Optimal. Leaf size=69 \[ -\frac {a \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac {b \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac {a \log (\cosh (c+d x))}{d \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2721, 801, 635, 203, 260} \[ -\frac {a \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac {b \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac {a \log (\cosh (c+d x))}{d \left (a^2+b^2\right )} \]
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 (c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x}{(a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a}{\left (a^2+b^2\right ) (a+x)}+\frac {-b^2-a x}{\left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac {a \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right ) d}-\frac {\operatorname {Subst}\left (\int \frac {-b^2-a x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac {a \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {a \operatorname {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {b \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {a \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 51, normalized size = 0.74 \[ \frac {a (\log (\cosh (c+d x))-\log (a+b \sinh (c+d x)))+2 b \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 92, normalized size = 1.33 \[ \frac {2 \, b \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - a \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + a \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.00, size = 85, normalized size = 1.23 \[ \frac {\frac {2 \, b \arctan \left (e^{\left (d x + c\right )}\right )}{a^{2} + b^{2}} + \frac {a \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{a^{2} + b^{2}} - \frac {a \log \left ({\left | b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - b \right |}\right )}{a^{2} + b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 113, normalized size = 1.64 \[ -\frac {2 a \ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right )}{d \left (2 a^{2}+2 b^{2}\right )}+\frac {2 a \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \left (2 a^{2}+2 b^{2}\right )}+\frac {4 b \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (2 a^{2}+2 b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 95, normalized size = 1.38 \[ -\frac {2 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} - \frac {a \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 130, normalized size = 1.88 \[ \frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )}{a\,d-b\,d\,1{}\mathrm {i}}-\frac {a\,\ln \left (8\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b^3-4\,a^2\,b+b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{d\,a^2+d\,b^2}+\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{-b\,d+a\,d\,1{}\mathrm {i}} \]
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 \[ \int \frac {\tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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