Optimal. Leaf size=327 \[ \frac {\tan (c+d x) (a (\sec (c+d x)+1))^{2/3} \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {5}{4};\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )}}{8 d \sqrt [3]{\frac {1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{4/3}}-\frac {5 \tan ^3(c+d x) (a (\sec (c+d x)+1))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {7}{4};\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )}}{8 d \sqrt [3]{\frac {1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{10/3}}+\frac {9 \sin (c+d x) \sec ^{\frac {2}{3}}(c+d x) (a (\sec (c+d x)+1))^{2/3}}{4 d}-\frac {3 a \sin (c+d x) \sec ^{\frac {5}{3}}(c+d x)}{2 d \sqrt [3]{a (\sec (c+d x)+1)}}-\frac {9 \tan (c+d x) (a (\sec (c+d x)+1))^{2/3}}{4 d \sqrt [3]{\frac {1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{7/3}} \]
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Rubi [C] time = 0.12, antiderivative size = 79, normalized size of antiderivative = 0.24, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3828, 3825, 133} \[ \frac {2 \sqrt [6]{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left (\frac {1}{2};-\frac {2}{3},-\frac {1}{6};\frac {3}{2};1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right )}{d (\sec (c+d x)+1)^{7/6}} \]
Warning: Unable to verify antiderivative.
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Rule 133
Rule 3825
Rule 3828
Rubi steps
\begin {align*} \int \sec ^{\frac {5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx &=\frac {(a+a \sec (c+d x))^{2/3} \int \sec ^{\frac {5}{3}}(c+d x) (1+\sec (c+d x))^{2/3} \, dx}{(1+\sec (c+d x))^{2/3}}\\ &=\frac {\left ((a+a \sec (c+d x))^{2/3} \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {(1-x)^{2/3} \sqrt [6]{2-x}}{\sqrt {x}} \, dx,x,1-\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} (1+\sec (c+d x))^{7/6}}\\ &=\frac {2 \sqrt [6]{2} F_1\left (\frac {1}{2};-\frac {2}{3},-\frac {1}{6};\frac {3}{2};1-\sec (c+d x),\frac {1}{2} (1-\sec (c+d x))\right ) (a+a \sec (c+d x))^{2/3} \tan (c+d x)}{d (1+\sec (c+d x))^{7/6}}\\ \end {align*}
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Mathematica [A] time = 8.33, size = 274, normalized size = 0.84 \[ \frac {(a (\sec (c+d x)+1))^{2/3} \left (\sqrt [3]{2} \tan \left (\frac {1}{2} (c+d x)\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {5}{4};\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )} \sqrt [3]{\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)}-5 \sqrt [3]{2} \tan ^3\left (\frac {1}{2} (c+d x)\right ) \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {7}{4};\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )} \sqrt [3]{\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)}-3 \left (\sin \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {3}{2} (c+d x)\right )\right ) \sec (c+d x) \sqrt [3]{\sec (c+d x)+1} \sec ^3\left (\frac {1}{2} (c+d x)\right )\right )}{8 d \sqrt [3]{\frac {1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{\frac {5}{3}}, x\right ) \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{\frac {5}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.30, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{\frac {5}{3}}\left (d x +c \right )\right ) \left (a +a \sec \left (d x +c \right )\right )^{\frac {2}{3}}\, dx \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {2}{3}} \sec \left (d x + c\right )^{\frac {5}{3}}\,{d x} \]
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 \[ \int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{2/3}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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