Optimal. Leaf size=69 \[ \frac {a \text {ArcTan}(\sinh (c+d x))}{\left (a^2+b^2\right ) d}-\frac {b \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {b \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right ) d} \]
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Rubi [A]
time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2747, 720, 31,
649, 210, 266} \begin {gather*} \frac {a \text {ArcTan}(\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac {b \log (a+b \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 266
Rule 649
Rule 720
Rule 2747
Rubi steps
\begin {align*} \int \frac {\text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {b \text {Subst}\left (\int \frac {1}{(a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {b \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac {b \text {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 {b \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {b \text {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) \text {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 {b \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 114, normalized size = 1.65 \begin {gather*} -\frac {b \left (\left (-a+\sqrt {-b^2}\right ) \log \left (\sqrt {-b^2}-b \sinh (c+d x)\right )-2 \sqrt {-b^2} \log (a+b \sinh (c+d x))+\left (a+\sqrt {-b^2}\right ) \log \left (\sqrt {-b^2}+b \sinh (c+d x)\right )\right )}{2 \sqrt {-b^2} \left (a^2+b^2\right ) d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.12, size = 88, normalized size = 1.28
method | result | size |
derivativedivides | \(\frac {\frac {b \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a^{2}+b^{2}}+\frac {-b \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 a \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}+b^{2}}}{d}\) | \(88\) |
default | \(\frac {\frac {b \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a^{2}+b^{2}}+\frac {-b \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 a \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}+b^{2}}}{d}\) | \(88\) |
risch | \(\frac {2 b \,d^{2} x}{a^{2} d^{2}+b^{2} d^{2}}+\frac {2 b d c}{a^{2} d^{2}+b^{2} d^{2}}-\frac {2 b x}{a^{2}+b^{2}}-\frac {2 b c}{d \left (a^{2}+b^{2}\right )}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{\left (a^{2}+b^{2}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}+i\right ) b}{\left (a^{2}+b^{2}\right ) d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{\left (a^{2}+b^{2}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}-i\right ) b}{\left (a^{2}+b^{2}\right ) d}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d \left (a^{2}+b^{2}\right )}\) | \(217\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 95, normalized size = 1.38 \begin {gather*} -\frac {2 \, a \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {b \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 {b \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 92, normalized size = 1.33 \begin {gather*} \frac {2 \, a \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + b \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 \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} \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}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 121, normalized size = 1.75 \begin {gather*} \frac {\frac {2 \, b^{2} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{2} b + b^{3}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} a}{a^{2} + b^{2}} - \frac {b \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{2} + b^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 129, normalized size = 1.87 \begin {gather*} \frac {b\,\ln \left (2\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-4\,b^3-a^2\,b+4\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+a^2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+8\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{d\,a^2+d\,b^2}-\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )}{b\,d+a\,d\,1{}\mathrm {i}}-\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{a\,d+b\,d\,1{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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