Optimal. Leaf size=113 \[ \frac {b^4 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^2}-\frac {1}{4 (a+b) (1-\text {sech}(x))}-\frac {1}{4 (a-b) (\text {sech}(x)+1)}+\frac {(2 a+3 b) \log (1-\text {sech}(x))}{4 (a+b)^2}+\frac {(2 a-3 b) \log (\text {sech}(x)+1)}{4 (a-b)^2}+\frac {\log (\cosh (x))}{a} \]
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Rubi [A] time = 0.19, antiderivative size = 113, 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 = {3885, 894} \[ \frac {b^4 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^2}-\frac {1}{4 (a+b) (1-\text {sech}(x))}-\frac {1}{4 (a-b) (\text {sech}(x)+1)}+\frac {(2 a+3 b) \log (1-\text {sech}(x))}{4 (a+b)^2}+\frac {(2 a-3 b) \log (\text {sech}(x)+1)}{4 (a-b)^2}+\frac {\log (\cosh (x))}{a} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3885
Rubi steps
\begin {align*} \int \frac {\coth ^3(x)}{a+b \text {sech}(x)} \, dx &=-\left (b^4 \operatorname {Subst}\left (\int \frac {1}{x (a+x) \left (b^2-x^2\right )^2} \, dx,x,b \text {sech}(x)\right )\right )\\ &=-\left (b^4 \operatorname {Subst}\left (\int \left (\frac {1}{4 b^3 (a+b) (b-x)^2}+\frac {2 a+3 b}{4 b^4 (a+b)^2 (b-x)}+\frac {1}{a b^4 x}-\frac {1}{a (a-b)^2 (a+b)^2 (a+x)}-\frac {1}{4 (a-b) b^3 (b+x)^2}+\frac {-2 a+3 b}{4 (a-b)^2 b^4 (b+x)}\right ) \, dx,x,b \text {sech}(x)\right )\right )\\ &=\frac {\log (\cosh (x))}{a}+\frac {(2 a+3 b) \log (1-\text {sech}(x))}{4 (a+b)^2}+\frac {(2 a-3 b) \log (1+\text {sech}(x))}{4 (a-b)^2}+\frac {b^4 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^2}-\frac {1}{4 (a+b) (1-\text {sech}(x))}-\frac {1}{4 (a-b) (1+\text {sech}(x))}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 112, normalized size = 0.99 \[ \frac {4 a \left (2 a \left (a^2-2 b^2\right ) \log (\sinh (x))+b \left (3 b^2-a^2\right ) \log \left (\tanh \left (\frac {x}{2}\right )\right )\right )+8 b^4 \log (a \cosh (x)+b)-a (a-b)^2 (a+b) \text {csch}^2\left (\frac {x}{2}\right )+a (a-b) (a+b)^2 \text {sech}^2\left (\frac {x}{2}\right )}{8 a (a-b)^2 (a+b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 1222, normalized size = 10.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 193, normalized size = 1.71 \[ \frac {b^{4} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac {{\left (2 \, a - 3 \, b\right )} \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {{\left (2 \, a + 3 \, b\right )} \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {a^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 2 \, a b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )} + 4 \, a b^{2}}{2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 119, normalized size = 1.05 \[ -\frac {\tanh ^{2}\left (\frac {x}{2}\right )}{8 \left (a -b \right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {b^{4} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )}{\left (a -b \right )^{2} \left (a +b \right )^{2} a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {1}{8 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) a}{\left (a +b \right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) b}{2 \left (a +b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 164, normalized size = 1.45 \[ \frac {b^{4} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac {{\left (2 \, a - 3 \, b\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {{\left (2 \, a + 3 \, b\right )} \log \left (e^{\left (-x\right )} - 1\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {b e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} + b e^{\left (-3 \, x\right )}}{a^{2} - b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} - b^{2}\right )} e^{\left (-4 \, x\right )}} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 339, normalized size = 3.00 \[ \frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (2\,a+3\,b\right )}{2\,a^2+4\,a\,b+2\,b^2}-\frac {x}{a}-\frac {\frac {2\,a}{a^2-b^2}-\frac {2\,b\,{\mathrm {e}}^x}{a^2-b^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,\left (a^4-a^2\,b^2\right )}{a\,{\left (a^2-b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{2\,x}-1}+\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (2\,a-3\,b\right )}{2\,a^2-4\,a\,b+2\,b^2}+\frac {b^4\,\ln \left (4\,a^9\,{\mathrm {e}}^{2\,x}+4\,a\,b^8+4\,a^9+7\,a^3\,b^6+14\,a^5\,b^4-17\,a^7\,b^2+8\,b^9\,{\mathrm {e}}^x+7\,a^3\,b^6\,{\mathrm {e}}^{2\,x}+14\,a^5\,b^4\,{\mathrm {e}}^{2\,x}-17\,a^7\,b^2\,{\mathrm {e}}^{2\,x}+8\,a^8\,b\,{\mathrm {e}}^x+4\,a\,b^8\,{\mathrm {e}}^{2\,x}+14\,a^2\,b^7\,{\mathrm {e}}^x+28\,a^4\,b^5\,{\mathrm {e}}^x-34\,a^6\,b^3\,{\mathrm {e}}^x\right )}{a^5-2\,a^3\,b^2+a\,b^4} \]
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 \[ \int \frac {\coth ^{3}{\relax (x )}}{a + b \operatorname {sech}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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