3.72 \(\int \frac {\sinh (x)}{1+\tanh (x)} \, dx\)

Optimal. Leaf size=17 \[ \frac {\cosh ^3(x)}{3}-\frac {\sinh ^3(x)}{3} \]

[Out]

1/3*cosh(x)^3-1/3*sinh(x)^3

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3518, 3108, 3107, 2565, 30, 2564} \[ \frac {\cosh ^3(x)}{3}-\frac {\sinh ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(1 + Tanh[x]),x]

[Out]

Cosh[x]^3/3 - Sinh[x]^3/3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 3107

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*sin[c + d*x]^n*(a*cos[c + d*x] + b*sin[c +
d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0]

Rule 3108

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_), x_Symbol] :> Dist[a^p*b^p, Int[(Cos[c + d*x]^m*Sin[c + d*x]^n)/(b*Cos[c + d*x] + a*Sin
[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[a^2 + b^2, 0] && ILtQ[p, 0]

Rule 3518

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[(Sin[e + f*x]
^m*(a*Cos[e + f*x] + b*Sin[e + f*x])^n)/Cos[e + f*x]^n, x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
ILtQ[n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {\sinh (x)}{1+\tanh (x)} \, dx &=\int \frac {\cosh (x) \sinh (x)}{\cosh (x)+\sinh (x)} \, dx\\ &=i \int \cosh (x) (-i \cosh (x)+i \sinh (x)) \sinh (x) \, dx\\ &=\int \left (\cosh ^2(x) \sinh (x)-\cosh (x) \sinh ^2(x)\right ) \, dx\\ &=\int \cosh ^2(x) \sinh (x) \, dx-\int \cosh (x) \sinh ^2(x) \, dx\\ &=-\left (i \operatorname {Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right )\right )+\operatorname {Subst}\left (\int x^2 \, dx,x,\cosh (x)\right )\\ &=\frac {\cosh ^3(x)}{3}-\frac {\sinh ^3(x)}{3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 19, normalized size = 1.12 \[ \frac {1}{12} \left (-4 \sinh ^3(x)+3 \cosh (x)+\cosh (3 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(1 + Tanh[x]),x]

[Out]

(3*Cosh[x] + Cosh[3*x] - 4*Sinh[x]^3)/12

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 23, normalized size = 1.35 \[ \frac {\cosh \relax (x)^{2} + \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}{3 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(1+tanh(x)),x, algorithm="fricas")

[Out]

1/3*(cosh(x)^2 + cosh(x)*sinh(x) + sinh(x)^2)/(cosh(x) + sinh(x))

________________________________________________________________________________________

giac [A]  time = 0.11, size = 11, normalized size = 0.65 \[ \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(1+tanh(x)),x, algorithm="giac")

[Out]

1/12*e^(-3*x) + 1/4*e^x

________________________________________________________________________________________

maple [B]  time = 0.07, size = 42, normalized size = 2.47 \[ -\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{2 \tanh \left (\frac {x}{2}\right )+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(1+tanh(x)),x)

[Out]

-1/2/(tanh(1/2*x)-1)+2/3/(tanh(1/2*x)+1)^3-1/(tanh(1/2*x)+1)^2+1/2/(tanh(1/2*x)+1)

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 11, normalized size = 0.65 \[ \frac {1}{12} \, e^{\left (-3 \, x\right )} + \frac {1}{4} \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(1+tanh(x)),x, algorithm="maxima")

[Out]

1/12*e^(-3*x) + 1/4*e^x

________________________________________________________________________________________

mupad [B]  time = 1.12, size = 11, normalized size = 0.65 \[ \frac {{\mathrm {e}}^{-3\,x}}{12}+\frac {{\mathrm {e}}^x}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(tanh(x) + 1),x)

[Out]

exp(-3*x)/12 + exp(x)/4

________________________________________________________________________________________

sympy [B]  time = 0.37, size = 48, normalized size = 2.82 \[ \frac {\sinh {\relax (x )} \tanh {\relax (x )}}{3 \tanh {\relax (x )} + 3} - \frac {\sinh {\relax (x )}}{3 \tanh {\relax (x )} + 3} + \frac {2 \cosh {\relax (x )} \tanh {\relax (x )}}{3 \tanh {\relax (x )} + 3} + \frac {\cosh {\relax (x )}}{3 \tanh {\relax (x )} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(1+tanh(x)),x)

[Out]

sinh(x)*tanh(x)/(3*tanh(x) + 3) - sinh(x)/(3*tanh(x) + 3) + 2*cosh(x)*tanh(x)/(3*tanh(x) + 3) + cosh(x)/(3*tan
h(x) + 3)

________________________________________________________________________________________