Optimal. Leaf size=132 \[ -\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{12 b^2 f}-\frac{\csc ^5(e+f x)}{5 b f \sqrt{b \sec (e+f x)}}+\frac{\csc ^3(e+f x)}{30 b f \sqrt{b \sec (e+f x)}}+\frac{\csc (e+f x)}{12 b f \sqrt{b \sec (e+f x)}} \]
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Rubi [A] time = 0.14399, antiderivative size = 132, 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 = {2623, 2625, 3771, 2641} \[ -\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{12 b^2 f}-\frac{\csc ^5(e+f x)}{5 b f \sqrt{b \sec (e+f x)}}+\frac{\csc ^3(e+f x)}{30 b f \sqrt{b \sec (e+f x)}}+\frac{\csc (e+f x)}{12 b f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2623
Rule 2625
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{\csc ^6(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx &=-\frac{\csc ^5(e+f x)}{5 b f \sqrt{b \sec (e+f x)}}-\frac{\int \csc ^4(e+f x) \sqrt{b \sec (e+f x)} \, dx}{10 b^2}\\ &=\frac{\csc ^3(e+f x)}{30 b f \sqrt{b \sec (e+f x)}}-\frac{\csc ^5(e+f x)}{5 b f \sqrt{b \sec (e+f x)}}-\frac{\int \csc ^2(e+f x) \sqrt{b \sec (e+f x)} \, dx}{12 b^2}\\ &=\frac{\csc (e+f x)}{12 b f \sqrt{b \sec (e+f x)}}+\frac{\csc ^3(e+f x)}{30 b f \sqrt{b \sec (e+f x)}}-\frac{\csc ^5(e+f x)}{5 b f \sqrt{b \sec (e+f x)}}-\frac{\int \sqrt{b \sec (e+f x)} \, dx}{24 b^2}\\ &=\frac{\csc (e+f x)}{12 b f \sqrt{b \sec (e+f x)}}+\frac{\csc ^3(e+f x)}{30 b f \sqrt{b \sec (e+f x)}}-\frac{\csc ^5(e+f x)}{5 b f \sqrt{b \sec (e+f x)}}-\frac{\left (\sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}\right ) \int \frac{1}{\sqrt{\cos (e+f x)}} \, dx}{24 b^2}\\ &=\frac{\csc (e+f x)}{12 b f \sqrt{b \sec (e+f x)}}+\frac{\csc ^3(e+f x)}{30 b f \sqrt{b \sec (e+f x)}}-\frac{\csc ^5(e+f x)}{5 b f \sqrt{b \sec (e+f x)}}-\frac{\sqrt{\cos (e+f x)} F\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{b \sec (e+f x)}}{12 b^2 f}\\ \end{align*}
Mathematica [A] time = 0.311429, size = 74, normalized size = 0.56 \[ \frac{-12 \csc ^5(e+f x)+2 \csc ^3(e+f x)+5 \csc (e+f x)-\frac{5 F\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{\sqrt{\cos (e+f x)}}}{60 b f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.178, size = 493, normalized size = 3.7 \begin{align*} \text{result too large to display} \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{\csc \left (f x + e\right )^{6}}{\left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}}}\,{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{\sqrt{b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{6}}{b^{2} \sec \left (f x + e\right )^{2}}, 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{\csc \left (f x + e\right )^{6}}{\left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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