Optimal. Leaf size=61 \[ \frac {b \tanh (c+d x) \sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x))}{d}-\frac {b \coth (c+d x) \sqrt {b \coth ^2(c+d x)}}{2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 61, 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, 3475} \[ \frac {b \tanh (c+d x) \sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x))}{d}-\frac {b \coth (c+d x) \sqrt {b \coth ^2(c+d x)}}{2 d} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \left (b \coth ^2(c+d x)\right )^{3/2} \, dx &=\left (b \sqrt {b \coth ^2(c+d x)} \tanh (c+d x)\right ) \int \coth ^3(c+d x) \, dx\\ &=-\frac {b \coth (c+d x) \sqrt {b \coth ^2(c+d x)}}{2 d}+\left (b \sqrt {b \coth ^2(c+d x)} \tanh (c+d x)\right ) \int \coth (c+d x) \, dx\\ &=-\frac {b \coth (c+d x) \sqrt {b \coth ^2(c+d x)}}{2 d}+\frac {b \sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x)) \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 56, normalized size = 0.92 \[ -\frac {\tanh ^3(c+d x) \left (b \coth ^2(c+d x)\right )^{3/2} \left (\coth ^2(c+d x)-2 \log (\tanh (c+d x))-2 \log (\cosh (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 823, normalized size = 13.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 90, normalized size = 1.48 \[ -\frac {{\left ({\left (d x + c\right )} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) - \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right ) + \frac {2 \, e^{\left (2 \, d x + 2 \, c\right )} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}\right )} b^{\frac {3}{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 53, normalized size = 0.87 \[ -\frac {\left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}} \left (\coth ^{2}\left (d x +c \right )+\ln \left (\coth \left (d x +c \right )-1\right )+\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 97, normalized size = 1.59 \[ -\frac {{\left (d x + c\right )} b^{\frac {3}{2}}}{d} - \frac {b^{\frac {3}{2}} \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {b^{\frac {3}{2}} \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, b^{\frac {3}{2}} e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \]
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 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{3/2} \,d x \]
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 \[ \int \left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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