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+24\,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}}}-24\,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 ) -59\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+55\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-24\,\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 )^{6}}{\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 )^{6} - 3 \, \cos \left (f x + e\right )^{4} + 3 \, \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(-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 )^{6}}{\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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