Optimal. Leaf size=105 \[ \frac{2 a^2 (3 c+4 d) \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{3/2} c \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{f}+\frac{2 a d \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{3 f} \]
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Rubi [A] time = 0.150098, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3917, 3915, 3774, 203, 3792} \[ \frac{2 a^2 (3 c+4 d) \tan (e+f x)}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{3/2} c \tan ^{-1}\left (\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right )}{f}+\frac{2 a d \tan (e+f x) \sqrt{a \sec (e+f x)+a}}{3 f} \]
Antiderivative was successfully verified.
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Rule 3917
Rule 3915
Rule 3774
Rule 203
Rule 3792
Rubi steps
\begin{align*} \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x)) \, dx &=\frac{2 a d \sqrt{a+a \sec (e+f x)} \tan (e+f x)}{3 f}+\frac{2}{3} \int \sqrt{a+a \sec (e+f x)} \left (\frac{3 a c}{2}+\frac{1}{2} a (3 c+4 d) \sec (e+f x)\right ) \, dx\\ &=\frac{2 a d \sqrt{a+a \sec (e+f x)} \tan (e+f x)}{3 f}+(a c) \int \sqrt{a+a \sec (e+f x)} \, dx+\frac{1}{3} (a (3 c+4 d)) \int \sec (e+f x) \sqrt{a+a \sec (e+f x)} \, dx\\ &=\frac{2 a^2 (3 c+4 d) \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+\frac{2 a d \sqrt{a+a \sec (e+f x)} \tan (e+f x)}{3 f}-\frac{\left (2 a^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,-\frac{a \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 (3 c+4 d) \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+\frac{2 a d \sqrt{a+a \sec (e+f x)} \tan (e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.561776, size = 102, normalized size = 0.97 \[ \frac{a \sec \left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{a (\sec (e+f x)+1)} \left (2 \sin \left (\frac{1}{2} (e+f x)\right ) ((3 c+5 d) \cos (e+f x)+d)+3 \sqrt{2} c \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (e+f x)\right )\right ) \cos ^{\frac{3}{2}}(e+f x)\right )}{3 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.229, size = 237, normalized size = 2.3 \begin{align*}{\frac{a}{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\,\sqrt{2}\sin \left ( fx+e \right ) \cos \left ( fx+e \right ){\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}c+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}c\sin \left ( fx+e \right ) -12\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}c-20\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}d+12\,c\cos \left ( fx+e \right ) +16\,d\cos \left ( fx+e \right ) +4\,d \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.90324, size = 1347, normalized size = 12.83 \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 = 1.20415, size = 819, normalized size = 7.8 \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 (a d +{\left (3 \, a c + 5 \, a d\right )} \cos \left (f x + e\right )\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 (a d +{\left (3 \, a c + 5 \, a d\right )} \cos \left (f x + e\right )\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*} \int \left (a \left (\sec{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}} \left (c + d \sec{\left (e + f x \right )}\right )\, dx \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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