Optimal. Leaf size=78 \[ \frac {\left (a^2-b^2\right ) \coth (x)}{a^3}+\frac {b \coth ^2(x)}{2 a^2}-\frac {\coth ^3(x)}{3 a}+\frac {b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}-\frac {b \left (a^2-b^2\right ) \log (a+b \tanh (x))}{a^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3597, 908}
\begin {gather*} \frac {b \coth ^2(x)}{2 a^2}+\frac {b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}-\frac {b \left (a^2-b^2\right ) \log (a+b \tanh (x))}{a^4}+\frac {\left (a^2-b^2\right ) \coth (x)}{a^3}-\frac {\coth ^3(x)}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 908
Rule 3597
Rubi steps
\begin {align*} \int \frac {\text {csch}^4(x)}{a+b \tanh (x)} \, dx &=-\left (b \text {Subst}\left (\int \frac {-b^2+x^2}{x^4 (a+x)} \, dx,x,b \tanh (x)\right )\right )\\ &=-\left (b \text {Subst}\left (\int \left (-\frac {b^2}{a x^4}+\frac {b^2}{a^2 x^3}+\frac {a^2-b^2}{a^3 x^2}+\frac {-a^2+b^2}{a^4 x}+\frac {a^2-b^2}{a^4 (a+x)}\right ) \, dx,x,b \tanh (x)\right )\right )\\ &=\frac {\left (a^2-b^2\right ) \coth (x)}{a^3}+\frac {b \coth ^2(x)}{2 a^2}-\frac {\coth ^3(x)}{3 a}+\frac {b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}-\frac {b \left (a^2-b^2\right ) \log (a+b \tanh (x))}{a^4}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 70, normalized size = 0.90 \begin {gather*} \frac {3 a^2 b \text {csch}^2(x)-2 \coth (x) \left (-2 a^3+3 a b^2+a^3 \text {csch}^2(x)\right )+6 b \left (a^2-b^2\right ) (\log (\sinh (x))-\log (a \cosh (x)+b \sinh (x)))}{6 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 146, normalized size = 1.87
method | result | size |
risch | \(-\frac {2 \left (-3 a b \,{\mathrm e}^{4 x}+3 b^{2} {\mathrm e}^{4 x}+6 a^{2} {\mathrm e}^{2 x}+3 a b \,{\mathrm e}^{2 x}-6 b^{2} {\mathrm e}^{2 x}-2 a^{2}+3 b^{2}\right )}{3 a^{3} \left ({\mathrm e}^{2 x}-1\right )^{3}}-\frac {b \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{2}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{4}}+\frac {b \ln \left ({\mathrm e}^{2 x}-1\right )}{a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 x}-1\right )}{a^{4}}\) | \(143\) |
default | \(-\frac {\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) a^{2}}{3}-a b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-3 a^{2} \tanh \left (\frac {x}{2}\right )+4 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{3}}-\frac {2 b \left (\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 b \tanh \left (\frac {x}{2}\right )+a \right )}{a^{4}}-\frac {1}{24 a \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-3 a^{2}+4 b^{2}}{8 a^{3} \tanh \left (\frac {x}{2}\right )}+\frac {b}{8 a^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {b \left (a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{4}}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 161 vs.
\(2 (74) = 148\).
time = 0.29, size = 161, normalized size = 2.06 \begin {gather*} -\frac {2 \, {\left (2 \, a^{2} - 3 \, b^{2} - 3 \, {\left (2 \, a^{2} - a b - 2 \, b^{2}\right )} e^{\left (-2 \, x\right )} - 3 \, {\left (a b + b^{2}\right )} e^{\left (-4 \, x\right )}\right )}}{3 \, {\left (3 \, a^{3} e^{\left (-2 \, x\right )} - 3 \, a^{3} e^{\left (-4 \, x\right )} + a^{3} e^{\left (-6 \, x\right )} - a^{3}\right )}} - \frac {{\left (a^{2} b - b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{4}} + \frac {{\left (a^{2} b - b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{4}} + \frac {{\left (a^{2} b - b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{4}} \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. 912 vs.
