Optimal. Leaf size=271 \[ \frac{2 a^{3/2} c^4 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^4 \tan (e+f x) (a-a \sec (e+f x))^3}{7 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \left (6 c^2+8 c d+3 d^2\right ) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 (4 c+3 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt{a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.172894, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3940, 88, 63, 206} \[ \frac{2 a^{3/2} c^4 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{2 d^4 \tan (e+f x) (a-a \sec (e+f x))^3}{7 a^2 f \sqrt{a \sec (e+f x)+a}}-\frac{2 d^2 \left (6 c^2+8 c d+3 d^2\right ) \tan (e+f x) (a-a \sec (e+f x))}{3 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt{a \sec (e+f x)+a}}+\frac{2 d^3 (4 c+3 d) \tan (e+f x) (a-a \sec (e+f x))^2}{5 a f \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 3940
Rule 88
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))^4 \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^4}{x \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \left (\frac{d (2 c+d) \left (2 c^2+2 c d+d^2\right )}{\sqrt{a-a x}}+\frac{c^4}{x \sqrt{a-a x}}-\frac{d^2 \left (6 c^2+8 c d+3 d^2\right ) \sqrt{a-a x}}{a}+\frac{d^3 (4 c+3 d) (a-a x)^{3/2}}{a^2}-\frac{d^4 (a-a x)^{5/2}}{a^3}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{2 d^2 \left (6 c^2+8 c d+3 d^2\right ) (a-a \sec (e+f x)) \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+\frac{2 d^3 (4 c+3 d) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 a f \sqrt{a+a \sec (e+f x)}}-\frac{2 d^4 (a-a \sec (e+f x))^3 \tan (e+f x)}{7 a^2 f \sqrt{a+a \sec (e+f x)}}-\frac{\left (a^2 c^4 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}-\frac{2 d^2 \left (6 c^2+8 c d+3 d^2\right ) (a-a \sec (e+f x)) \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+\frac{2 d^3 (4 c+3 d) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 a f \sqrt{a+a \sec (e+f x)}}-\frac{2 d^4 (a-a \sec (e+f x))^3 \tan (e+f x)}{7 a^2 f \sqrt{a+a \sec (e+f x)}}+\frac{\left (2 a c^4 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 a d (2 c+d) \left (2 c^2+2 c d+d^2\right ) \tan (e+f x)}{f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^{3/2} c^4 \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right ) \tan (e+f x)}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{2 d^2 \left (6 c^2+8 c d+3 d^2\right ) (a-a \sec (e+f x)) \tan (e+f x)}{3 f \sqrt{a+a \sec (e+f x)}}+\frac{2 d^3 (4 c+3 d) (a-a \sec (e+f x))^2 \tan (e+f x)}{5 a f \sqrt{a+a \sec (e+f x)}}-\frac{2 d^4 (a-a \sec (e+f x))^3 \tan (e+f x)}{7 a^2 f \sqrt{a+a \sec (e+f x)}}\\ \end{align*}
Mathematica [C] time = 14.6181, size = 589, normalized size = 2.17 \[ \frac{\cos ^4(e+f x) \sec \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} (c+d \sec (e+f x))^4 \left (\frac{8}{105} d \left (105 c^2 d+105 c^3+56 c d^2+12 d^3\right ) \sin \left (\frac{1}{2} (e+f x)\right )+\frac{4}{105} \sec (e+f x) \left (105 c^2 d^2 \sin \left (\frac{1}{2} (e+f x)\right )+56 c d^3 \sin \left (\frac{1}{2} (e+f x)\right )+12 d^4 \sin \left (\frac{1}{2} (e+f x)\right )\right )+\frac{4}{35} \sec ^2(e+f x) \left (14 c d^3 \sin \left (\frac{1}{2} (e+f x)\right )+3 d^4 \sin \left (\frac{1}{2} (e+f x)\right )\right )+\frac{2}{7} d^4 \sin \left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x)\right )}{f (c \cos (e+f x)+d)^4}-\frac{8 \left (-3-2 \sqrt{2}\right ) c^4 \cos ^4\left (\frac{1}{4} (e+f x)\right ) \sqrt{\frac{\left (10-7 \sqrt{2}\right ) \cos \left (\frac{1}{2} (e+f x)\right )-5 \sqrt{2}+7}{\cos \left (\frac{1}{2} (e+f x)\right )+1}} \sqrt{\frac{-\left (\sqrt{2}-2\right ) \cos \left (\frac{1}{2} (e+f x)\right )+\sqrt{2}-1}{\cos \left (\frac{1}{2} (e+f x)\right )+1}} \left (\left (\sqrt{2}-2\right ) \cos \left (\frac{1}{2} (e+f x)\right )-\sqrt{2}+1\right ) \cos ^3(e+f x) \sqrt{-\tan ^2\left (\frac{1}{4} (e+f x)\right )-2 \sqrt{2}+3} \sec \left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sec (e+f x)+1)} \sqrt{\left (\left (2+\sqrt{2}\right ) \cos \left (\frac{1}{2} (e+f x)\right )-\sqrt{2}-1\right ) \sec ^2\left (\frac{1}{4} (e+f x)\right )} (c+d \sec (e+f x))^4 \left (\text{EllipticF}\left (\sin ^{-1}\left (\frac{\tan \left (\frac{1}{4} (e+f x)\right )}{\sqrt{3-2 \sqrt{2}}}\right ),17-12 \sqrt{2}\right )+2 \Pi \left (-3+2 \sqrt{2};-\sin ^{-1}\left (\frac{\tan \left (\frac{1}{4} (e+f x)\right )}{\sqrt{3-2 \sqrt{2}}}\right )|17-12 \sqrt{2}\right )\right )}{f (c \cos (e+f x)+d)^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.413, size = 546, normalized size = 2. \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 = 1.69881, size = 1181, normalized size = 4.36 \begin{align*} \left [\frac{105 \,{\left (c^{4} \cos \left (f x + e\right )^{4} + c^{4} \cos \left (f x + e\right )^{3}\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 (15 \, d^{4} + 4 \,{\left (105 \, c^{3} d + 105 \, c^{2} d^{2} + 56 \, c d^{3} + 12 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + 2 \,{\left (105 \, c^{2} d^{2} + 56 \, c d^{3} + 12 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (14 \, c d^{3} + 3 \, d^{4}\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 )}{105 \,{\left (f \cos \left (f x + e\right )^{4} + f \cos \left (f x + e\right )^{3}\right )}}, -\frac{2 \,{\left (105 \,{\left (c^{4} \cos \left (f x + e\right )^{4} + c^{4} \cos \left (f x + e\right )^{3}\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 (15 \, d^{4} + 4 \,{\left (105 \, c^{3} d + 105 \, c^{2} d^{2} + 56 \, c d^{3} + 12 \, d^{4}\right )} \cos \left (f x + e\right )^{3} + 2 \,{\left (105 \, c^{2} d^{2} + 56 \, c d^{3} + 12 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + 6 \,{\left (14 \, c d^{3} + 3 \, d^{4}\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 )}}{105 \,{\left (f \cos \left (f x + e\right )^{4} + f \cos \left (f x + e\right )^{3}\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 \sqrt{a \left (\sec{\left (e + f x \right )} + 1\right )} \left (c + d \sec{\left (e + f x \right )}\right )^{4}\, 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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