Optimal. Leaf size=66 \[ -\frac {\sqrt [6]{2} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2652, 2651} \[ -\frac {\sqrt [6]{2} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2651
Rule 2652
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{a+a \sin (c+d x)}} \, dx &=\frac {\sqrt [3]{1+\sin (c+d x)} \int \frac {1}{\sqrt [3]{1+\sin (c+d x)}} \, dx}{\sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac {\sqrt [6]{2} \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 70, normalized size = 1.06 \[ \frac {3 \sqrt {2} \cos (c+d x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\sin ^2\left (\frac {1}{4} (2 c+2 d x+\pi )\right )\right )}{d \sqrt {1-\sin (c+d x)} \sqrt [3]{a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +a \sin \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{\sqrt [3]{a \sin {\left (c + d x \right )} + a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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