Optimal. Leaf size=327 \[ -\frac {3 a \sec ^{\frac {5}{3}}(c+d x) \sin (c+d x)}{2 d \sqrt [3]{a (1+\sec (c+d x))}}+\frac {9 \sec ^{\frac {2}{3}}(c+d x) (a (1+\sec (c+d x)))^{2/3} \sin (c+d x)}{4 d}-\frac {9 (a (1+\sec (c+d x)))^{2/3} \tan (c+d x)}{4 d \sqrt [3]{\frac {1}{1+\cos (c+d x)}} (1+\sec (c+d x))^{7/3}}+\frac {\, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {5}{4};\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )} (a (1+\sec (c+d x)))^{2/3} \tan (c+d x)}{8 d \sqrt [3]{\frac {1}{1+\cos (c+d x)}} (1+\sec (c+d x))^{4/3}}-\frac {5 \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {7}{4};\tan ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt [3]{\cos (c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right )} (a (1+\sec (c+d x)))^{2/3} \tan ^3(c+d x)}{8 d \sqrt [3]{\frac {1}{1+\cos (c+d x)}} (1+\sec (c+d x))^{10/3}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.09, 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 = {3913, 3910,
138} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 3910
Rule 3913
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 ) \text {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 = 9.73, size = 274, normalized size = 0.84 \begin {gather*} \frac {(a (1+\sec (c+d x)))^{2/3} \left (-3 \sec ^3\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sqrt [3]{1+\sec (c+d x)} \left (\sin \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {3}{2} (c+d x)\right )\right )+\sqrt [3]{2} \, _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)} \tan \left (\frac {1}{2} (c+d x)\right )-5 \sqrt [3]{2} \, _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)} \tan ^3\left (\frac {1}{2} (c+d x)\right )\right )}{8 d \sqrt [3]{\frac {1}{1+\cos (c+d x)}} (1+\sec (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, 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 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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