Optimal. Leaf size=75 \[ \frac{3 \sqrt{2} \tan (c+d x) F_1\left (\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{d \sqrt{1-\sec (c+d x)} \sqrt [3]{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.042687, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3779, 3778, 136} \[ \frac{3 \sqrt{2} \tan (c+d x) F_1\left (\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{d \sqrt{1-\sec (c+d x)} \sqrt [3]{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3779
Rule 3778
Rule 136
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{a+a \sec (c+d x)}} \, dx &=\frac{\sqrt [3]{1+\sec (c+d x)} \int \frac{1}{\sqrt [3]{1+\sec (c+d x)}} \, dx}{\sqrt [3]{a+a \sec (c+d x)}}\\ &=-\frac{\tan (c+d x) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x (1+x)^{5/6}} \, dx,x,\sec (c+d x)\right )}{d \sqrt{1-\sec (c+d x)} \sqrt [6]{1+\sec (c+d x)} \sqrt [3]{a+a \sec (c+d x)}}\\ &=\frac{3 \sqrt{2} F_1\left (\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{d \sqrt{1-\sec (c+d x)} \sqrt [3]{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [B] time = 4.51008, size = 718, normalized size = 9.57 \[ \frac{45 \cos (c+d x) \tan \left (\frac{1}{2} (c+d x)\right ) (\sec (c+d x)+1)^2 F_1\left (\frac{1}{2};-\frac{1}{3},1;\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 (9 F_1\left (\frac{1}{2};-\frac{1}{3},1;\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 \tan ^2\left (\frac{1}{2} (c+d x)\right ) \left (3 F_1\left (\frac{3}{2};-\frac{1}{3},2;\frac{5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )+F_1\left (\frac{3}{2};\frac{2}{3},1;\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 )\right )}{d \sqrt [3]{a (\sec (c+d x)+1)} \left (40 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \tan ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (3 F_1\left (\frac{3}{2};-\frac{1}{3},2;\frac{5}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )+F_1\left (\frac{3}{2};\frac{2}{3},1;\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+135 F_1\left (\frac{1}{2};-\frac{1}{3},1;\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 \left (-\tan ^2(c+d x)+3 \sec (c+d x)-3 \sin (c+d x) \tan (c+d x)+3\right )+6 \sin ^2\left (\frac{1}{2} (c+d x)\right ) \sec ^2(c+d x) F_1\left (\frac{1}{2};-\frac{1}{3},1;\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 (-24 \cos (c+d x) \tan ^2\left (\frac{1}{2} (c+d x)\right ) \left (9 F_1\left (\frac{5}{2};-\frac{1}{3},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 )+3 F_1\left (\frac{5}{2};\frac{2}{3},2;\frac{7}{2};\tan ^2\left (\frac{1}{2} (c+d x)\right ),-\tan ^2\left (\frac{1}{2} (c+d x)\right )\right )-F_1\left (\frac{5}{2};\frac{5}{3},1;\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 )-15 (-10 \cos (c+d x)+3 \cos (2 (c+d x))+1) F_1\left (\frac{3}{2};-\frac{1}{3},2;\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 (-10 \cos (c+d x)+3 \cos (2 (c+d x))+1) F_1\left (\frac{3}{2};\frac{2}{3},1;\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 )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{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{1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}\,{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{1}{\sqrt [3]{a \sec{\left (c + d x \right )} + a}}\, 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{1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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