Optimal. Leaf size=31 \[ \frac {\sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x)) \tanh (c+d x)}{d} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3739, 3556}
\begin {gather*} \frac {\tanh (c+d x) \sqrt {b \coth ^2(c+d x)} \log (\sinh (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3739
Rubi steps
\begin {align*} \int \sqrt {b \coth ^2(c+d x)} \, dx &=\left (\sqrt {b \coth ^2(c+d x)} \tanh (c+d x)\right ) \int \coth (c+d x) \, dx\\ &=\frac {\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 = 0.03, size = 39, normalized size = 1.26 \begin {gather*} \frac {\sqrt {b \coth ^2(c+d x)} (\log (\cosh (c+d x))+\log (\tanh (c+d x))) \tanh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.64, size = 45, normalized size = 1.45
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \left (\coth ^{2}\left (d x +c \right )\right )}\, \left (\ln \left (\coth \left (d x +c \right )-1\right )+\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )}\) | \(45\) |
default | \(-\frac {\sqrt {b \left (\coth ^{2}\left (d x +c \right )\right )}\, \left (\ln \left (\coth \left (d x +c \right )-1\right )+\ln \left (\coth \left (d x +c \right )+1\right )\right )}{2 d \coth \left (d x +c \right )}\) | \(45\) |
risch | \(\frac {\sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right ) x}{1+{\mathrm e}^{2 d x +2 c}}-\frac {2 \sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right ) \left (d x +c \right )}{\left (1+{\mathrm e}^{2 d x +2 c}\right ) d}+\frac {\sqrt {\frac {b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}}\, \left ({\mathrm e}^{2 d x +2 c}-1\right ) \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{\left (1+{\mathrm e}^{2 d x +2 c}\right ) d}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 54, normalized size = 1.74 \begin {gather*} -\frac {{\left (d x + c\right )} \sqrt {b}}{d} - \frac {\sqrt {b} \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\sqrt {b} \log \left (e^{\left (-d x - c\right )} - 1\right )}{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. 125 vs.
\(2 (29) = 58\).
time = 0.43, size = 125, normalized size = 4.03 \begin {gather*} -\frac {{\left (d x e^{\left (2 \, d x + 2 \, c\right )} - d x - {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\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}}}{d e^{\left (2 \, d x + 2 \, c\right )} + d} \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 \sqrt {b \coth ^{2}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 54, normalized size = 1.74 \begin {gather*} -\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 )\right )} \sqrt {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 = -0.03 \begin {gather*} \int \sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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