Optimal. Leaf size=85 \[ \frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\left (a^2-b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^2}+\frac {2 a b \tan ^{-1}(\sinh (x))}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cosh (x))}{\left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.10, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2721, 801, 635, 203, 260} \[ \frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\left (a^2-b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^2}+\frac {2 a b \tan ^{-1}(\sinh (x))}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cosh (x))}{\left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 801
Rule 2721
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx &=-\operatorname {Subst}\left (\int \frac {x}{(a+x)^2 \left (-b^2-x^2\right )} \, dx,x,b \sinh (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a}{\left (a^2+b^2\right ) (a+x)^2}+\frac {a^2-b^2}{\left (a^2+b^2\right )^2 (a+x)}+\frac {-2 a b^2-\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (x)\right )\\ &=-\frac {\left (a^2-b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^2}+\frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\operatorname {Subst}\left (\int \frac {-2 a b^2-\left (a^2-b^2\right ) x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{\left (a^2+b^2\right )^2}\\ &=-\frac {\left (a^2-b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^2}+\frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \operatorname {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{\left (a^2+b^2\right )^2}\\ &=\frac {2 a b \tan ^{-1}(\sinh (x))}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cosh (x))}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^2}+\frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 146, normalized size = 1.72 \[ \frac {a \left (2 \left (\left (b^2-a^2\right ) \log (a+b \sinh (x))+a^2+b^2\right )+(a-i b)^2 \log (-\sinh (x)+i)+(a+i b)^2 \log (\sinh (x)+i)\right )+b \sinh (x) \left (2 \left (b^2-a^2\right ) \log (a+b \sinh (x))+(a-i b)^2 \log (-\sinh (x)+i)+(a+i b)^2 \log (\sinh (x)+i)\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.53, size = 423, normalized size = 4.98 \[ -\frac {4 \, {\left (a b^{2} \cosh \relax (x)^{2} + a b^{2} \sinh \relax (x)^{2} + 2 \, a^{2} b \cosh \relax (x) - a b^{2} + 2 \, {\left (a b^{2} \cosh \relax (x) + a^{2} b\right )} \sinh \relax (x)\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) + 2 \, {\left (a^{3} + a b^{2}\right )} \cosh \relax (x) + {\left (a^{2} b - b^{3} - {\left (a^{2} b - b^{3}\right )} \cosh \relax (x)^{2} - {\left (a^{2} b - b^{3}\right )} \sinh \relax (x)^{2} - 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \relax (x) - 2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, {\left (b \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) - {\left (a^{2} b - b^{3} - {\left (a^{2} b - b^{3}\right )} \cosh \relax (x)^{2} - {\left (a^{2} b - b^{3}\right )} \sinh \relax (x)^{2} - 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \relax (x) - 2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \cosh \relax (x)\right )} \sinh \relax (x)\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (a^{3} + a b^{2}\right )} \sinh \relax (x)}{a^{4} b + 2 \, a^{2} b^{3} + b^{5} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cosh \relax (x)^{2} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sinh \relax (x)^{2} - 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cosh \relax (x) - 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cosh \relax (x)\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 199, normalized size = 2.34 \[ \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )} - b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} - 4 \, a^{3}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 248, normalized size = 2.92 \[ \frac {2 \tanh \left (\frac {x}{2}\right ) a^{2} b}{\left (a^{2}+b^{2}\right )^{2} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}+\frac {2 \tanh \left (\frac {x}{2}\right ) b^{3}}{\left (a^{2}+b^{2}\right )^{2} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}-\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right ) a^{2}}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right ) b^{2}}{\left (a^{2}+b^{2}\right )^{2}}+\frac {2 \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) a^{2}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}-\frac {2 \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) b^{2}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {8 a b \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 155, normalized size = 1.82 \[ -\frac {4 \, a b \arctan \left (e^{\left (-x\right )}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, a e^{\left (-x\right )}}{a^{2} b + b^{3} + 2 \, {\left (a^{3} + a b^{2}\right )} e^{\left (-x\right )} - {\left (a^{2} b + b^{3}\right )} e^{\left (-2 \, x\right )}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.75, size = 190, normalized size = 2.24 \[ \frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}{a^2+a\,b\,2{}\mathrm {i}-b^2}-\frac {\ln \left (b^5\,{\mathrm {e}}^{2\,x}-a^4\,b-b^5+a^2\,b^3+2\,a^5\,{\mathrm {e}}^x-a^2\,b^3\,{\mathrm {e}}^{2\,x}+2\,a\,b^4\,{\mathrm {e}}^x+a^4\,b\,{\mathrm {e}}^{2\,x}-2\,a^3\,b^2\,{\mathrm {e}}^x\right )\,\left (a^2-b^2\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {2\,a\,b\,{\mathrm {e}}^x}{\left (a^2\,b+b^3\right )\,\left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )}+\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}} \]
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 \[ \int \frac {\tanh {\relax (x )}}{\left (a + b \sinh {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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