Optimal. Leaf size=123 \[ -\frac{b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac{7 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{20 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
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Rubi [A] time = 0.143419, 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 = {2625, 3771, 2639} \[ -\frac{b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac{7 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{20 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2625
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\csc ^6(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx &=-\frac{b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac{7}{10} \int \frac{\csc ^4(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}+\frac{7}{20} \int \frac{\csc ^2(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac{7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{7}{40} \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac{7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{7 \int \sqrt{\cos (e+f x)} \, dx}{40 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac{7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{7 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{20 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.141438, size = 86, normalized size = 0.7 \[ -\frac{\tan (e+f x) \left (12 \csc ^6(e+f x)+2 \csc ^4(e+f x)+7 \csc ^2(e+f x)+21 \sqrt{\cos (e+f x)} \csc (e+f x) E\left (\left .\frac{1}{2} (e+f x)\right |2\right )-21\right )}{60 f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.187, size = 918, normalized size = 7.5 \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}}{\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{\sqrt{b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{6}}{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{\csc \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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