\(2 (74) = 148\).
time = 0.36, size = 912, normalized size = 11.69 \begin {gather*} \frac {6 \, {\left (a^{2} b - a b^{2}\right )} \cosh \left (x\right )^{4} + 24 \, {\left (a^{2} b - a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + 6 \, {\left (a^{2} b - a b^{2}\right )} \sinh \left (x\right )^{4} + 4 \, a^{3} - 6 \, a b^{2} - 6 \, {\left (2 \, a^{3} + a^{2} b - 2 \, a b^{2}\right )} \cosh \left (x\right )^{2} - 6 \, {\left (2 \, a^{3} + a^{2} b - 2 \, a b^{2} - 6 \, {\left (a^{2} b - a b^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} - 3 \, {\left ({\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{6} + 6 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} + {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )^{6} - 3 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{4} - 3 \, {\left (a^{2} b - b^{3} - 5 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{3} - 3 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - a^{2} b + b^{3} + 3 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{4} + a^{2} b - b^{3} - 6 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 6 \, {\left ({\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{5} - 2 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 3 \, {\left ({\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{6} + 6 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{5} + {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )^{6} - 3 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{4} - 3 \, {\left (a^{2} b - b^{3} - 5 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{3} - 3 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - a^{2} b + b^{3} + 3 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{4} + a^{2} b - b^{3} - 6 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 6 \, {\left ({\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{5} - 2 \, {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{3} + {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 12 \, {\left (2 \, {\left (a^{2} b - a b^{2}\right )} \cosh \left (x\right )^{3} - {\left (2 \, a^{3} + a^{2} b - 2 \, a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{3 \, {\left (a^{4} \cosh \left (x\right )^{6} + 6 \, a^{4} \cosh \left (x\right ) \sinh \left (x\right )^{5} + a^{4} \sinh \left (x\right )^{6} - 3 \, a^{4} \cosh \left (x\right )^{4} + 3 \, a^{4} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a^{4} \cosh \left (x\right )^{2} - a^{4}\right )} \sinh \left (x\right )^{4} - a^{4} + 4 \, {\left (5 \, a^{4} \cosh \left (x\right )^{3} - 3 \, a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, a^{4} \cosh \left (x\right )^{4} - 6 \, a^{4} \cosh \left (x\right )^{2} + a^{4}\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a^{4} \cosh \left (x\right )^{5} - 2 \, a^{4} \cosh \left (x\right )^{3} + a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )\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 {\operatorname {csch}^{4}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs.
\(2 (74) = 148\).
time = 0.42, size = 202, normalized size = 2.59 \begin {gather*} -\frac {{\left (a^{3} b + a^{2} b^{2} - a b^{3} - b^{4}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{5} + a^{4} b} + \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{a^{4}} - \frac {11 \, a^{2} b e^{\left (6 \, x\right )} - 11 \, b^{3} e^{\left (6 \, x\right )} - 45 \, a^{2} b e^{\left (4 \, x\right )} + 12 \, a b^{2} e^{\left (4 \, x\right )} + 33 \, b^{3} e^{\left (4 \, x\right )} + 24 \, a^{3} e^{\left (2 \, x\right )} + 45 \, a^{2} b e^{\left (2 \, x\right )} - 24 \, a b^{2} e^{\left (2 \, x\right )} - 33 \, b^{3} e^{\left (2 \, x\right )} - 8 \, a^{3} - 11 \, a^{2} b + 12 \, a b^{2} + 11 \, b^{3}}{6 \, a^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 123, normalized size = 1.58 \begin {gather*} \frac {2\,b\,\left (a-b\right )}{a^3\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {2\,\left (2\,a-b\right )}{a^2\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {8}{3\,a\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {b\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a+b\right )\,\left (a-b\right )}{a^4}+\frac {b\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a+b\right )\,\left (a-b\right )}{a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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