Optimal. Leaf size=50 \[ x \tanh ^2(c+d x) \left (b \coth ^3(c+d x)\right )^{2/3}-\frac {\tanh (c+d x) \left (b \coth ^3(c+d x)\right )^{2/3}}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3658, 3473, 8} \[ x \tanh ^2(c+d x) \left (b \coth ^3(c+d x)\right )^{2/3}-\frac {\tanh (c+d x) \left (b \coth ^3(c+d x)\right )^{2/3}}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \left (b \coth ^3(c+d x)\right )^{2/3} \, dx &=\left (\left (b \coth ^3(c+d x)\right )^{2/3} \tanh ^2(c+d x)\right ) \int \coth ^2(c+d x) \, dx\\ &=-\frac {\left (b \coth ^3(c+d x)\right )^{2/3} \tanh (c+d x)}{d}+\left (\left (b \coth ^3(c+d x)\right )^{2/3} \tanh ^2(c+d x)\right ) \int 1 \, dx\\ &=-\frac {\left (b \coth ^3(c+d x)\right )^{2/3} \tanh (c+d x)}{d}+x \left (b \coth ^3(c+d x)\right )^{2/3} \tanh ^2(c+d x)\\ \end {align*}
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Mathematica [C] time = 0.03, size = 41, normalized size = 0.82 \[ -\frac {\tanh (c+d x) \left (b \coth ^3(c+d x)\right )^{2/3} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 392, normalized size = 7.84 \[ \frac {{\left (d x \cosh \left (d x + c\right )^{2} + {\left (d x e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d x e^{\left (2 \, d x + 2 \, c\right )} + d x\right )} \sinh \left (d x + c\right )^{2} - d x + {\left (d x \cosh \left (d x + c\right )^{2} - d x - 2\right )} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, {\left (d x \cosh \left (d x + c\right )^{2} - d x - 2\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d x \cosh \left (d x + c\right ) e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 2\right )} \left (\frac {b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {2}{3}}}{d \cosh \left (d x + c\right )^{2} + {\left (d e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d e^{\left (2 \, d x + 2 \, c\right )} + d\right )} \sinh \left (d x + c\right )^{2} + {\left (d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d \cosh \left (d x + c\right ) e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - d} \]
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 \[ \int \left (b \coth \left (d x + c\right )^{3}\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 119, normalized size = 2.38 \[ \frac {\left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {2}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} x}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}-\frac {2 \left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {2}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 34, normalized size = 0.68 \[ \frac {{\left (d x + c\right )} b^{\frac {2}{3}}}{d} + \frac {2 \, b^{\frac {2}{3}}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
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 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.76, size = 90, normalized size = 1.80 \[ \begin {cases} \tilde {\infty } b^{\frac {2}{3}} x & \text {for}\: c = \log {\left (- e^{- d x} \right )} \vee c = \log {\left (e^{- d x} \right )} \\x \left (b \coth ^{3}{\relax (c )}\right )^{\frac {2}{3}} & \text {for}\: d = 0 \\b^{\frac {2}{3}} x \left (\frac {1}{\tanh ^{3}{\left (c + d x \right )}}\right )^{\frac {2}{3}} \tanh ^{2}{\left (c + d x \right )} - \frac {b^{\frac {2}{3}} \left (\frac {1}{\tanh ^{3}{\left (c + d x \right )}}\right )^{\frac {2}{3}} \tanh {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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