Optimal. Leaf size=24 \[ \frac {\text {csch}(x)}{3 (a \cosh (x)+a)}-\frac {2 \coth (x)}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2672, 3767, 8} \[ \frac {\text {csch}(x)}{3 (a \cosh (x)+a)}-\frac {2 \coth (x)}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2672
Rule 3767
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(x)}{a+a \cosh (x)} \, dx &=\frac {\text {csch}(x)}{3 (a+a \cosh (x))}+\frac {2 \int \text {csch}^2(x) \, dx}{3 a}\\ &=\frac {\text {csch}(x)}{3 (a+a \cosh (x))}-\frac {(2 i) \operatorname {Subst}(\int 1 \, dx,x,-i \coth (x))}{3 a}\\ &=-\frac {2 \coth (x)}{3 a}+\frac {\text {csch}(x)}{3 (a+a \cosh (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 30, normalized size = 1.25 \[ -\frac {(2 \cosh (x)+\cosh (2 x)) \text {csch}\left (\frac {x}{2}\right ) \text {sech}^3\left (\frac {x}{2}\right )}{12 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.62, size = 94, normalized size = 3.92 \[ -\frac {4 \, {\left (2 \, \cosh \relax (x) + 2 \, \sinh \relax (x) + 1\right )}}{3 \, {\left (a \cosh \relax (x)^{4} + a \sinh \relax (x)^{4} + 2 \, a \cosh \relax (x)^{3} + 2 \, {\left (2 \, a \cosh \relax (x) + a\right )} \sinh \relax (x)^{3} + 6 \, {\left (a \cosh \relax (x)^{2} + a \cosh \relax (x)\right )} \sinh \relax (x)^{2} - 2 \, a \cosh \relax (x) + 2 \, {\left (2 \, a \cosh \relax (x)^{3} + 3 \, a \cosh \relax (x)^{2} - a\right )} \sinh \relax (x) - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 35, normalized size = 1.46 \[ -\frac {1}{2 \, a {\left (e^{x} - 1\right )}} + \frac {3 \, e^{\left (2 \, x\right )} + 12 \, e^{x} + 5}{6 \, a {\left (e^{x} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 29, normalized size = 1.21 \[ \frac {\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}-2 \tanh \left (\frac {x}{2}\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )}}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.30, size = 59, normalized size = 2.46 \[ -\frac {8 \, e^{\left (-x\right )}}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} - \frac {4}{3 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.92, size = 89, normalized size = 3.71 \[ \frac {\frac {{\mathrm {e}}^{2\,x}}{6\,a}+\frac {1}{6\,a}+\frac {{\mathrm {e}}^x}{a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}+\frac {\frac {1}{2\,a}+\frac {{\mathrm {e}}^x}{6\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {1}{2\,a\,\left ({\mathrm {e}}^x-1\right )}+\frac {1}{6\,a\,\left ({\mathrm {e}}^x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {csch}^{2}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________