Optimal. Leaf size=31 \[ \frac {\tanh (c+d x) \sqrt [3]{b \coth ^3(c+d x)} \log (\sinh (c+d x))}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 31, 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 = {3658, 3475} \[ \frac {\tanh (c+d x) \sqrt [3]{b \coth ^3(c+d x)} \log (\sinh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \sqrt [3]{b \coth ^3(c+d x)} \, dx &=\left (\sqrt [3]{b \coth ^3(c+d x)} \tanh (c+d x)\right ) \int \coth (c+d x) \, dx\\ &=\frac {\sqrt [3]{b \coth ^3(c+d x)} \log (\sinh (c+d x)) \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 39, normalized size = 1.26 \[ \frac {\tanh (c+d x) \sqrt [3]{b \coth ^3(c+d x)} (\log (\tanh (c+d x))+\log (\cosh (c+d x)))}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 148, normalized size = 4.77 \[ -\frac {{\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x - {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )\right )} \left (\frac {b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {1}{3}}}{d e^{\left (2 \, d x + 2 \, c\right )} + d} \]
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 \[ \int \left (b \coth \left (d x + c\right )^{3}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 192, normalized size = 6.19 \[ \frac {\left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {1}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right ) x}{1+{\mathrm e}^{2 d x +2 c}}-\frac {2 \left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {1}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (d x +c \right )}{\left (1+{\mathrm e}^{2 d x +2 c}\right ) d}+\frac {\left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {1}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right ) \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{\left (1+{\mathrm e}^{2 d x +2 c}\right ) d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 51, normalized size = 1.65 \[ \frac {{\left (d x + c\right )} b^{\frac {1}{3}}}{d} + \frac {b^{\frac {1}{3}} \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {b^{\frac {1}{3}} \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{1/3} \,d x \]
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 \[ \int \sqrt [3]{b \coth ^{3}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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