Optimal. Leaf size=101 \[ -\frac{2 a^3 c \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{f}-\frac{2 a^2 c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.123469, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {3904, 3887, 459, 321, 203} \[ -\frac{2 a^3 c \tan ^3(e+f x)}{3 f (a \sec (e+f x)+a)^{3/2}}+\frac{2 a^{3/2} c \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{f}-\frac{2 a^2 c \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3887
Rule 459
Rule 321
Rule 203
Rubi steps
\begin{align*} \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x)) \, dx &=-\left ((a c) \int \sqrt{a+a \sec (e+f x)} \tan ^2(e+f x) \, dx\right )\\ &=\frac{\left (2 a^3 c\right ) \operatorname{Subst}\left (\int \frac{x^2 \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^3 c \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}+\frac{\left (2 a^3 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a x^2} \, dx,x,-\frac{\tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{f}\\ &=-\frac{2 a^2 c \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{2 a^3 c \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}-\frac{\left (2 a^2 c\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 \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a+a \sec (e+f x)}}\right )}{f}-\frac{2 a^2 c \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{2 a^3 c \tan ^3(e+f x)}{3 f (a+a \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.669873, size = 96, normalized size = 0.95 \[ -\frac{2 a c \tan \left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \left ((2 \cos (e+f x)+1) \sqrt{\sec (e+f x)-1}-3 \cos (e+f x) \tan ^{-1}\left (\sqrt{\sec (e+f x)-1}\right )\right )}{3 f \sqrt{\sec (e+f x)-1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.222, size = 212, normalized size = 2.1 \begin{align*}{\frac{ac}{6\,f\cos \left ( fx+e \right ) \sin \left ( fx+e \right ) }\sqrt{{\frac{a \left ( 1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}} \left ( 3\,{\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 ) +3\,\sqrt{2}{\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}\sin \left ( fx+e \right ) +8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-4\,\cos \left ( fx+e \right ) -4 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.91235, size = 1347, normalized size = 13.34 \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 = 1.1727, size = 792, normalized size = 7.84 \begin{align*} \left [\frac{3 \,{\left (a c \cos \left (f x + e\right )^{2} + a c \cos \left (f x + e\right )\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 (2 \, a c \cos \left (f x + e\right ) + a c\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{3 \,{\left (f \cos \left (f x + e\right )^{2} + f \cos \left (f x + e\right )\right )}}, -\frac{2 \,{\left (3 \,{\left (a c \cos \left (f x + e\right )^{2} + a c \cos \left (f x + e\right )\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 (2 \, a c \cos \left (f x + e\right ) + a c\right )} \sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )\right )}}{3 \,{\left (f \cos \left (f x + e\right )^{2} + f \cos \left (f x + e\right )\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 \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 ^{2}{\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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