Optimal. Leaf size=241 \[ \frac{2 a^2 \tan (e+f x) \left (d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)+2 \left (243 c^2 d+36 c^3+189 c d^2+52 d^3\right )\right )}{105 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} c^3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c+d \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 (6 c+13 d) \tan (e+f x) (c+d \sec (e+f x))^2}{35 f \sqrt{a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.199716, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3940, 153, 147, 63, 206} \[ \frac{2 a^2 \tan (e+f x) \left (d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)+2 \left (243 c^2 d+36 c^3+189 c d^2+52 d^3\right )\right )}{105 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^{5/2} c^3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (c+d \sec (e+f x))^3}{7 f \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 (6 c+13 d) \tan (e+f x) (c+d \sec (e+f x))^2}{35 f \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 3940
Rule 153
Rule 147
Rule 63
Rule 206
Rubi steps
\begin{align*} \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(a+a x) (c+d x)^3}{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^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt{a+a \sec (e+f x)}}+\frac{(2 a \tan (e+f x)) \operatorname{Subst}\left (\int \frac{(c+d x)^2 \left (-\frac{7 a^2 c}{2}-\frac{1}{2} a^2 (6 c+13 d) x\right )}{x \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{7 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt{a+a \sec (e+f x)}}-\frac{(4 \tan (e+f x)) \operatorname{Subst}\left (\int \frac{(c+d x) \left (\frac{35 a^3 c^2}{4}+\frac{1}{4} a^3 \left (24 c^2+111 c d+52 d^2\right ) x\right )}{x \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{35 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt{a+a \sec (e+f x)}}-\frac{\left (a^3 c^3 \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^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt{a+a \sec (e+f x)}}+\frac{\left (2 a^2 c^3 \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^{5/2} c^3 \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 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt{a+a \sec (e+f x)}}+\frac{2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt{a+a \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.86925, size = 219, normalized size = 0.91 \[ \frac{a \sec \left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x) \sqrt{a (\sec (e+f x)+1)} \left (2 \sin \left (\frac{1}{2} (e+f x)\right ) \left (9 \left (175 c^2 d+35 c^3+154 c d^2+52 d^3\right ) \cos (e+f x)+2 d \left (105 c^2+189 c d+52 d^2\right ) \cos (2 (e+f x))+525 c^2 d \cos (3 (e+f x))+210 c^2 d+105 c^3 \cos (3 (e+f x))+378 c d^2 \cos (3 (e+f x))+378 c d^2+104 d^3 \cos (3 (e+f x))+164 d^3\right )+420 \sqrt{2} c^3 \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (e+f x)\right )\right ) \cos ^{\frac{7}{2}}(e+f x)\right )}{420 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.316, size = 539, normalized size = 2.2 \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 = 1.34508, size = 1230, normalized size = 5.1 \begin{align*} \left [\frac{105 \,{\left (a c^{3} \cos \left (f x + e\right )^{4} + a c^{3} \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 \, a d^{3} +{\left (105 \, a c^{3} + 525 \, a c^{2} d + 378 \, a c d^{2} + 104 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} +{\left (105 \, a c^{2} d + 189 \, a c d^{2} + 52 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \,{\left (21 \, a c d^{2} + 13 \, a d^{3}\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 (a c^{3} \cos \left (f x + e\right )^{4} + a c^{3} \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 \, a d^{3} +{\left (105 \, a c^{3} + 525 \, a c^{2} d + 378 \, a c d^{2} + 104 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} +{\left (105 \, a c^{2} d + 189 \, a c d^{2} + 52 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \,{\left (21 \, a c d^{2} + 13 \, a d^{3}\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(-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 [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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