Optimal. Leaf size=63 \[ -\frac{b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}-\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
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Rubi [A] time = 0.0580531, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2625, 3771, 2639} \[ -\frac{b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}-\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2625
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\csc ^2(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx &=-\frac{b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}-\frac{1}{2} \int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx\\ &=-\frac{b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}-\frac{\int \sqrt{\cos (e+f x)} \, dx}{2 \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ &=-\frac{b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}-\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.151907, size = 48, normalized size = 0.76 \[ \frac{-\cot (e+f x)-\frac{E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{\sqrt{\cos (e+f x)}}}{f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.144, size = 316, normalized size = 5. \begin{align*} -{\frac{ \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{bf \left ( \sin \left ( fx+e \right ) \right ) ^{5}} \left ( 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 ) -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}}}+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 ) -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}}}+\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 \frac{\csc \left (f x + e\right )^{2}}{\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 )^{2}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (e + f x \right )}}{\sqrt{b \sec{\left (e + f x \right )}}}\, dx \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 )^{2}}{\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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