Optimal. Leaf size=90 \[ -\frac {b \text {ArcTan}(\sinh (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\log (\sinh (c+d x))}{a d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right ) d} \]
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Rubi [A]
time = 0.13, antiderivative size = 90, 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 = {2916, 12, 908,
649, 209, 266} \begin {gather*} -\frac {b \text {ArcTan}(\sinh (c+d x))}{d \left (a^2+b^2\right )}-\frac {b^2 \log (a+b \sinh (c+d x))}{a d \left (a^2+b^2\right )}-\frac {a \log (\cosh (c+d x))}{d \left (a^2+b^2\right )}+\frac {\log (\sinh (c+d x))}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 266
Rule 649
Rule 908
Rule 2916
Rubi steps
\begin {align*} \int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {b \text {Subst}\left (\int \frac {b}{x (a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac {b^2 \text {Subst}\left (\int \frac {1}{x (a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=-\frac {b^2 \text {Subst}\left (\int \left (-\frac {1}{a b^2 x}+\frac {1}{a \left (a^2+b^2\right ) (a+x)}+\frac {b^2+a x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {\log (\sinh (c+d x))}{a d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right ) d}-\frac {\text {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 {\log (\sinh (c+d x))}{a d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right ) d}-\frac {a \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 {b^2 \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 {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 {\log (\sinh (c+d x))}{a d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.13, size = 92, normalized size = 1.02 \begin {gather*} -\frac {\frac {\log (i-\sinh (c+d x))}{a+i b}-\frac {2 \log (\sinh (c+d x))}{a}+\frac {\log (i+\sinh (c+d x))}{a-i b}+\frac {2 b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.34, size = 108, normalized size = 1.20
method | result | size |
derivativedivides | \(\frac {\frac {-a \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-2 b \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}+b^{2}}-\frac {b^{2} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right ) a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(108\) |
default | \(\frac {\frac {-a \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-2 b \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}+b^{2}}-\frac {b^{2} \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right ) a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\) | \(108\) |
risch | \(\frac {2 a \,d^{2} x}{a^{2} d^{2}+b^{2} d^{2}}+\frac {2 a d c}{a^{2} d^{2}+b^{2} d^{2}}-\frac {2 x}{a}-\frac {2 c}{a d}+\frac {2 b^{2} x}{a \left (a^{2}+b^{2}\right )}+\frac {2 b^{2} c}{a d \left (a^{2}+b^{2}\right )}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{\left (a^{2}+b^{2}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}-i\right ) a}{\left (a^{2}+b^{2}\right ) d}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{\left (a^{2}+b^{2}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}+i\right ) a}{\left (a^{2}+b^{2}\right ) d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a d \left (a^{2}+b^{2}\right )}\) | \(267\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 138, normalized size = 1.53 \begin {gather*} -\frac {b^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{3} + a b^{2}\right )} d} + \frac {2 \, b \arctan \left (e^{\left (-d x - c\right )}\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} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 134, normalized size = 1.49 \begin {gather*} -\frac {2 \, a b \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + b^{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 ) + a^{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^{2}\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{{\left (a^{3} + a 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 {csch}{\left (c + d x \right )} \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 = 0.47, size = 147, normalized size = 1.63 \begin {gather*} -\frac {\frac {2 \, b^{3} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{3} b + a 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 )} b}{a^{2} + b^{2}} + \frac {a \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{2} + b^{2}} - \frac {2 \, \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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