Optimal. Leaf size=128 \[ \frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}-\frac{16 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{2 b \sin ^5(e+f x) \sqrt{b \sec (e+f x)}}{f} \]
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Rubi [A] time = 0.148598, antiderivative size = 128, 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 = {2624, 2627, 3771, 2639} \[ \frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}-\frac{16 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{2 b \sin ^5(e+f x) \sqrt{b \sec (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 2624
Rule 2627
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx &=\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}-\left (10 b^2\right ) \int \frac{\sin ^4(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=\frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}-\frac{1}{3} \left (20 b^2\right ) \int \frac{\sin ^2(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}+\frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}-\frac{1}{3} \left (8 b^2\right ) \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx\\ &=\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}+\frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}-\frac{\left (8 b^2\right ) \int \sqrt{\cos (e+f x)} \, dx}{3 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{16 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}+\frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)} \sin ^5(e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.150728, size = 70, normalized size = 0.55 \[ -\frac{b \sqrt{b \sec (e+f x)} \left (-158 \sin (e+f x)-13 \sin (3 (e+f x))+\sin (5 (e+f x))+384 \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{72 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.233, size = 330, normalized size = 2.6 \begin{align*}{\frac{2\,\cos \left ( fx+e \right ) }{9\,f\sin \left ( fx+e \right ) } \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{6}-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 ) +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{\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 ) +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}}}-5\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+19\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-24\,\cos \left ( fx+e \right ) +9 \right ) \left ({\frac{b}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{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 \left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}} \sin \left (f x + e\right )^{6}\,{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 (b \cos \left (f x + e\right )^{6} - 3 \, b \cos \left (f x + e\right )^{4} + 3 \, b \cos \left (f x + e\right )^{2} - b\right )} \sqrt{b \sec \left (f x + e\right )} \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 \left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}} \sin \left (f x + e\right )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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