Optimal. Leaf size=78 \[ -\frac {3 \tan (c+d x) F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};\sec (c+d x),-\sec (c+d x)\right ) (e \sec (c+d x))^{2/3}}{2 d \sqrt {1-\sec (c+d x)} \sqrt {a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.16, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3828, 3827, 130, 510} \[ -\frac {3 \tan (c+d x) F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};\sec (c+d x),-\sec (c+d x)\right ) (e \sec (c+d x))^{2/3}}{2 d \sqrt {1-\sec (c+d x)} \sqrt {a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 3827
Rule 3828
Rubi steps
\begin {align*} \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+a \sec (c+d x)}} \, dx &=\frac {\sqrt {1+\sec (c+d x)} \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {1+\sec (c+d x)}} \, dx}{\sqrt {a+a \sec (c+d x)}}\\ &=-\frac {(e \tan (c+d x)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt [3]{e x} (1+x)} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=-\frac {(3 \tan (c+d x)) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-\frac {x^3}{e}} \left (1+\frac {x^3}{e}\right )} \, dx,x,\sqrt [3]{e \sec (c+d x)}\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ &=-\frac {3 F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};\sec (c+d x),-\sec (c+d x)\right ) (e \sec (c+d x))^{2/3} \tan (c+d x)}{2 d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [B] time = 9.25, size = 750, normalized size = 9.62 \[ \frac {90 \cos ^2(c+d x) \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sec (c+d x)+1)} F_1\left (\frac {1}{2};\frac {1}{6},\frac {1}{3};\frac {3}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\tan ^2\left (\frac {1}{2} (c+d x)\right ) \left (F_1\left (\frac {3}{2};\frac {7}{6},\frac {1}{3};\frac {5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 F_1\left (\frac {3}{2};\frac {1}{6},\frac {4}{3};\frac {5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )+9 F_1\left (\frac {1}{2};\frac {1}{6},\frac {1}{3};\frac {3}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) (e \sec (c+d x))^{2/3}}{a d \left (270 \cos ^2\left (\frac {1}{2} (c+d x)\right ) (2 \cos (c+d x)+1) F_1\left (\frac {1}{2};\frac {1}{6},\frac {1}{3};\frac {3}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ){}^2+10 \cos (c+d x) \tan ^4\left (\frac {1}{2} (c+d x)\right ) \left (F_1\left (\frac {3}{2};\frac {7}{6},\frac {1}{3};\frac {5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 F_1\left (\frac {3}{2};\frac {1}{6},\frac {4}{3};\frac {5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ){}^2+3 \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {1}{2};\frac {1}{6},\frac {1}{3};\frac {3}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (-10 (-9 \cos (c+d x)+\cos (2 (c+d x))+2) \cos ^2\left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {3}{2};\frac {1}{6},\frac {4}{3};\frac {5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+5 (-9 \cos (c+d x)+\cos (2 (c+d x))+2) \cos ^2\left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {3}{2};\frac {7}{6},\frac {1}{3};\frac {5}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-6 \sin ^2\left (\frac {1}{2} (c+d x)\right ) \cos (c+d x) \left (16 F_1\left (\frac {5}{2};\frac {1}{6},\frac {7}{3};\frac {7}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-4 F_1\left (\frac {5}{2};\frac {7}{6},\frac {4}{3};\frac {7}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+7 F_1\left (\frac {5}{2};\frac {13}{6},\frac {1}{3};\frac {7}{2};\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [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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maple [F] time = 1.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sec \left (d x +c \right )\right )^{\frac {2}{3}}}{\sqrt {a +a \sec \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}}{\sqrt {a \sec \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{2/3}}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {\left (e \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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