Optimal. Leaf size=50 \[ \frac {x \coth ^2(c+d x)}{\left (b \coth ^3(c+d x)\right )^{2/3}}-\frac {\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}} \]
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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} \[ \frac {x \coth ^2(c+d x)}{\left (b \coth ^3(c+d x)\right )^{2/3}}-\frac {\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\left (b \coth ^3(c+d x)\right )^{2/3}} \, dx &=\frac {\coth ^2(c+d x) \int \tanh ^2(c+d x) \, dx}{\left (b \coth ^3(c+d x)\right )^{2/3}}\\ &=-\frac {\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}}+\frac {\coth ^2(c+d x) \int 1 \, dx}{\left (b \coth ^3(c+d x)\right )^{2/3}}\\ &=-\frac {\coth (c+d x)}{d \left (b \coth ^3(c+d x)\right )^{2/3}}+\frac {x \coth ^2(c+d x)}{\left (b \coth ^3(c+d x)\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 40, normalized size = 0.80 \[ \frac {\coth (c+d x) \left (\tanh ^{-1}(\tanh (c+d x)) \coth (c+d x)-1\right )}{d \left (b \coth ^3(c+d x)\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 287, normalized size = 5.74 \[ -\frac {{\left (d x \cosh \left (d x + c\right )^{2} - {\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x\right )} \sinh \left (d x + c\right )^{2} + d x - {\left (d x \cosh \left (d x + c\right )^{2} + d x + 2\right )} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, {\left (d x \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - d x \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 2\right )} \left (\frac {b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (6 \, d x + 6 \, c\right )} - 3 \, e^{\left (4 \, d x + 4 \, c\right )} + 3 \, e^{\left (2 \, d x + 2 \, c\right )} - 1}\right )^{\frac {1}{3}}}{b d \cosh \left (d x + c\right )^{2} + {\left (b d e^{\left (2 \, d x + 2 \, c\right )} + b d\right )} \sinh \left (d x + c\right )^{2} + b d + {\left (b d \cosh \left (d x + c\right )^{2} + b d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (b d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + b d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \]
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 \[ \int \frac {1}{\left (b \coth \left (d x + c\right )^{3}\right )^{\frac {2}{3}}}\,{d x} \]
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 \[ \frac {\left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} x}{\left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {2}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {2+2 \,{\mathrm e}^{2 d x +2 c}}{\left (\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{3}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\right )^{\frac {2}{3}} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 37, normalized size = 0.74 \[ \frac {d x + c}{b^{\frac {2}{3}} d} - \frac {2}{{\left (b^{\frac {2}{3}} e^{\left (-2 \, d x - 2 \, c\right )} + b^{\frac {2}{3}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^3\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \coth ^{3}{\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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