Optimal. Leaf size=74 \[ \frac {a^2 \log (a+b \sinh (c+d x))}{b d \left (a^2+b^2\right )}-\frac {a \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac {b \log (\cosh (c+d x))}{d \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2837, 12, 1629, 635, 203, 260} \[ \frac {a^2 \log (a+b \sinh (c+d x))}{b d \left (a^2+b^2\right )}-\frac {a \tan ^{-1}(\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac {b \log (\cosh (c+d x))}{d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 260
Rule 635
Rule 1629
Rule 2837
Rubi steps
\begin {align*} \int \frac {\sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {b \operatorname {Subst}\left (\int \frac {x^2}{b^2 (a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^2}{(a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{b d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {a^2}{\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 )}{b d}\\ &=\frac {a^2 \log (a+b \sinh (c+d x))}{b \left (a^2+b^2\right ) d}-\frac {b \operatorname {Subst}\left (\int \frac {a-x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {a^2 \log (a+b \sinh (c+d x))}{b \left (a^2+b^2\right ) d}+\frac {b \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 {(a b) \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 {a \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {b \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {a^2 \log (a+b \sinh (c+d x))}{b \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 78, normalized size = 1.05 \[ \frac {2 a^2 \log (a+b \sinh (c+d x))+b (b+i a) \log (-\sinh (c+d x)+i)+b (b-i a) \log (\sinh (c+d x)+i)}{2 b d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 111, normalized size = 1.50 \[ -\frac {{\left (a^{2} + b^{2}\right )} d x + 2 \, a b \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - a^{2} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - b^{2} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{{\left (a^{2} b + b^{3}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 95, normalized size = 1.28 \[ \frac {\frac {a^{2} \log \left ({\left | b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - b \right |}\right )}{a^{2} b + b^{3}} - \frac {d x}{b} - \frac {2 \, a \arctan \left (e^{\left (d x + c\right )}\right )}{a^{2} + b^{2}} + \frac {b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{a^{2} + b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 153, normalized size = 2.07 \[ -\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d b}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d b}+\frac {a^{2} \ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right )}{d b \left (a^{2}+b^{2}\right )}+\frac {4 b \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \left (4 a^{2}+4 b^{2}\right )}-\frac {8 a \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \left (4 a^{2}+4 b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 110, normalized size = 1.49 \[ \frac {a^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{2} b + b^{3}\right )} d} + \frac {2 \, a \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {b \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {d x + c}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 174, normalized size = 2.35 \[ \frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )}{b\,d+a\,d\,1{}\mathrm {i}}-\frac {x}{b}+\frac {a^2\,\ln \left (a^2\,b^3-b^5-a^4\,b+2\,a^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-2\,a^3\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{d\,a^2\,b+d\,b^3}+\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{a\,d+b\,d\,1{}\mathrm {i}} \]
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 \[ \int \frac {\sinh {\left (c + d x \right )} \tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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