Optimal. Leaf size=65 \[ \frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 65, 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} \[ \frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\left (b \coth ^2(c+d x)\right )^{3/2}} \, dx &=\frac {\coth (c+d x) \int \tanh ^3(c+d x) \, dx}{b \sqrt {b \coth ^2(c+d x)}}\\ &=-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}}+\frac {\coth (c+d x) \int \tanh (c+d x) \, dx}{b \sqrt {b \coth ^2(c+d x)}}\\ &=\frac {\coth (c+d x) \log (\cosh (c+d x))}{b d \sqrt {b \coth ^2(c+d x)}}-\frac {\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 48, normalized size = 0.74 \[ \frac {2 \coth (c+d x) \log (\cosh (c+d x))-\tanh (c+d x)}{2 b d \sqrt {b \coth ^2(c+d x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 817, normalized size = 12.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 104, normalized size = 1.60 \[ -\frac {\frac {d x + c}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{\sqrt {b} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )} - \frac {2 \, e^{\left (2 \, d x + 2 \, c\right )}}{\sqrt {b} {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2} \mathrm {sgn}\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 79, normalized size = 1.22 \[ -\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )-1\right ) \left (\coth ^{2}\left (d x +c \right )\right )+\ln \left (\coth \left (d x +c \right )+1\right ) \left (\coth ^{2}\left (d x +c \right )\right )-2 \ln \left (\coth \left (d x +c \right )\right ) \left (\coth ^{2}\left (d x +c \right )\right )+1\right )}{2 d \left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 84, normalized size = 1.29 \[ -\frac {2 \, \sqrt {b} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-4 \, d x - 4 \, c\right )} + b^{2}\right )} d} - \frac {d x + c}{b^{\frac {3}{2}} d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{\frac {3}{2}} d} \]
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 \[ \int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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