Optimal. Leaf size=76 \[ -\frac {3 \tan (c+d x) F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};\sec (c+d x),-\sec (c+d x)\right ) \sqrt [3]{e \sec (c+d x)}}{d \sqrt {1-\sec (c+d x)} \sqrt {a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.14, antiderivative size = 76, 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, 429} \[ -\frac {3 \tan (c+d x) F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};\sec (c+d x),-\sec (c+d x)\right ) \sqrt [3]{e \sec (c+d x)}}{d \sqrt {1-\sec (c+d x)} \sqrt {a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 429
Rule 3827
Rule 3828
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx &=\frac {\sqrt {1+\sec (c+d x)} \int \frac {\sqrt [3]{e \sec (c+d x)}}{\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} (e x)^{2/3} (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 {1}{\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 {1}{3};\frac {1}{2},1;\frac {4}{3};\sec (c+d x),-\sec (c+d x)\right ) \sqrt [3]{e \sec (c+d x)} \tan (c+d x)}{d \sqrt {1-\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [B] time = 8.12, size = 749, normalized size = 9.86 \[ \frac {720 e \sin \left (\frac {1}{2} (c+d x)\right ) \cos \left (\frac {1}{2} (c+d x)\right ) (\cos (c+d x)+1)^2 F_1\left (\frac {1}{2};-\frac {1}{6},\frac {2}{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 (9 F_1\left (\frac {1}{2};-\frac {1}{6},\frac {2}{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 )-\tan ^2\left (\frac {1}{2} (c+d x)\right ) \left (4 F_1\left (\frac {3}{2};-\frac {1}{6},\frac {5}{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 )+F_1\left (\frac {3}{2};\frac {5}{6},\frac {2}{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 )\right )}{d \sqrt {a (\sec (c+d x)+1)} (e \sec (c+d x))^{2/3} \left (4320 (4 \cos (c+d x)-1) \cos ^6\left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {1}{2};-\frac {1}{6},\frac {2}{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+160 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \cos (c+d x) \left (4 F_1\left (\frac {3}{2};-\frac {1}{6},\frac {5}{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 )+F_1\left (\frac {3}{2};\frac {5}{6},\frac {2}{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+12 \sin ^2\left (\frac {1}{2} (c+d x)\right ) F_1\left (\frac {1}{2};-\frac {1}{6},\frac {2}{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 (20 (14 \cos (c+d x)+5 \cos (2 (c+d x))-2 \cos (3 (c+d x))+7) F_1\left (\frac {3}{2};-\frac {1}{6},\frac {5}{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 (14 \cos (c+d x)+5 \cos (2 (c+d x))-2 \cos (3 (c+d x))+7) F_1\left (\frac {3}{2};\frac {5}{6},\frac {2}{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 )-24 \sin ^2\left (\frac {1}{2} (c+d x)\right ) \cos (c+d x) \left (40 F_1\left (\frac {5}{2};-\frac {1}{6},\frac {8}{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 )+8 F_1\left (\frac {5}{2};\frac {5}{6},\frac {5}{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 )-5 F_1\left (\frac {5}{2};\frac {11}{6},\frac {2}{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 = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sec \left (d x +c \right )\right )^{\frac {1}{3}}}{\sqrt {a +a \sec \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {1}{3}}}{\sqrt {a \sec \left (d x + c\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{1/3}}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
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 \[ \int \frac {\sqrt [3]{e \sec {\left (c + d x \right )}}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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