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 = 0.0202028, antiderivative size = 31, normalized size of antiderivative = 1., 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 3658
Rule 3475
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*}
Mathematica [A] time = 0.0258951, 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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Maple [B] time = 0.092, size = 192, normalized size = 6.2 \begin{align*}{\frac{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) x}{1+{{\rm e}^{2\,dx+2\,c}}}\sqrt [3]{{\frac{b \left ( 1+{{\rm e}^{2\,dx+2\,c}} \right ) ^{3}}{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) ^{3}}}}}-2\,{\frac{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) \left ( dx+c \right ) }{ \left ( 1+{{\rm e}^{2\,dx+2\,c}} \right ) d}\sqrt [3]{{\frac{b \left ( 1+{{\rm e}^{2\,dx+2\,c}} \right ) ^{3}}{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) ^{3}}}}}+{\frac{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) \ln \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) }{ \left ( 1+{{\rm e}^{2\,dx+2\,c}} \right ) d}\sqrt [3]{{\frac{b \left ( 1+{{\rm e}^{2\,dx+2\,c}} \right ) ^{3}}{ \left ({{\rm e}^{2\,dx+2\,c}}-1 \right ) ^{3}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74058, size = 69, normalized size = 2.23 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13365, size = 363, normalized size = 11.71 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b \coth ^{3}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \coth \left (d x + c\right )^{3}\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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