Optimal. Leaf size=64 \[ -\frac {b x}{a^2-b^2}-\frac {\coth (x)}{b}+\frac {a^3 \log (a+b \coth (x))}{b^2 \left (a^2-b^2\right )}+\frac {a \log (\sinh (x))}{a^2-b^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3647, 3707,
3698, 31, 3556} \begin {gather*} -\frac {b x}{a^2-b^2}+\frac {a \log (\sinh (x))}{a^2-b^2}+\frac {a^3 \log (a+b \coth (x))}{b^2 \left (a^2-b^2\right )}-\frac {\coth (x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 3556
Rule 3647
Rule 3698
Rule 3707
Rubi steps
\begin {align*} \int \frac {\coth ^3(x)}{a+b \coth (x)} \, dx &=-\frac {\coth (x)}{b}-\frac {\int \frac {-a-b \coth (x)+a \coth ^2(x)}{a+b \coth (x)} \, dx}{b}\\ &=-\frac {b x}{a^2-b^2}-\frac {\coth (x)}{b}+\frac {a \int \coth (x) \, dx}{a^2-b^2}+\frac {a^3 \int \frac {1-\coth ^2(x)}{a+b \coth (x)} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {b x}{a^2-b^2}-\frac {\coth (x)}{b}+\frac {a \log (\sinh (x))}{a^2-b^2}+\frac {a^3 \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \coth (x)\right )}{b^2 \left (a^2-b^2\right )}\\ &=-\frac {b x}{a^2-b^2}-\frac {\coth (x)}{b}+\frac {a^3 \log (a+b \coth (x))}{b^2 \left (a^2-b^2\right )}+\frac {a \log (\sinh (x))}{a^2-b^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 64, normalized size = 1.00 \begin {gather*} \frac {b^3 x+b \left (a^2-b^2\right ) \coth (x)+a \left (a^2-b^2\right ) \log (\sinh (x))-a^3 \log (b \cosh (x)+a \sinh (x))}{b^2 (-a+b) (a+b)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 67, normalized size = 1.05
method | result | size |
derivativedivides | \(-\frac {\coth \left (x \right )}{b}+\frac {a^{3} \ln \left (a +b \coth \left (x \right )\right )}{b^{2} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (1+\coth \left (x \right )\right )}{2 a -2 b}-\frac {\ln \left (\coth \left (x \right )-1\right )}{2 b +2 a}\) | \(67\) |
default | \(-\frac {\coth \left (x \right )}{b}+\frac {a^{3} \ln \left (a +b \coth \left (x \right )\right )}{b^{2} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (1+\coth \left (x \right )\right )}{2 a -2 b}-\frac {\ln \left (\coth \left (x \right )-1\right )}{2 b +2 a}\) | \(67\) |
risch | \(\frac {x}{a +b}+\frac {2 a x}{b^{2}}-\frac {2 a^{3} x}{b^{2} \left (a^{2}-b^{2}\right )}-\frac {2}{b \left ({\mathrm e}^{2 x}-1\right )}-\frac {a \ln \left ({\mathrm e}^{2 x}-1\right )}{b^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{b^{2} \left (a^{2}-b^{2}\right )}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 82, normalized size = 1.28 \begin {gather*} \frac {a^{3} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{2} b^{2} - b^{4}} + \frac {x}{a + b} - \frac {a \log \left (e^{\left (-x\right )} + 1\right )}{b^{2}} - \frac {a \log \left (e^{\left (-x\right )} - 1\right )}{b^{2}} + \frac {2}{b e^{\left (-2 \, x\right )} - b} \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. 271 vs.
\(2 (64) = 128\).
time = 0.38, size = 271, normalized size = 4.23 \begin {gather*} \frac {{\left (a b^{2} + b^{3}\right )} x \cosh \left (x\right )^{2} + 2 \, {\left (a b^{2} + b^{3}\right )} x \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a b^{2} + b^{3}\right )} x \sinh \left (x\right )^{2} + 2 \, a^{2} b - 2 \, b^{3} - {\left (a b^{2} + b^{3}\right )} x - {\left (a^{3} \cosh \left (x\right )^{2} + 2 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right ) + a^{3} \sinh \left (x\right )^{2} - a^{3}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left (a^{3} - a b^{2} - {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{3} - a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{2} b^{2} - b^{4} - {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) - {\left (a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 636 vs.
