Optimal. Leaf size=203 \[ -\frac{35 \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{64 \sqrt{2} a^2 c^{5/2} f}-\frac{35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}-\frac{35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac{7 \tan (e+f x)}{6 f \left (a^2 \sec (e+f x)+a^2\right ) (c-c \sec (e+f x))^{5/2}}+\frac{\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}} \]
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Rubi [A] time = 0.388975, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3960, 3796, 3795, 203} \[ -\frac{35 \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{64 \sqrt{2} a^2 c^{5/2} f}-\frac{35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}-\frac{35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac{7 \tan (e+f x)}{6 f \left (a^2 \sec (e+f x)+a^2\right ) (c-c \sec (e+f x))^{5/2}}+\frac{\tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 (c-c \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3960
Rule 3796
Rule 3795
Rule 203
Rubi steps
\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}} \, dx &=\frac{\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac{7 \int \frac{\sec (e+f x)}{(a+a \sec (e+f x)) (c-c \sec (e+f x))^{5/2}} \, dx}{6 a}\\ &=\frac{\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac{7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}+\frac{35 \int \frac{\sec (e+f x)}{(c-c \sec (e+f x))^{5/2}} \, dx}{12 a^2}\\ &=-\frac{35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac{\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac{7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}+\frac{35 \int \frac{\sec (e+f x)}{(c-c \sec (e+f x))^{3/2}} \, dx}{32 a^2 c}\\ &=-\frac{35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac{\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac{7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac{35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}+\frac{35 \int \frac{\sec (e+f x)}{\sqrt{c-c \sec (e+f x)}} \, dx}{128 a^2 c^2}\\ &=-\frac{35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac{\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac{7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac{35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}-\frac{35 \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 )}{64 a^2 c^2 f}\\ &=-\frac{35 \tan ^{-1}\left (\frac{\sqrt{c} \tan (e+f x)}{\sqrt{2} \sqrt{c-c \sec (e+f x)}}\right )}{64 \sqrt{2} a^2 c^{5/2} f}-\frac{35 \tan (e+f x)}{48 a^2 f (c-c \sec (e+f x))^{5/2}}+\frac{\tan (e+f x)}{3 f (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^{5/2}}+\frac{7 \tan (e+f x)}{6 f \left (a^2+a^2 \sec (e+f x)\right ) (c-c \sec (e+f x))^{5/2}}-\frac{35 \tan (e+f x)}{64 a^2 c f (c-c \sec (e+f x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 2.38116, size = 434, normalized size = 2.14 \[ \frac{\cot ^4(e+f x) \left (\frac{105 i \sqrt{2} \left (-1+e^{i (e+f x)}\right )^5 \left (1+e^{i (e+f x)}\right )^4 \tanh ^{-1}\left (\frac{1+e^{i (e+f x)}}{\sqrt{2} \sqrt{1+e^{2 i (e+f x)}}}\right )}{\left (1+e^{2 i (e+f x)}\right )^{9/2}}-3648 \csc \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \sin ^9(e+f x) \csc \left (\frac{1}{2} (e+f x)\right ) \csc ^5(2 (e+f x))-5504 \sin \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \sin \left (\frac{1}{2} (e+f x)\right ) \sin ^9(e+f x) \csc ^5(2 (e+f x))-13312 \sin ^2\left (\frac{1}{2} (e+f x)\right ) \sin ^8(e+f x) \csc ^5(2 (e+f x))+2752 \cos \left (\frac{e}{2}\right ) \cos \left (\frac{f x}{2}\right ) \sin ^5\left (\frac{1}{2} (e+f x)\right ) \cos ^4\left (\frac{1}{2} (e+f x)\right ) \sec ^5(e+f x)+256 \sin ^5\left (\frac{1}{2} (e+f x)\right ) \cos \left (\frac{1}{2} (e+f x)\right ) \sec ^5(e+f x)+114 \cot \left (\frac{e}{2}\right ) \tan ^4(e+f x) \sec (e+f x)-192 \cot \left (\frac{e}{2}\right ) \sin ^2\left (\frac{1}{2} (e+f x)\right ) \cos ^4\left (\frac{1}{2} (e+f x)\right ) \sec ^5(e+f x)+192 \csc \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \sin \left (\frac{1}{2} (e+f x)\right ) \cos ^4\left (\frac{1}{2} (e+f x)\right ) \sec ^5(e+f x)\right )}{384 a^2 c^2 f \sqrt{c-c \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.242, size = 551, normalized size = 2.7 \begin{align*} \text{result too large to display} \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 = 0.759568, size = 1239, normalized size = 6.1 \begin{align*} \left [-\frac{105 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) + 1\right )} \sqrt{-c} \log \left (\frac{2 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt{-c} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} +{\left (3 \, c \cos \left (f x + e\right ) + c\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 ) + 4 \,{\left (43 \, \cos \left (f x + e\right )^{4} - 161 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )^{2} + 105 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{768 \,{\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\right )} \sin \left (f x + e\right )}, \frac{105 \, \sqrt{2}{\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) + 1\right )} \sqrt{c} \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 ) - 2 \,{\left (43 \, \cos \left (f x + e\right )^{4} - 161 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )^{2} + 105 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{384 \,{\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\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(-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 [A] time = 1.64403, size = 265, normalized size = 1.31 \begin{align*} -\frac{\sqrt{2}{\left (105 \, \sqrt{c} \arctan \left (\frac{\sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c}}{\sqrt{c}}\right ) + \frac{8 \,{\left ({\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{3}{2}} c^{2} - 9 \, \sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c} c^{3}\right )}}{c^{3}} - \frac{3 \,{\left (13 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{3}{2}} c + 11 \, \sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c} c^{2}\right )}}{c^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4}}\right )}}{384 \, a^{2} c^{3} f \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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