Optimal. Leaf size=77 \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{f}-\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{f}+\frac{2 b \sqrt{b \sec (e+f x)}}{f} \]
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Rubi [A] time = 0.0533208, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2622, 321, 329, 212, 206, 203} \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{f}-\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{f}+\frac{2 b \sqrt{b \sec (e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 2622
Rule 321
Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \csc (e+f x) (b \sec (e+f x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^{3/2}}{-1+\frac{x^2}{b^2}} \, dx,x,b \sec (e+f x)\right )}{b f}\\ &=\frac{2 b \sqrt{b \sec (e+f x)}}{f}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (-1+\frac{x^2}{b^2}\right )} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac{2 b \sqrt{b \sec (e+f x)}}{f}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^4}{b^2}} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{f}\\ &=\frac{2 b \sqrt{b \sec (e+f x)}}{f}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{b-x^2} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{f}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{b+x^2} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{f}\\ &=-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{f}-\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{f}+\frac{2 b \sqrt{b \sec (e+f x)}}{f}\\ \end{align*}
Mathematica [A] time = 0.914073, size = 85, normalized size = 1.1 \[ \frac{(b \sec (e+f x))^{3/2} \left (4 \sqrt{\sec (e+f x)}+\log \left (1-\sqrt{\sec (e+f x)}\right )-\log \left (\sqrt{\sec (e+f x)}+1\right )-2 \tan ^{-1}\left (\sqrt{\sec (e+f x)}\right )\right )}{2 f \sec ^{\frac{3}{2}}(e+f x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.148, size = 235, normalized size = 3.1 \begin{align*}{\frac{ \left ( -1+\cos \left ( fx+e \right ) \right ) ^{3} \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{2\,f \left ( \sin \left ( fx+e \right ) \right ) ^{6}} \left ( 4\,\cos \left ( fx+e \right ) \sqrt{-{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}}}+\cos \left ( fx+e \right ) \ln \left ( -2\,{\frac{1}{ \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \left ( 2\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sqrt{-{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}}}- \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,\cos \left ( fx+e \right ) -2\,\sqrt{-{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}}}-1 \right ) } \right ) +\cos \left ( fx+e \right ) \arctan \left ({\frac{1}{2}{\frac{1}{\sqrt{-{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}}}}}} \right ) +4\,\sqrt{-{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}}} \right ) \left ({\frac{b}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{3}{2}}} \left ( -{\frac{\cos \left ( fx+e \right ) }{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.05017, size = 737, normalized size = 9.57 \begin{align*} \left [\frac{2 \, \sqrt{-b} b \arctan \left (\frac{\sqrt{-b} \sqrt{\frac{b}{\cos \left (f x + e\right )}}{\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) + \sqrt{-b} b \log \left (\frac{b \cos \left (f x + e\right )^{2} + 4 \,{\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt{-b} \sqrt{\frac{b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, b \sqrt{\frac{b}{\cos \left (f x + e\right )}}}{4 \, f}, \frac{2 \, b^{\frac{3}{2}} \arctan \left (\frac{\sqrt{\frac{b}{\cos \left (f x + e\right )}}{\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt{b}}\right ) + b^{\frac{3}{2}} \log \left (\frac{b \cos \left (f x + e\right )^{2} - 4 \,{\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt{b} \sqrt{\frac{b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, b \sqrt{\frac{b}{\cos \left (f x + e\right )}}}{4 \, f}\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 [A] time = 1.36877, size = 107, normalized size = 1.39 \begin{align*} \frac{b^{4}{\left (\frac{\arctan \left (\frac{\sqrt{b \cos \left (f x + e\right )}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{\arctan \left (\frac{\sqrt{b \cos \left (f x + e\right )}}{\sqrt{b}}\right )}{b^{\frac{5}{2}}} + \frac{2}{\sqrt{b \cos \left (f x + e\right )} b^{2}}\right )} \mathrm{sgn}\left (\cos \left (f x + e\right )\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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