Optimal. Leaf size=12 \[ -\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3770} \[ -\frac {\tanh ^{-1}(\cosh (a+b x))}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3770
Rubi steps
\begin {align*} \int \text {csch}(a+b x) \, dx &=-\frac {\tanh ^{-1}(\cosh (a+b x))}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.02, size = 38, normalized size = 3.17 \[ \frac {\log \left (\sinh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}-\frac {\log \left (\cosh \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.51, size = 38, normalized size = 3.17 \[ -\frac {\log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.13, size = 27, normalized size = 2.25 \[ -\frac {\log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 15, normalized size = 1.25 \[ \frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 14, normalized size = 1.17 \[ \frac {\log \left (\tanh \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 27, normalized size = 2.25 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________