\(2 (49) = 98\).
time = 1.32, size = 636, normalized size = 9.94 \begin {gather*} \begin {cases} \tilde {\infty } \left (x - \frac {1}{\tanh {\left (x \right )}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {5 x \tanh ^{2}{\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} - 2 b \tanh {\left (x \right )}} - \frac {5 x \tanh {\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} - 2 b \tanh {\left (x \right )}} - \frac {2 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh ^{2}{\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} - 2 b \tanh {\left (x \right )}} + \frac {2 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh {\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} - 2 b \tanh {\left (x \right )}} + \frac {2 \log {\left (\tanh {\left (x \right )} \right )} \tanh ^{2}{\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} - 2 b \tanh {\left (x \right )}} - \frac {2 \log {\left (\tanh {\left (x \right )} \right )} \tanh {\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} - 2 b \tanh {\left (x \right )}} - \frac {3 \tanh {\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} - 2 b \tanh {\left (x \right )}} + \frac {2}{2 b \tanh ^{2}{\left (x \right )} - 2 b \tanh {\left (x \right )}} & \text {for}\: a = - b \\\frac {x \tanh ^{2}{\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} + 2 b \tanh {\left (x \right )}} + \frac {x \tanh {\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} + 2 b \tanh {\left (x \right )}} + \frac {2 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh ^{2}{\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} + 2 b \tanh {\left (x \right )}} + \frac {2 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh {\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} + 2 b \tanh {\left (x \right )}} - \frac {2 \log {\left (\tanh {\left (x \right )} \right )} \tanh ^{2}{\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} + 2 b \tanh {\left (x \right )}} - \frac {2 \log {\left (\tanh {\left (x \right )} \right )} \tanh {\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} + 2 b \tanh {\left (x \right )}} - \frac {3 \tanh {\left (x \right )}}{2 b \tanh ^{2}{\left (x \right )} + 2 b \tanh {\left (x \right )}} - \frac {2}{2 b \tanh ^{2}{\left (x \right )} + 2 b \tanh {\left (x \right )}} & \text {for}\: a = b \\\frac {x - \log {\left (\tanh {\left (x \right )} + 1 \right )} + \log {\left (\tanh {\left (x \right )} \right )} - \frac {1}{2 \tanh ^{2}{\left (x \right )}}}{a} & \text {for}\: b = 0 \\\frac {x - \frac {1}{\tanh {\left (x \right )}}}{b} & \text {for}\: a = 0 \\\frac {a^{3} \log {\left (\tanh {\left (x \right )} + \frac {b}{a} \right )} \tanh {\left (x \right )}}{a^{2} b^{2} \tanh {\left (x \right )} - b^{4} \tanh {\left (x \right )}} - \frac {a^{3} \log {\left (\tanh {\left (x \right )} \right )} \tanh {\left (x \right )}}{a^{2} b^{2} \tanh {\left (x \right )} - b^{4} \tanh {\left (x \right )}} - \frac {a^{2} b}{a^{2} b^{2} \tanh {\left (x \right )} - b^{4} \tanh {\left (x \right )}} + \frac {a b^{2} x \tanh {\left (x \right )}}{a^{2} b^{2} \tanh {\left (x \right )} - b^{4} \tanh {\left (x \right )}} - \frac {a b^{2} \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh {\left (x \right )}}{a^{2} b^{2} \tanh {\left (x \right )} - b^{4} \tanh {\left (x \right )}} + \frac {a b^{2} \log {\left (\tanh {\left (x \right )} \right )} \tanh {\left (x \right )}}{a^{2} b^{2} \tanh {\left (x \right )} - b^{4} \tanh {\left (x \right )}} - \frac {b^{3} x \tanh {\left (x \right )}}{a^{2} b^{2} \tanh {\left (x \right )} - b^{4} \tanh {\left (x \right )}} + \frac {b^{3}}{a^{2} b^{2} \tanh {\left (x \right )} - b^{4} \tanh {\left (x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 76, normalized size = 1.19 \begin {gather*} \frac {a^{3} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a^{2} b^{2} - b^{4}} - \frac {x}{a - b} - \frac {a \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{b^{2}} - \frac {2}{b {\left (e^{\left (2 \, x\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.51, size = 74, normalized size = 1.16 \begin {gather*} -\frac {2}{b\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {x}{a-b}-\frac {a^3\,\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{b^4-a^2\,b^2}-\frac {a\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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