Optimal. Leaf size=99 \[ -\frac{a^2 \tan (e+f x) \log (1-\sec (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f (c-c \sec (e+f x))^{3/2}} \]
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Rubi [A] time = 0.275484, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {3954, 3952} \[ -\frac{a^2 \tan (e+f x) \log (1-\sec (e+f x))}{c f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{f (c-c \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3954
Rule 3952
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx &=-\frac{a \sqrt{a+a \sec (e+f x)} \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac{a \int \frac{\sec (e+f x) \sqrt{a+a \sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}} \, dx}{c}\\ &=-\frac{a \sqrt{a+a \sec (e+f x)} \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac{a^2 \log (1-\sec (e+f x)) \tan (e+f x)}{c f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.655738, size = 134, normalized size = 1.35 \[ -\frac{a \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} \left (-2 \log \left (1-e^{i (e+f x)}\right )+\log \left (1+e^{2 i (e+f x)}\right )+\left (2 \log \left (1-e^{i (e+f x)}\right )-\log \left (1+e^{2 i (e+f x)}\right )\right ) \cos (e+f x)+2\right )}{c f (\cos (e+f x)-1) \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.308, size = 251, normalized size = 2.5 \begin{align*} -{\frac{a \left ( -1+\cos \left ( fx+e \right ) \right ) }{f\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) } \left ( \cos \left ( fx+e \right ) \ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +\cos \left ( fx+e \right ) \ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -2\,\cos \left ( fx+e \right ) \ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) -\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) +2\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -\cos \left ( fx+e \right ) -1 \right ) \sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\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 [A] time = 1.59513, size = 165, normalized size = 1.67 \begin{align*} \frac{\frac{\sqrt{-a} a \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c^{\frac{3}{2}}} + \frac{\sqrt{-a} a \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c^{\frac{3}{2}}} - \frac{2 \, \sqrt{-a} a \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{\frac{3}{2}}} + \frac{\sqrt{-a} a{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{c^{\frac{3}{2}} \sin \left (f x + e\right )^{2}}}{f} \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{{\left (a \sec \left (f x + e\right )^{2} + a \sec \left (f x + e\right )\right )} \sqrt{a \sec \left (f x + e\right ) + a} \sqrt{-c \sec \left (f x + e\right ) + c}}{c^{2} \sec \left (f x + e\right )^{2} - 2 \, c^{2} \sec \left (f x + e\right ) + c^{2}}, 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(-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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