Optimal. Leaf size=841 \[ \frac{3 \sqrt [3]{2} \sqrt [4]{3} (2 A-5 C) 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 ) \sqrt [3]{\sec (c+d x)+1} \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}} \tan (c+d x)}{7 a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{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^{3/4} \left (1-\sqrt{3}\right ) (2 A-5 C) \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 ) \sqrt [3]{\sec (c+d x)+1} \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}} \tan (c+d x)}{7\ 2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{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 F_1\left (-\frac{1}{6};\frac{1}{2},1;\frac{5}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right ) \tan (c+d x)}{a d \sqrt{1-\sec (c+d x)} (\sec (c+d x) a+a)^{2/3}}-\frac{3 (2 A-5 C) \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3}}-\frac{3 \left (1+\sqrt{3}\right ) (2 A-5 C) \sqrt [3]{\sec (c+d x)+1} \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )}-\frac{3 (A+C) \tan (c+d x)}{7 d (\sec (c+d x) a+a)^{5/3}} \]
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Rubi [A] time = 0.926865, antiderivative size = 841, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {4053, 3924, 3779, 3778, 136, 3828, 3827, 51, 63, 308, 225, 1881} \[ \frac{3 \sqrt [3]{2} \sqrt [4]{3} (2 A-5 C) 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 ) \sqrt [3]{\sec (c+d x)+1} \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}} \tan (c+d x)}{7 a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{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^{3/4} \left (1-\sqrt{3}\right ) (2 A-5 C) 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 ) \sqrt [3]{\sec (c+d x)+1} \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}} \tan (c+d x)}{7\ 2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{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 F_1\left (-\frac{1}{6};\frac{1}{2},1;\frac{5}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right ) \tan (c+d x)}{a d \sqrt{1-\sec (c+d x)} (\sec (c+d x) a+a)^{2/3}}-\frac{3 (2 A-5 C) \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3}}-\frac{3 \left (1+\sqrt{3}\right ) (2 A-5 C) \sqrt [3]{\sec (c+d x)+1} \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )}-\frac{3 (A+C) \tan (c+d x)}{7 d (\sec (c+d x) a+a)^{5/3}} \]
Antiderivative was successfully verified.
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Rule 4053
Rule 3924
Rule 3779
Rule 3778
Rule 136
Rule 3828
Rule 3827
Rule 51
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))^{5/3}} \, dx &=-\frac{3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac{3 \int \frac{-\frac{7 a A}{3}+\frac{1}{3} a (2 A-5 C) \sec (c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx}{7 a^2}\\ &=-\frac{3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac{A \int \frac{1}{(a+a \sec (c+d x))^{2/3}} \, dx}{a}-\frac{(2 A-5 C) \int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx}{7 a}\\ &=-\frac{3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac{\left (A (1+\sec (c+d x))^{2/3}\right ) \int \frac{1}{(1+\sec (c+d x))^{2/3}} \, dx}{a (a+a \sec (c+d x))^{2/3}}-\frac{\left ((2 A-5 C) (1+\sec (c+d x))^{2/3}\right ) \int \frac{\sec (c+d x)}{(1+\sec (c+d x))^{2/3}} \, dx}{7 a (a+a \sec (c+d x))^{2/3}}\\ &=-\frac{3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac{\left (A \sqrt [6]{1+\sec (c+d x)} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x (1+x)^{7/6}} \, dx,x,\sec (c+d x)\right )}{a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}+\frac{\left ((2 A-5 C) \sqrt [6]{1+\sec (c+d x)} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} (1+x)^{7/6}} \, dx,x,\sec (c+d x)\right )}{7 a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac{3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac{3 (2 A-5 C) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac{3 \sqrt{2} A F_1\left (-\frac{1}{6};\frac{1}{2},1;\frac{5}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}-\frac{\left ((2 A-5 C) \sqrt [6]{1+\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 )}{7 a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac{3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac{3 (2 A-5 C) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac{3 \sqrt{2} A F_1\left (-\frac{1}{6};\frac{1}{2},1;\frac{5}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}-\frac{\left (6 (2 A-5 C) \sqrt [6]{1+\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 )}{7 a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac{3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac{3 (2 A-5 C) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac{3 \sqrt{2} A F_1\left (-\frac{1}{6};\frac{1}{2},1;\frac{5}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}+\frac{\left (3 (2 A-5 C) \sqrt [6]{1+\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 )}{7 a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}+\frac{\left (3\ 2^{2/3} \left (1-\sqrt{3}\right ) (2 A-5 C) \sqrt [6]{1+\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 )}{7 a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac{3 (A+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac{3 (2 A-5 C) \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac{3 \sqrt{2} A F_1\left (-\frac{1}{6};\frac{1}{2},1;\frac{5}{6};\frac{1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) \tan (c+d x)}{a d \sqrt{1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}-\frac{3 \left (1+\sqrt{3}\right ) (2 A-5 C) \sqrt [3]{1+\sec (c+d x)} \tan (c+d x)}{7 a d (a+a \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} (2 A-5 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]{1+\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)}{7 a d (1-\sec (c+d x)) (a+a \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 ) (2 A-5 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]{1+\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)}{7\ 2^{2/3} a d (1-\sec (c+d x)) (a+a \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 = 9.65701, size = 0, normalized size = 0. \[ \int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.16, 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{5}{3}}}}\, dx \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 [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{5}{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{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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