Optimal. Leaf size=791 \[ \frac{3^{3/4} \left (1-\sqrt{3}\right ) (A+2 C) \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac{3 \sqrt{2} A \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{5 a d \sqrt{1-\sec (c+d x)}}-\frac{3 (A+C) \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}-\frac{3 \left (1+\sqrt{3}\right ) (A+2 C) \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a}}{a d (\sec (c+d x)+1)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )}+\frac{3 \sqrt [3]{2} \sqrt [4]{3} (A+2 C) \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}} \]
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Rubi [A] time = 0.878553, antiderivative size = 791, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {4053, 3924, 3779, 3778, 136, 3828, 3827, 63, 308, 225, 1881} \[ \frac{3 \sqrt{2} A \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a} F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{5 a d \sqrt{1-\sec (c+d x)}}-\frac{3 (A+C) \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}-\frac{3 \left (1+\sqrt{3}\right ) (A+2 C) \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a}}{a d (\sec (c+d x)+1)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )}+\frac{3^{3/4} \left (1-\sqrt{3}\right ) (A+2 C) \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac{3 \sqrt [3]{2} \sqrt [4]{3} (A+2 C) \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 4053
Rule 3924
Rule 3779
Rule 3778
Rule 136
Rule 3828
Rule 3827
Rule 63
Rule 308
Rule 225
Rule 1881
Rubi steps
\begin{align*} \int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx &=-\frac{3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}-\frac{3 \int \sqrt [3]{a+a \sec (c+d x)} \left (-\frac{a A}{3}-\frac{1}{3} a (A+2 C) \sec (c+d x)\right ) \, dx}{a^2}\\ &=-\frac{3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac{A \int \sqrt [3]{a+a \sec (c+d x)} \, dx}{a}+\frac{(A+2 C) \int \sec (c+d x) \sqrt [3]{a+a \sec (c+d x)} \, dx}{a}\\ &=-\frac{3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac{\left (A \sqrt [3]{a+a \sec (c+d x)}\right ) \int \sqrt [3]{1+\sec (c+d x)} \, dx}{a \sqrt [3]{1+\sec (c+d x)}}+\frac{\left ((A+2 C) \sqrt [3]{a+a \sec (c+d x)}\right ) \int \sec (c+d x) \sqrt [3]{1+\sec (c+d x)} \, dx}{a \sqrt [3]{1+\sec (c+d x)}}\\ &=-\frac{3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}-\frac{\left (A \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x \sqrt [6]{1+x}} \, dx,x,\sec (c+d x)\right )}{a d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}-\frac{\left ((A+2 C) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt [6]{1+x}} \, dx,x,\sec (c+d x)\right )}{a d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=-\frac{3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac{3 \sqrt{2} A F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{5 a d \sqrt{1-\sec (c+d x)}}-\frac{\left (6 (A+2 C) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{a d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=-\frac{3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac{3 \sqrt{2} A F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{5 a d \sqrt{1-\sec (c+d x)}}+\frac{\left (3 (A+2 C) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{2^{2/3} \left (-1+\sqrt{3}\right )-2 x^4}{\sqrt{2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{a d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}+\frac{\left (3\ 2^{2/3} \left (1-\sqrt{3}\right ) (A+2 C) \sqrt [3]{a+a \sec (c+d x)} \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 )}{a d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=-\frac{3 (A+C) \tan (c+d x)}{d (a+a \sec (c+d x))^{2/3}}+\frac{3 \sqrt{2} A F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{5 a d \sqrt{1-\sec (c+d x)}}-\frac{3 \left (1+\sqrt{3}\right ) (A+2 C) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{a d (1+\sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}+\frac{3 \sqrt [3]{2} \sqrt [4]{3} (A+2 C) E\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 ) \sqrt [3]{a+a \sec (c+d x)} \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)}{a d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \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}}}+\frac{3^{3/4} \left (1-\sqrt{3}\right ) (A+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 ) \sqrt [3]{a+a \sec (c+d x)} \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)}{2^{2/3} a d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \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 [F] time = 17.8809, size = 0, normalized size = 0. \[ \int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.176, size = 0, normalized size = 0. \begin{align*} \int{(A+C \left ( \sec \left ( dx+c \right ) \right ) ^{2}) \left ( a+a\sec \left ( dx+c \right ) \right ) ^{-{\frac{2}{3}}}}\, 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{C \sec \left (d x + c\right )^{2} + A}{{\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 \frac{A + C \sec ^{2}{\left (c + d x \right )}}{\left (a \left (\sec{\left (c + d x \right )} + 1\right )\right )^{\frac{2}{3}}}\, 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{C \sec \left (d x + c\right )^{2} + A}{{\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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