Optimal. Leaf size=38 \[ -\frac {3 x}{8}-\frac {1}{8 (1-\coth (x))}+\frac {1}{4 (\coth (x)+1)}+\frac {1}{8 (\coth (x)+1)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3487, 44, 207} \[ -\frac {3 x}{8}-\frac {1}{8 (1-\coth (x))}+\frac {1}{4 (\coth (x)+1)}+\frac {1}{8 (\coth (x)+1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 207
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{1+\coth (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{(1-x)^2 (1+x)^3} \, dx,x,\coth (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{8 (-1+x)^2}+\frac {1}{4 (1+x)^3}+\frac {1}{4 (1+x)^2}-\frac {3}{8 \left (-1+x^2\right )}\right ) \, dx,x,\coth (x)\right )\\ &=-\frac {1}{8 (1-\coth (x))}+\frac {1}{8 (1+\coth (x))^2}+\frac {1}{4 (1+\coth (x))}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\coth (x)\right )\\ &=-\frac {3 x}{8}-\frac {1}{8 (1-\coth (x))}+\frac {1}{8 (1+\coth (x))^2}+\frac {1}{4 (1+\coth (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 30, normalized size = 0.79 \[ \frac {1}{32} (-12 x+8 \sinh (2 x)-\sinh (4 x)-4 \cosh (2 x)+\cosh (4 x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 50, normalized size = 1.32 \[ \frac {3 \, \cosh \relax (x)^{3} + 9 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} - 6 \, {\left (2 \, x + 1\right )} \cosh \relax (x) + 3 \, {\left (\cosh \relax (x)^{2} - 4 \, x + 2\right )} \sinh \relax (x)}{32 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 30, normalized size = 0.79 \[ \frac {1}{32} \, {\left (9 \, e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-4 \, x\right )} - \frac {3}{8} \, x + \frac {1}{16} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.09, size = 70, normalized size = 1.84 \[ \frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{4 \tanh \left (\frac {x}{2}\right )-4}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 22, normalized size = 0.58 \[ -\frac {3}{8} \, x + \frac {1}{16} \, e^{\left (2 \, x\right )} - \frac {3}{16} \, e^{\left (-2 \, x\right )} + \frac {1}{32} \, e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.24, size = 22, normalized size = 0.58 \[ \frac {{\mathrm {e}}^{2\,x}}{16}-\frac {3\,{\mathrm {e}}^{-2\,x}}{16}-\frac {3\,x}{8}+\frac {{\mathrm {e}}^{-4\,x}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________