Optimal. Leaf size=766 \[ -\frac{125\ 3^{3/4} \left (1-\sqrt{3}\right ) \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{28\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac{375 \left (1+\sqrt{3}\right ) \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a}}{28 a^2 d (\sec (c+d x)+1)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )}-\frac{375 \sqrt [4]{3} \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{14\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac{3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a \sec (c+d x)+a)^{2/3}}-\frac{33 \tan (c+d x)}{28 d (a \sec (c+d x)+a)^{5/3}} \]
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Rubi [A] time = 1.05862, antiderivative size = 766, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {3824, 4008, 4000, 3828, 3827, 63, 308, 225, 1881} \[ \frac{375 \left (1+\sqrt{3}\right ) \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a}}{28 a^2 d (\sec (c+d x)+1)^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )}-\frac{125\ 3^{3/4} \left (1-\sqrt{3}\right ) \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{28\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}-\frac{375 \sqrt [4]{3} \tan (c+d x) \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} \sqrt [3]{a \sec (c+d x)+a} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{2}-\left (1-\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{14\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt{3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}}}+\frac{3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a \sec (c+d x)+a)^{2/3}}-\frac{33 \tan (c+d x)}{28 d (a \sec (c+d x)+a)^{5/3}} \]
Antiderivative was successfully verified.
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Rule 3824
Rule 4008
Rule 4000
Rule 3828
Rule 3827
Rule 63
Rule 308
Rule 225
Rule 1881
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx &=\frac{3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac{3 \int \frac{\sec ^2(c+d x) \left (2 a-\frac{5}{3} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^{5/3}} \, dx}{4 a}\\ &=-\frac{33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac{3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}-\frac{9 \int \frac{\sec (c+d x) \left (-\frac{55 a^2}{9}+\frac{35}{9} a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^{2/3}} \, dx}{28 a^3}\\ &=-\frac{33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac{3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}-\frac{125 \int \sec (c+d x) \sqrt [3]{a+a \sec (c+d x)} \, dx}{28 a^2}\\ &=-\frac{33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac{3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}-\frac{\left (125 \sqrt [3]{a+a \sec (c+d x)}\right ) \int \sec (c+d x) \sqrt [3]{1+\sec (c+d x)} \, dx}{28 a^2 \sqrt [3]{1+\sec (c+d x)}}\\ &=-\frac{33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac{3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}+\frac{\left (125 \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 )}{28 a^2 d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=-\frac{33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac{3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}+\frac{\left (375 \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 )}{14 a^2 d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=-\frac{33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac{3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}-\frac{\left (375 \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 )}{28 a^2 d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}-\frac{\left (375 \left (1-\sqrt{3}\right ) \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 )}{14 \sqrt [3]{2} a^2 d \sqrt{1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=-\frac{33 \tan (c+d x)}{28 d (a+a \sec (c+d x))^{5/3}}+\frac{3 \sec ^2(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a+a \sec (c+d x))^{2/3}}+\frac{375 \left (1+\sqrt{3}\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{28 a^2 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{375 \sqrt [4]{3} 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)}{14\ 2^{2/3} a^2 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{125\ 3^{3/4} \left (1-\sqrt{3}\right ) 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)}{28\ 2^{2/3} a^2 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 [C] time = 0.55807, size = 111, normalized size = 0.14 \[ \frac{\tan (c+d x) \left (3 \left (7 \sec ^2(c+d x)+90 \sec (c+d x)+79\right )-250\ 2^{5/6} \cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \sqrt [6]{\sec (c+d x)+1} \text{Hypergeometric2F1}\left (\frac{1}{6},\frac{1}{2},\frac{3}{2},\frac{1}{2} (1-\sec (c+d x))\right )\right )}{28 d (a (\sec (c+d x)+1))^{5/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.132, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sec \left ( dx+c \right ) \right ) ^{4} \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{1}{3}} \sec \left (d x + c\right )^{4}}{a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}}, x\right ) \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{\sec ^{4}{\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{\sec \left (d x + c\right )^{4}}{{\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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