Optimal. Leaf size=731 \[ \frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {36 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {57 \left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{7 a^2 d (1+\sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}+\frac {57 \sqrt [3]{2} \sqrt [4]{3} E\left (\text {ArcCos}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7 a^2 d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}+\frac {19\ 3^{3/4} \left (1-\sqrt {3}\right ) F\left (\text {ArcCos}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{a+a \sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7\ 2^{2/3} a^2 d (1-\sec (c+d x)) (1+\sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}} \]
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Rubi [A]
time = 0.61, antiderivative size = 731, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3884, 4085,
3913, 3912, 65, 314, 231, 1895} \begin {gather*} \frac {19\ 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 (\text {ArcCos}\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 )}{7\ 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 {57 \sqrt [3]{2} \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 (\text {ArcCos}\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 )}{7 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 {57 \left (1+\sqrt {3}\right ) \tan (c+d x) \sqrt [3]{a \sec (c+d x)+a}}{7 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 {36 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 65
Rule 231
Rule 314
Rule 1895
Rule 3884
Rule 3912
Rule 3913
Rule 4085
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx &=\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {3 \int \frac {\sec (c+d x) \left (-\frac {5 a}{3}+\frac {7}{3} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^{2/3}} \, dx}{7 a^2}\\ &=\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {36 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}+\frac {19 \int \sec (c+d x) \sqrt [3]{a+a \sec (c+d x)} \, dx}{7 a^2}\\ &=\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {36 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}+\frac {\left (19 \sqrt [3]{a+a \sec (c+d x)}\right ) \int \sec (c+d x) \sqrt [3]{1+\sec (c+d x)} \, dx}{7 a^2 \sqrt [3]{1+\sec (c+d x)}}\\ &=\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {36 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {\left (19 \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt [6]{1+x}} \, dx,x,\sec (c+d x)\right )}{7 a^2 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {36 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {\left (114 \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{7 a^2 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {36 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}+\frac {\left (57 \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {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^2 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}+\frac {\left (57\ 2^{2/3} \left (1-\sqrt {3}\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{7 a^2 d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{5/6}}\\ &=\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {36 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {57 \left (1+\sqrt {3}\right ) \sqrt [3]{a+a \sec (c+d x)} \tan (c+d x)}{7 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 {57 \sqrt [3]{2} \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)}{7 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 {19\ 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)}{7\ 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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.29, size = 98, normalized size = 0.13 \begin {gather*} \frac {\left (-33-36 \sec (c+d x)+38\ 2^{5/6} \cos ^2\left (\frac {1}{2} (c+d x)\right ) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x))\right ) \sec (c+d x) \sqrt [6]{1+\sec (c+d x)}\right ) \tan (c+d x)}{7 d (a (1+\sec (c+d x)))^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\sec ^{3}\left (d x +c \right )}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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