Optimal. Leaf size=446 \[ -\frac{3^{3/4} (5 B+2 C) \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} (a \sec (c+d x)+a)^{2/3} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 \sqrt [3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac{3 \sqrt{2} A \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left (\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{7 d \sqrt{1-\sec (c+d x)}}+\frac{3 (5 B+2 C) \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{10 d (\sec (c+d x)+1)}+\frac{3 C \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d} \]
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Rubi [A] time = 0.722644, antiderivative size = 446, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4054, 3924, 3779, 3778, 136, 3828, 3827, 50, 63, 225} \[ \frac{3 \sqrt{2} A \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left (\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{7 d \sqrt{1-\sec (c+d x)}}+\frac{3 (5 B+2 C) \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{10 d (\sec (c+d x)+1)}-\frac{3^{3/4} (5 B+2 C) \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} (a \sec (c+d x)+a)^{2/3} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 \sqrt [3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac{3 C \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d} \]
Antiderivative was successfully verified.
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Rule 4054
Rule 3924
Rule 3779
Rule 3778
Rule 136
Rule 3828
Rule 3827
Rule 50
Rule 63
Rule 225
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{3 C (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{5 d}+\frac{3 \int (a+a \sec (c+d x))^{2/3} \left (\frac{5 a A}{3}+\frac{1}{3} a (5 B+2 C) \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac{3 C (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{5 d}+A \int (a+a \sec (c+d x))^{2/3} \, dx+\frac{1}{5} (5 B+2 C) \int \sec (c+d x) (a+a \sec (c+d x))^{2/3} \, dx\\ &=\frac{3 C (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{5 d}+\frac{\left (A (a+a \sec (c+d x))^{2/3}\right ) \int (1+\sec (c+d x))^{2/3} \, dx}{(1+\sec (c+d x))^{2/3}}+\frac{\left ((5 B+2 C) (a+a \sec (c+d x))^{2/3}\right ) \int \sec (c+d x) (1+\sec (c+d x))^{2/3} \, dx}{5 (1+\sec (c+d x))^{2/3}}\\ &=\frac{3 C (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{5 d}-\frac{\left (A (a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{1+x}}{\sqrt{1-x} x} \, dx,x,\sec (c+d x)\right )}{d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}}-\frac{\left ((5 B+2 C) (a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt [6]{1+x}}{\sqrt{1-x}} \, dx,x,\sec (c+d x)\right )}{5 d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}}\\ &=\frac{3 C (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{5 d}+\frac{3 \sqrt{2} A F_1\left (\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{7 d \sqrt{1-\sec (c+d x)}}+\frac{3 (5 B+2 C) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{10 d (1+\sec (c+d x))}-\frac{\left ((5 B+2 C) (a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} (1+x)^{5/6}} \, dx,x,\sec (c+d x)\right )}{10 d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}}\\ &=\frac{3 C (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{5 d}+\frac{3 \sqrt{2} A F_1\left (\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{7 d \sqrt{1-\sec (c+d x)}}+\frac{3 (5 B+2 C) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{10 d (1+\sec (c+d x))}-\frac{\left (3 (5 B+2 C) (a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{5 d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}}\\ &=\frac{3 C (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{5 d}+\frac{3 \sqrt{2} A F_1\left (\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{7 d \sqrt{1-\sec (c+d x)}}+\frac{3 (5 B+2 C) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{10 d (1+\sec (c+d x))}-\frac{3^{3/4} (5 B+2 C) F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right ) (a+a \sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt{\frac{2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{10 \sqrt [3]{2} d (1-\sec (c+d x)) (1+\sec (c+d x)) \sqrt{-\frac{\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}\\ \end{align*}
Mathematica [B] time = 21.0617, size = 5449, normalized size = 12.22 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.182, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sec \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( A+B\sec \left ( dx+c \right ) +C \left ( \sec \left ( dx+c \right ) \right ) ^{2} \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{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{2}{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 \left (a \left (\sec{\left (c + d x \right )} + 1\right )\right )^{\frac{2}{3}} \left (A + B \sec{\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\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{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{2}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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