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 = 0.03, 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 = {3739, 3554,
3556} \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 3554
Rule 3556
Rule 3739
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 = 0.10, size = 48, normalized size = 0.74 \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.41, size = 79, normalized size = 1.22
method | result | size |
derivativedivides | \(-\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )+1\right ) \left (\coth ^{2}\left (d x +c \right )\right )+\ln \left (\coth \left (d x +c \right )-1\right ) \left (\coth ^{2}\left (d x +c \right )\right )-2 \ln \left (\coth \left (d x +c \right )\right ) \left (\coth ^{2}\left (d x +c \right )\right )+1\right )}{2 d \left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}}}\) | \(79\) |
default | \(-\frac {\coth \left (d x +c \right ) \left (\ln \left (\coth \left (d x +c \right )+1\right ) \left (\coth ^{2}\left (d x +c \right )\right )+\ln \left (\coth \left (d x +c \right )-1\right ) \left (\coth ^{2}\left (d x +c \right )\right )-2 \ln \left (\coth \left (d x +c \right )\right ) \left (\coth ^{2}\left (d x +c \right )\right )+1\right )}{2 d \left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}}}\) | \(79\) |
risch | \(\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) x}{b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}}-\frac {2 \left (1+{\mathrm e}^{2 d x +2 c}\right ) \left (d x +c \right )}{b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, d}+\frac {2 \,{\mathrm e}^{2 d x +2 c}}{b \left (1+{\mathrm e}^{2 d x +2 c}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, d}+\frac {\left (1+{\mathrm e}^{2 d x +2 c}\right ) \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{b \left ({\mathrm e}^{2 d x +2 c}-1\right ) \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, d}\) | \(274\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 84, normalized size = 1.29 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 817 vs.
\(2 (59) = 118\).
time = 0.46, size = 817, normalized size = 12.57 \begin {gather*} \frac {{\left (d x \cosh \left (d x + c\right )^{4} - {\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x\right )} \sinh \left (d x + c\right )^{4} - 4 \, {\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 )^{3} + 2 \, {\left (d x - 1\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d x \cosh \left (d x + c\right )^{2} + d x - {\left (3 \, d x \cosh \left (d x + c\right )^{2} + d x - 1\right )} e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \sinh \left (d x + c\right )^{2} + d x - {\left (d x \cosh \left (d x + c\right )^{4} + 2 \, {\left (d x - 1\right )} \cosh \left (d x + c\right )^{2} + d x\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left ({\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \sinh \left (d x + c\right )^{4} - \cosh \left (d x + c\right )^{4} + 4 \, {\left (\cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} - \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 2 \, {\left (3 \, \cosh \left (d x + c\right )^{2} - {\left (3 \, \cosh \left (d x + c\right )^{2} + 1\right )} e^{\left (2 \, d x + 2 \, c\right )} + 1\right )} \sinh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{4} + 2 \, \cosh \left (d x + c\right )^{2} + 1\right )} e^{\left (2 \, d x + 2 \, c\right )} - 4 \, {\left (\cosh \left (d x + c\right )^{3} - {\left (\cosh \left (d x + c\right )^{3} + \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )} + \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 1\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (d x \cosh \left (d x + c\right )^{3} + {\left (d x - 1\right )} \cosh \left (d x + c\right ) - {\left (d x \cosh \left (d x + c\right )^{3} + {\left (d x - 1\right )} \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b}{e^{\left (4 \, d x + 4 \, c\right )} - 2 \, e^{\left (2 \, d x + 2 \, c\right )} + 1}}}{b^{2} d \cosh \left (d x + c\right )^{4} + 2 \, b^{2} d \cosh \left (d x + c\right )^{2} + {\left (b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (b^{2} d \cosh \left (d x + c\right ) e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + b^{2} d + 2 \, {\left (3 \, b^{2} d \cosh \left (d x + c\right )^{2} + b^{2} d + {\left (3 \, b^{2} d \cosh \left (d x + c\right )^{2} + b^{2} d\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right )^{2} + {\left (b^{2} d \cosh \left (d x + c\right )^{4} + 2 \, b^{2} d \cosh \left (d x + c\right )^{2} + b^{2} d\right )} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, {\left (b^{2} d \cosh \left (d x + c\right )^{3} + b^{2} d \cosh \left (d x + c\right ) + {\left (b^{2} d \cosh \left (d x + c\right )^{3} + b^{2} d \cosh \left (d x + c\right )\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \sinh \left (d x + c\right )} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 104, normalized size = 1.60 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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