Optimal. Leaf size=168 \[ \frac{10 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{c^{3/2} f}-\frac{10 a^3 \tan (e+f x)}{c f \sqrt{c-c \sec (e+f x)}}-\frac{5 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{3 c f \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x) (a \sec (e+f x)+a)^2}{f (c-c \sec (e+f x))^{3/2}} \]
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Rubi [A] time = 0.336487, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3957, 3956, 3795, 203} \[ \frac{10 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{c^{3/2} f}-\frac{10 a^3 \tan (e+f x)}{c f \sqrt{c-c \sec (e+f x)}}-\frac{5 \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{3 c f \sqrt{c-c \sec (e+f x)}}-\frac{a \tan (e+f x) (a \sec (e+f x)+a)^2}{f (c-c \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3957
Rule 3956
Rule 3795
Rule 203
Rubi steps
\begin{align*} \int \frac{\sec (e+f x) (a+a \sec (e+f x))^3}{(c-c \sec (e+f x))^{3/2}} \, dx &=-\frac{a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac{(5 a) \int \frac{\sec (e+f x) (a+a \sec (e+f x))^2}{\sqrt{c-c \sec (e+f x)}} \, dx}{2 c}\\ &=-\frac{a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac{5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f \sqrt{c-c \sec (e+f x)}}-\frac{\left (5 a^2\right ) \int \frac{\sec (e+f x) (a+a \sec (e+f x))}{\sqrt{c-c \sec (e+f x)}} \, dx}{c}\\ &=-\frac{a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac{10 a^3 \tan (e+f x)}{c f \sqrt{c-c \sec (e+f x)}}-\frac{5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f \sqrt{c-c \sec (e+f x)}}-\frac{\left (10 a^3\right ) \int \frac{\sec (e+f x)}{\sqrt{c-c \sec (e+f x)}} \, dx}{c}\\ &=-\frac{a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac{10 a^3 \tan (e+f x)}{c f \sqrt{c-c \sec (e+f x)}}-\frac{5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f \sqrt{c-c \sec (e+f x)}}+\frac{\left (20 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{2 c+x^2} \, dx,x,\frac{c \tan (e+f x)}{\sqrt{c-c \sec (e+f x)}}\right )}{c f}\\ &=\frac{10 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{c^{3/2} f}-\frac{a (a+a \sec (e+f x))^2 \tan (e+f x)}{f (c-c \sec (e+f x))^{3/2}}-\frac{10 a^3 \tan (e+f x)}{c f \sqrt{c-c \sec (e+f x)}}-\frac{5 \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{3 c f \sqrt{c-c \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 2.96077, size = 324, normalized size = 1.93 \[ -\frac{a^3 \csc \left (\frac{e}{2}\right ) e^{-\frac{1}{2} i (e+f x)} \tan ^3\left (\frac{1}{2} (e+f x)\right ) \sec ^2\left (\frac{1}{2} (e+f x)\right ) (\sec (e+f x)+1)^3 \left (-\frac{i \left (-1+e^{i e}\right ) e^{\frac{i f x}{2}} \left (-24 e^{i (e+f x)}+34 e^{2 i (e+f x)}-24 e^{3 i (e+f x)}+19 e^{4 i (e+f x)}+19\right ) \sqrt{\sec (e+f x)}}{2 \left (-1+e^{i (e+f x)}\right )^2 \left (1+e^{2 i (e+f x)}\right )}-15 \sin \left (\frac{e}{2}\right ) \sqrt{\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \sqrt{1+e^{2 i (e+f x)}} \sec \left (\frac{1}{2} (e+f x)\right ) \tanh ^{-1}\left (\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right )\right )}{3 c f (\sec (e+f x)-1) \sec ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.232, size = 157, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}\sin \left ( fx+e \right ) }{3\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( 15\,\arctan \left ({\frac{1}{\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} \left ( \cos \left ( fx+e \right ) \right ) ^{2}-15\,\arctan \left ({\frac{1}{\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}+38\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-24\,\cos \left ( fx+e \right ) -2 \right ) \left ({\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{3}{2}}}} \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 = 0.655823, size = 1067, normalized size = 6.35 \begin{align*} \left [\frac{15 \, \sqrt{2}{\left (a^{3} c \cos \left (f x + e\right )^{2} - a^{3} c \cos \left (f x + e\right )\right )} \sqrt{-\frac{1}{c}} \log \left (\frac{2 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sqrt{-\frac{1}{c}} +{\left (3 \, \cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right )}{{\left (\cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 2 \,{\left (19 \, a^{3} \cos \left (f x + e\right )^{3} + 7 \, a^{3} \cos \left (f x + e\right )^{2} - 13 \, a^{3} \cos \left (f x + e\right ) - a^{3}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \,{\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}, -\frac{2 \,{\left (\frac{15 \, \sqrt{2}{\left (a^{3} c \cos \left (f x + e\right )^{2} - a^{3} c \cos \left (f x + e\right )\right )} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt{c} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right )}{\sqrt{c}} -{\left (19 \, a^{3} \cos \left (f x + e\right )^{3} + 7 \, a^{3} \cos \left (f x + e\right )^{2} - 13 \, a^{3} \cos \left (f x + e\right ) - a^{3}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}\right )}}{3 \,{\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right )\right )} \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] time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \frac{\sec{\left (e + f x \right )}}{- c \sqrt{- c \sec{\left (e + f x \right )} + c} \sec{\left (e + f x \right )} + c \sqrt{- c \sec{\left (e + f x \right )} + c}}\, dx + \int \frac{3 \sec ^{2}{\left (e + f x \right )}}{- c \sqrt{- c \sec{\left (e + f x \right )} + c} \sec{\left (e + f x \right )} + c \sqrt{- c \sec{\left (e + f x \right )} + c}}\, dx + \int \frac{3 \sec ^{3}{\left (e + f x \right )}}{- c \sqrt{- c \sec{\left (e + f x \right )} + c} \sec{\left (e + f x \right )} + c \sqrt{- c \sec{\left (e + f x \right )} + c}}\, dx + \int \frac{\sec ^{4}{\left (e + f x \right )}}{- c \sqrt{- c \sec{\left (e + f x \right )} + c} \sec{\left (e + f x \right )} + c \sqrt{- c \sec{\left (e + f x \right )} + c}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.82238, size = 300, normalized size = 1.79 \begin{align*} \frac{2 \, a^{3} c^{2}{\left (\frac{15 \, \sqrt{2} \arctan \left (\frac{\sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c}}{\sqrt{c}}\right )}{c^{\frac{7}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )} + \frac{2 \, \sqrt{2}{\left (6 \, c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 7 \, c\right )}}{{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{3}{2}} c^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )} + \frac{3 \, \sqrt{2} \sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c}}{c^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2}}\right )}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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