Optimal. Leaf size=124 \[ -\frac{7 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \csc ^3(e+f x) \sqrt{b \sec (e+f x)}}{3 f}-\frac{7 b \csc (e+f x) \sqrt{b \sec (e+f x)}}{6 f}+\frac{7 b \sin (e+f x) \sqrt{b \sec (e+f x)}}{2 f} \]
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Rubi [A] time = 0.128983, antiderivative size = 124, 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 = {2625, 3768, 3771, 2639} \[ -\frac{7 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \csc ^3(e+f x) \sqrt{b \sec (e+f x)}}{3 f}-\frac{7 b \csc (e+f x) \sqrt{b \sec (e+f x)}}{6 f}+\frac{7 b \sin (e+f x) \sqrt{b \sec (e+f x)}}{2 f} \]
Antiderivative was successfully verified.
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Rule 2625
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx &=-\frac{b \csc ^3(e+f x) \sqrt{b \sec (e+f x)}}{3 f}+\frac{7}{6} \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx\\ &=-\frac{7 b \csc (e+f x) \sqrt{b \sec (e+f x)}}{6 f}-\frac{b \csc ^3(e+f x) \sqrt{b \sec (e+f x)}}{3 f}+\frac{7}{4} \int (b \sec (e+f x))^{3/2} \, dx\\ &=-\frac{7 b \csc (e+f x) \sqrt{b \sec (e+f x)}}{6 f}-\frac{b \csc ^3(e+f x) \sqrt{b \sec (e+f x)}}{3 f}+\frac{7 b \sqrt{b \sec (e+f x)} \sin (e+f x)}{2 f}-\frac{1}{4} \left (7 b^2\right ) \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{7 b \csc (e+f x) \sqrt{b \sec (e+f x)}}{6 f}-\frac{b \csc ^3(e+f x) \sqrt{b \sec (e+f x)}}{3 f}+\frac{7 b \sqrt{b \sec (e+f x)} \sin (e+f x)}{2 f}-\frac{\left (7 b^2\right ) \int \sqrt{\cos (e+f x)} \, dx}{4 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{7 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{7 b \csc (e+f x) \sqrt{b \sec (e+f x)}}{6 f}-\frac{b \csc ^3(e+f x) \sqrt{b \sec (e+f x)}}{3 f}+\frac{7 b \sqrt{b \sec (e+f x)} \sin (e+f x)}{2 f}\\ \end{align*}
Mathematica [A] time = 0.206314, size = 77, normalized size = 0.62 \[ -\frac{b \sin (e+f x) \sqrt{b \sec (e+f x)} \left (2 \csc ^4(e+f x)+7 \csc ^2(e+f x)+21 \sqrt{\cos (e+f x)} \csc (e+f x) E\left (\left .\frac{1}{2} (e+f x)\right |2\right )-21\right )}{6 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.158, size = 622, normalized size = 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 \left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}} \csc \left (f x + e\right )^{4}\,{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 (\sqrt{b \sec \left (f x + e\right )} b \csc \left (f x + e\right )^{4} \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}} \csc \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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