Optimal. Leaf size=179 \[ \frac{g^2 \cot (e+f x) \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}{a c f}-\frac{2 g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right )}{\sqrt{a} c f}+\frac{g^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \tan (e+f x)}{\sqrt{2} \sqrt{a \sec (e+f x)+a} \sqrt{g \sec (e+f x)}}\right )}{\sqrt{2} \sqrt{a} c f} \]
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Rubi [A] time = 0.357504, antiderivative size = 242, normalized size of antiderivative = 1.35, number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3964, 98, 157, 63, 217, 203, 93, 205} \[ \frac{2 g^{5/2} \tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{g \sec (e+f x)}}{\sqrt{g} \sqrt{c-c \sec (e+f x)}}\right )}{\sqrt{c} f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{g^{5/2} \tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{g \sec (e+f x)}}{\sqrt{g} \sqrt{c-c \sec (e+f x)}}\right )}{\sqrt{2} \sqrt{c} f \sqrt{a \sec (e+f x)+a} \sqrt{c-c \sec (e+f x)}}-\frac{g^2 \tan (e+f x) \sqrt{g \sec (e+f x)}}{f \sqrt{a \sec (e+f x)+a} (c-c \sec (e+f x))} \]
Antiderivative was successfully verified.
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Rule 3964
Rule 98
Rule 157
Rule 63
Rule 217
Rule 203
Rule 93
Rule 205
Rubi steps
\begin{align*} \int \frac{(g \sec (e+f x))^{5/2}}{\sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))} \, dx &=-\frac{(a c g \tan (e+f x)) \operatorname{Subst}\left (\int \frac{(g x)^{3/2}}{(a+a x) (c-c x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{g^2 \sqrt{g \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac{(g \tan (e+f x)) \operatorname{Subst}\left (\int \frac{\frac{1}{2} a c g^2+a c g^2 x}{\sqrt{g x} (a+a x) \sqrt{c-c x}} \, dx,x,\sec (e+f x)\right )}{c f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{g^2 \sqrt{g \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac{\left (g^3 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{g x} \sqrt{c-c x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}-\frac{\left (a g^3 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{g x} (a+a x) \sqrt{c-c x}} \, dx,x,\sec (e+f x)\right )}{2 f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{g^2 \sqrt{g \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac{\left (2 g^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{c x^2}{g}}} \, dx,x,\sqrt{g \sec (e+f x)}\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}-\frac{\left (a g^3 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{a g+2 a c x^2} \, dx,x,\frac{\sqrt{g \sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}}\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{g^2 \sqrt{g \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))}-\frac{g^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{g \sec (e+f x)}}{\sqrt{g} \sqrt{c-c \sec (e+f x)}}\right ) \tan (e+f x)}{\sqrt{2} \sqrt{c} f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}+\frac{\left (2 g^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{c x^2}{g}} \, dx,x,\frac{\sqrt{g \sec (e+f x)}}{\sqrt{c-c \sec (e+f x)}}\right )}{f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ &=-\frac{g^2 \sqrt{g \sec (e+f x)} \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)} (c-c \sec (e+f x))}+\frac{2 g^{5/2} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{g \sec (e+f x)}}{\sqrt{g} \sqrt{c-c \sec (e+f x)}}\right ) \tan (e+f x)}{\sqrt{c} f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}-\frac{g^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{g \sec (e+f x)}}{\sqrt{g} \sqrt{c-c \sec (e+f x)}}\right ) \tan (e+f x)}{\sqrt{2} \sqrt{c} f \sqrt{a+a \sec (e+f x)} \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.30084, size = 328, normalized size = 1.83 \[ -\frac{\sin ^3(e+f x) \sqrt{\sec (e+f x)+1} (g \sec (e+f x))^{5/2} \left (8 \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1}+\sqrt{\tan ^2(e+f x)} \left (16 \log (\sec (e+f x)+1)-16 \log \left (\sec ^{\frac{3}{2}}(e+f x)+\sqrt{\sec (e+f x)}+\sqrt{\tan ^2(e+f x)} \sqrt{\sec (e+f x)+1}\right )+\sqrt{2} \left (\log \left (-3 \sec ^2(e+f x)-2 \sec (e+f x)-2 \sqrt{2} \sqrt{\tan ^2(e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\sec (e+f x)}+1\right )-\log \left (-3 \sec ^2(e+f x)-2 \sec (e+f x)+2 \sqrt{2} \sqrt{\tan ^2(e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\sec (e+f x)}+1\right )\right )\right )\right )}{8 c f (\cos (e+f x)-1) (\cos (e+f x)+1)^2 (\sec (e+f x)-1) \sec ^{\frac{5}{2}}(e+f x) \sqrt{a (\sec (e+f x)+1)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.334, size = 294, normalized size = 1.6 \begin{align*} -{\frac{ \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{2\,fca \left ( \sin \left ( fx+e \right ) \right ) ^{6}} \left ( -\cos \left ( fx+e \right ){\it Arcsinh} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \sqrt{2}+2\,\cos \left ( fx+e \right ){\it Artanh} \left ( 1/2\,\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}} \left ( \cos \left ( fx+e \right ) +1-\sin \left ( fx+e \right ) \right ) \right ) -2\,\cos \left ( fx+e \right ){\it Artanh} \left ( 1/2\,\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}} \left ( \cos \left ( fx+e \right ) +1+\sin \left ( fx+e \right ) \right ) \right ) -2\,\sin \left ( fx+e \right ) \sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}}+{\it Arcsinh} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }} \right ) \sqrt{2}-2\,{\it Artanh} \left ( 1/2\,\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}} \left ( \cos \left ( fx+e \right ) +1-\sin \left ( fx+e \right ) \right ) \right ) +2\,{\it Artanh} \left ( 1/2\,\sqrt{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1}} \left ( \cos \left ( fx+e \right ) +1+\sin \left ( fx+e \right ) \right ) \right ) \right ) \left ({\frac{g}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.1389, size = 1890, normalized size = 10.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.913317, size = 1401, normalized size = 7.83 \begin{align*} \left [\frac{\sqrt{2} a g^{2} \sqrt{\frac{g}{a}} \log \left (\frac{2 \, \sqrt{2} \sqrt{\frac{g}{a}} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - g \cos \left (f x + e\right )^{2} + 2 \, g \cos \left (f x + e\right ) + 3 \, g}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 2 \, a g^{2} \sqrt{\frac{g}{a}} \log \left (\frac{g \cos \left (f x + e\right )^{3} + 4 \,{\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{g}{a}} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{g}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, g \cos \left (f x + e\right )^{2} + 8 \, g}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2}}\right ) \sin \left (f x + e\right ) + 4 \, g^{2} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{4 \, a c f \sin \left (f x + e\right )}, -\frac{\sqrt{2} a g^{2} \sqrt{-\frac{g}{a}} \arctan \left (\frac{\sqrt{2} \sqrt{-\frac{g}{a}} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{g \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \, a g^{2} \sqrt{-\frac{g}{a}} \arctan \left (\frac{2 \, \sqrt{-\frac{g}{a}} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{g \cos \left (f x + e\right )^{2} - g \cos \left (f x + e\right ) - 2 \, g}\right ) \sin \left (f x + e\right ) - 2 \, g^{2} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{g}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{2 \, a c f \sin \left (f x + e\right )}\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 -\frac{\left (g \sec \left (f x + e\right )\right )^{\frac{5}{2}}}{\sqrt{a \sec \left (f x + e\right ) + a}{\left (c \sec \left (f x + e\right ) - c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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