Optimal. Leaf size=58 \[ \frac {3 \cosh (c+d x) \sqrt [3]{b \sinh (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-\sinh ^2(c+d x)\right )}{b d \sqrt {\cosh ^2(c+d x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2643} \[ \frac {3 \cosh (c+d x) \sqrt [3]{b \sinh (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-\sinh ^2(c+d x)\right )}{b d \sqrt {\cosh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin {align*} \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx &=\frac {3 \cosh (c+d x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-\sinh ^2(c+d x)\right ) \sqrt [3]{b \sinh (c+d x)}}{b d \sqrt {\cosh ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 55, normalized size = 0.95 \[ \frac {3 \sqrt {\cosh ^2(c+d x)} \tanh (c+d x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};-\sinh ^2(c+d x)\right )}{d (b \sinh (c+d x))^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b \sinh \left (d x + c\right )\right )^{\frac {1}{3}}}{b \sinh \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \sinh \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {2}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{2/3}} \,d x \]
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 \[ \int \frac {1}{\left (b \sinh {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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