Optimal. Leaf size=142 \[ \frac{2 a^4 c^2 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^3 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c^2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{f}-\frac{2 a^2 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.170043, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3904, 3887, 459, 302, 203} \[ \frac{2 a^4 c^2 \tan ^5(e+f x)}{5 f (a \sec (e+f x)+a)^{5/2}}+\frac{2 a^3 c^2 \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c^2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{f}-\frac{2 a^2 c^2 \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3887
Rule 459
Rule 302
Rule 203
Rubi steps
\begin{align*} \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \frac{\tan ^4(e+f x)}{\sqrt{a+a \sec (e+f x)}} \, dx\\ &=-\frac{\left (2 a^4 c^2\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (2+a x^2\right )}{1+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{f}\\ &=\frac{2 a^4 c^2 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac{\left (2 a^4 c^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{f}\\ &=\frac{2 a^4 c^2 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac{\left (2 a^4 c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^2}+\frac{x^2}{a}+\frac{1}{a^2 \left (1+a x^2\right )}\right ) \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{f}\\ &=-\frac{2 a^2 c^2 \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^3 c^2 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}+\frac{2 a^4 c^2 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}-\frac{\left (2 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{f}\\ &=\frac{2 a^{3/2} c^2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{f}-\frac{2 a^2 c^2 \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^3 c^2 \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}+\frac{2 a^4 c^2 \tan ^5(e+f x)}{5 f (a+a \sec (e+f x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.84158, size = 112, normalized size = 0.79 \[ -\frac{a c^2 \tan \left (\frac{1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt{a (\sec (e+f x)+1)} \left ((2 \cos (e+f x)+17 \cos (2 (e+f x))+11) \sqrt{\sec (e+f x)-1}-30 \cos ^2(e+f x) \tan ^{-1}\left (\sqrt{\sec (e+f x)-1}\right )\right )}{15 f \sqrt{\sec (e+f x)-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.247, size = 232, normalized size = 1.6 \begin{align*}{\frac{{c}^{2}a}{30\,f \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) }\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( 15\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}} \right ) \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{3/2}\sqrt{2}\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+15\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }\sqrt{-2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}} \right ) \left ( -2\,{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }} \right ) ^{3/2}\sqrt{2}\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +68\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-64\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-16\,\cos \left ( fx+e \right ) +12 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 1.19576, size = 900, normalized size = 6.34 \begin{align*} \left [\frac{15 \,{\left (a c^{2} \cos \left (f x + e\right )^{3} + a c^{2} \cos \left (f x + e\right )^{2}\right )} \sqrt{-a} \log \left (\frac{2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) - 2 \,{\left (17 \, a c^{2} \cos \left (f x + e\right )^{2} + a c^{2} \cos \left (f x + e\right ) - 3 \, a c^{2}\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{15 \,{\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}, -\frac{2 \,{\left (15 \,{\left (a c^{2} \cos \left (f x + e\right )^{3} + a c^{2} \cos \left (f x + e\right )^{2}\right )} \sqrt{a} \arctan \left (\frac{\sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt{a} \sin \left (f x + e\right )}\right ) +{\left (17 \, a c^{2} \cos \left (f x + e\right )^{2} + a c^{2} \cos \left (f x + e\right ) - 3 \, a c^{2}\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )\right )}}{15 \,{\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}\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*} c^{2} \left (\int a \sqrt{a \sec{\left (e + f x \right )} + a}\, dx + \int - a \sqrt{a \sec{\left (e + f x \right )} + a} \sec{\left (e + f x \right )}\, dx + \int - a \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )}\, dx + \int a \sqrt{a \sec{\left (e + f x \right )} + a} \sec ^{3}{\left (e + f x \right )}\, dx\right ) \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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