Optimal. Leaf size=79 \[ -\frac {b \left (a^2-b^2\right ) \log (a+b \coth (x))}{a^4}-\frac {b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}+\frac {\left (a^2-b^2\right ) \tanh (x)}{a^3}+\frac {b \tanh ^2(x)}{2 a^2}-\frac {\tanh ^3(x)}{3 a} \]
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Rubi [A]
time = 0.07, antiderivative size = 79, 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 \tanh ^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 \coth (x))}{a^4}+\frac {\left (a^2-b^2\right ) \tanh (x)}{a^3}-\frac {\tanh ^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 {sech}^4(x)}{a+b \coth (x)} \, dx &=-\left (b \text {Subst}\left (\int \frac {-b^2+x^2}{x^4 (a+x)} \, dx,x,b \coth (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 \coth (x)\right )\right )\\ &=-\frac {b \left (a^2-b^2\right ) \log (a+b \coth (x))}{a^4}-\frac {b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}+\frac {\left (a^2-b^2\right ) \tanh (x)}{a^3}+\frac {b \tanh ^2(x)}{2 a^2}-\frac {\tanh ^3(x)}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 68, normalized size = 0.86 \begin {gather*} \frac {-6 b \left (-a^2+b^2\right ) (\log (\cosh (x))-\log (b \cosh (x)+a \sinh (x)))+\left (4 a^3-6 a b^2\right ) \tanh (x)+a^2 \text {sech}^2(x) (-3 b+2 a \tanh (x))}{6 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 151, normalized size = 1.91
method | result | size |
risch | \(-\frac {2 \left (3 a b \,{\mathrm e}^{4 x}-3 b^{2} {\mathrm e}^{4 x}+6 a^{2} {\mathrm e}^{2 x}+3 a b \,{\mathrm e}^{2 x}-6 b^{2} {\mathrm e}^{2 x}+2 a^{2}-3 b^{2}\right )}{3 a^{3} \left (1+{\mathrm e}^{2 x}\right )^{3}}-\frac {b \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{a^{2}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{a^{4}}+\frac {b \ln \left (1+{\mathrm e}^{2 x}\right )}{a^{2}}-\frac {b^{3} \ln \left (1+{\mathrm e}^{2 x}\right )}{a^{4}}\) | \(145\) |
default | \(-\frac {b \left (a^{2}-b^{2}\right ) \ln \left (b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 a \tanh \left (\frac {x}{2}\right )+b \right )}{a^{4}}-\frac {2 \left (\frac {\left (-a^{3}+a \,b^{2}\right ) \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )-a^{2} b \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )+\left (-\frac {2}{3} a^{3}+2 a \,b^{2}\right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )-a^{2} b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+\left (-a^{3}+a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{\left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-\frac {b \left (a^{2}-b^{2}\right ) \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )}{2}\right )}{a^{4}}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 133, normalized size = 1.68 \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 (-2 \, 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. 909 vs.
\(2 (75) = 150\).
time = 0.37, size = 909, normalized size = 11.51 \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 (b \cosh \left (x\right ) + a \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 \, \cosh \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 {sech}^{4}{\left (x \right )}}{a + b \coth {\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. 201 vs.
\(2 (75) = 150\).
time = 0.42, size = 201, normalized size = 2.54 \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 (e^{\left (2 \, x\right )} + 1\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 = 1.45, size = 123, normalized size = 1.56 \begin {gather*} \frac {8}{3\,a\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {2\,\left (2\,a-b\right )}{a^2\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}-\frac {2\,b\,\left (a-b\right )}{a^3\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {b\,\ln \left (b-a+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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