Optimal. Leaf size=90 \[ -\frac{3 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \csc (e+f x) \sqrt{b \sec (e+f x)}}{f}+\frac{3 b \sin (e+f x) \sqrt{b \sec (e+f x)}}{f} \]
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Rubi [A] time = 0.0823653, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2625, 3768, 3771, 2639} \[ -\frac{3 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \csc (e+f x) \sqrt{b \sec (e+f x)}}{f}+\frac{3 b \sin (e+f x) \sqrt{b \sec (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 2625
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx &=-\frac{b \csc (e+f x) \sqrt{b \sec (e+f x)}}{f}+\frac{3}{2} \int (b \sec (e+f x))^{3/2} \, dx\\ &=-\frac{b \csc (e+f x) \sqrt{b \sec (e+f x)}}{f}+\frac{3 b \sqrt{b \sec (e+f x)} \sin (e+f x)}{f}-\frac{1}{2} \left (3 b^2\right ) \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{b \csc (e+f x) \sqrt{b \sec (e+f x)}}{f}+\frac{3 b \sqrt{b \sec (e+f x)} \sin (e+f x)}{f}-\frac{\left (3 b^2\right ) \int \sqrt{\cos (e+f x)} \, dx}{2 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{3 b^2 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \csc (e+f x) \sqrt{b \sec (e+f x)}}{f}+\frac{3 b \sqrt{b \sec (e+f x)} \sin (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.147256, size = 57, normalized size = 0.63 \[ \frac{b \sqrt{b \sec (e+f x)} \left (3 \sin (e+f x)-\csc (e+f x)-3 \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.146, size = 322, normalized size = 3.6 \begin{align*} -{\frac{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}\cos \left ( fx+e \right ) }{f \left ( \sin \left ( fx+e \right ) \right ) ^{5}} \left ( 3\,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 ) -3\,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}}}+3\,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 ) -3\,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}}}+3\,\cos \left ( fx+e \right ) -2 \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}} \csc \left (f x + e\right )^{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 (\sqrt{b \sec \left (f x + e\right )} b \csc \left (f x + e\right )^{2} \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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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