Optimal. Leaf size=123 \[ -\frac{2 b \sin ^5(e+f x)}{11 f \sqrt{b \sec (e+f x)}}-\frac{20 b \sin ^3(e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{40 b \sin (e+f x)}{77 f \sqrt{b \sec (e+f x)}}+\frac{80 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{77 f} \]
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Rubi [A] time = 0.143108, 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, 2641} \[ -\frac{2 b \sin ^5(e+f x)}{11 f \sqrt{b \sec (e+f x)}}-\frac{20 b \sin ^3(e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{40 b \sin (e+f x)}{77 f \sqrt{b \sec (e+f x)}}+\frac{80 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{77 f} \]
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 ^6(e+f x) \, dx &=-\frac{2 b \sin ^5(e+f x)}{11 f \sqrt{b \sec (e+f x)}}+\frac{10}{11} \int \sqrt{b \sec (e+f x)} \sin ^4(e+f x) \, dx\\ &=-\frac{20 b \sin ^3(e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^5(e+f x)}{11 f \sqrt{b \sec (e+f x)}}+\frac{60}{77} \int \sqrt{b \sec (e+f x)} \sin ^2(e+f x) \, dx\\ &=-\frac{40 b \sin (e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{20 b \sin ^3(e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^5(e+f x)}{11 f \sqrt{b \sec (e+f x)}}+\frac{40}{77} \int \sqrt{b \sec (e+f x)} \, dx\\ &=-\frac{40 b \sin (e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{20 b \sin ^3(e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^5(e+f x)}{11 f \sqrt{b \sec (e+f x)}}+\frac{1}{77} \left (40 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx\\ &=\frac{80 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{77 f}-\frac{40 b \sin (e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{20 b \sin ^3(e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^5(e+f x)}{11 f \sqrt{b \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.150176, size = 73, normalized size = 0.59 \[ \frac{\sqrt{b \sec (e+f x)} \left (-435 \sin (2 (e+f x))+68 \sin (4 (e+f x))-7 \sin (6 (e+f x))+1280 \sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{1232 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.253, size = 165, normalized size = 1.3 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( fx+e \right ) \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{77\,f \left ( \sin \left ( fx+e \right ) \right ) ^{3}} \left ( 7\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}-7\, \left ( \cos \left ( fx+e \right ) \right ) ^{5}+40\,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 ) -24\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+24\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}+37\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-37\,\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 )^{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 (\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 )}, 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 )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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