Optimal. Leaf size=100 \[ \frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}} \]
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Rubi [A] time = 0.180779, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3910, 3911, 31} \[ \frac{a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{2 a^3 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3910
Rule 3911
Rule 31
Rubi steps
\begin{align*} \int \frac{(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx &=-\frac{2 a^3 \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}+\frac{a^2 \int \frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}} \, dx}{c^2}\\ &=-\frac{2 a^3 \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}+\frac{\left (a^3 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{-c+c x} \, dx,x,\cos (e+f x)\right )}{c f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{2 a^3 \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}+\frac{a^3 \log (1-\cos (e+f x)) \tan (e+f x)}{c^2 f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.36979, size = 155, normalized size = 1.55 \[ \frac{a^2 \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} \left (6 \log \left (1-e^{i (e+f x)}\right )+\left (-8 \log \left (1-e^{i (e+f x)}\right )+4 i f x-8\right ) \cos (e+f x)+\left (2 \log \left (1-e^{i (e+f x)}\right )-i f x\right ) \cos (2 (e+f x))-3 i f x+4\right )}{2 c^2 f (\cos (e+f x)-1)^2 \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.266, size = 229, normalized size = 2.3 \begin{align*} -{\frac{{a}^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{2\,f\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( 4\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}-2\,\ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}-8\,\cos \left ( fx+e \right ) \ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+4\,\cos \left ( fx+e \right ) \ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) +4\,\ln \left ( -{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) -2\,\cos \left ( fx+e \right ) -2\,\ln \left ( 2\, \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \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{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51431, size = 188, normalized size = 1.88 \begin{align*} \frac{\frac{4 \, \sqrt{-a} a^{2} \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{\frac{5}{2}}} - \frac{2 \, \sqrt{-a} a^{2} \log \left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}{c^{\frac{5}{2}}} - \frac{{\left (\sqrt{-a} a^{2} \sqrt{c} - \frac{2 \, \sqrt{-a} a^{2} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}{c^{3} \sin \left (f x + e\right )^{4}}}{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^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}\right )} \sqrt{a \sec \left (f x + e\right ) + a} \sqrt{-c \sec \left (f x + e\right ) + c}}{c^{3} \sec \left (f x + e\right )^{3} - 3 \, c^{3} \sec \left (f x + e\right )^{2} + 3 \, c^{3} \sec \left (f x + e\right ) - c^{3}}, 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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