Optimal. Leaf size=78 \[ -\frac{3 \tan (c+d x) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\sec (c+d x),-\sec (c+d x)\right ) (e \sec (c+d x))^{2/3}}{2 d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.164334, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3828, 3827, 130, 510} \[ -\frac{3 \tan (c+d x) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\sec (c+d x),-\sec (c+d x)\right ) (e \sec (c+d x))^{2/3}}{2 d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3828
Rule 3827
Rule 130
Rule 510
Rubi steps
\begin{align*} \int \frac{(e \sec (c+d x))^{2/3}}{\sqrt{a+a \sec (c+d x)}} \, dx &=\frac{\sqrt{1+\sec (c+d x)} \int \frac{(e \sec (c+d x))^{2/3}}{\sqrt{1+\sec (c+d x)}} \, dx}{\sqrt{a+a \sec (c+d x)}}\\ &=-\frac{(e \tan (c+d x)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt [3]{e x} (1+x)} \, dx,x,\sec (c+d x)\right )}{d \sqrt{1-\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}\\ &=-\frac{(3 \tan (c+d x)) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-\frac{x^3}{e}} \left (1+\frac{x^3}{e}\right )} \, dx,x,\sqrt [3]{e \sec (c+d x)}\right )}{d \sqrt{1-\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}\\ &=-\frac{3 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\sec (c+d x),-\sec (c+d x)\right ) (e \sec (c+d x))^{2/3} \tan (c+d x)}{2 d \sqrt{1-\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [B] time = 6.74803, size = 760, normalized size = 9.74 \[ \frac{90 \sin \left (\frac{1}{2} (c+d x)\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \cos ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} F_1\left (\frac{1}{2};\frac{1}{6},\frac{1}{3};\frac{3}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right ) \left (\tan ^2\left (\frac{1}{2} (c+d x)\right ) \left (F_1\left (\frac{3}{2};\frac{7}{6},\frac{1}{3};\frac{5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )-2 F_1\left (\frac{3}{2};\frac{1}{6},\frac{4}{3};\frac{5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{6},\frac{1}{3};\frac{3}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )\right ) (e \sec (c+d x))^{2/3}}{a d \left (270 (2 \cos (c+d x)+1) \cos ^4\left (\frac{1}{2} (c+d x)\right ) F_1\left (\frac{1}{2};\frac{1}{6},\frac{1}{3};\frac{3}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right ){}^2-3 \tan ^2\left (\frac{1}{2} (c+d x)\right ) F_1\left (\frac{1}{2};\frac{1}{6},\frac{1}{3};\frac{3}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right ) \left (10 (-9 \cos (c+d x)+\cos (2 (c+d x))+2) \cos ^2\left (\frac{1}{2} (c+d x)\right ) F_1\left (\frac{3}{2};\frac{1}{6},\frac{4}{3};\frac{5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )-5 (-9 \cos (c+d x)+\cos (2 (c+d x))+2) \cos ^2\left (\frac{1}{2} (c+d x)\right ) F_1\left (\frac{3}{2};\frac{7}{6},\frac{1}{3};\frac{5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )+6 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \cos (c+d x) \left (16 F_1\left (\frac{5}{2};\frac{1}{6},\frac{7}{3};\frac{7}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )-4 F_1\left (\frac{5}{2};\frac{7}{6},\frac{4}{3};\frac{7}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )+7 F_1\left (\frac{5}{2};\frac{13}{6},\frac{1}{3};\frac{7}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )\right )\right )+10 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \cos (c+d x) \tan ^2\left (\frac{1}{2} (c+d x)\right ) \left (F_1\left (\frac{3}{2};\frac{7}{6},\frac{1}{3};\frac{5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )-2 F_1\left (\frac{3}{2};\frac{1}{6},\frac{4}{3};\frac{5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )\right ){}^2\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.171, size = 0, normalized size = 0. \begin{align*} \int{ \left ( e\sec \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}}{\frac{1}{\sqrt{a+a\sec \left ( dx+c \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sec \left (d x + c\right )\right )^{\frac{2}{3}}}{\sqrt{a \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sec{\left (c + d x \right )}\right )^{\frac{2}{3}}}{\sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sec \left (d x + c\right )\right )^{\frac{2}{3}}}{\sqrt{a \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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