3.1043 \(\int \sinh (x) (\cosh (x)+\sinh (x)) \, dx\)

Optimal. Leaf size=22 \[ -\frac {x}{2}+\frac {\sinh ^2(x)}{2}+\frac {1}{2} \sinh (x) \cosh (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3089, 2564, 30, 2635, 8} \[ -\frac {x}{2}+\frac {\sinh ^2(x)}{2}+\frac {1}{2} \sinh (x) \cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]*(Cosh[x] + Sinh[x]),x]

[Out]

-x/2 + (Cosh[x]*Sinh[x])/2 + Sinh[x]^2/2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 3089

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \[ -\frac {x}{2}+\frac {1}{4} \sinh (2 x)+\frac {\cosh ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]*(Cosh[x] + Sinh[x]),x]

[Out]

-1/2*x + Cosh[x]^2/2 + Sinh[2*x]/4

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 29, normalized size = 1.32 \[ -\frac {{\left (2 \, x - 1\right )} \cosh \relax (x) - {\left (2 \, x + 1\right )} \sinh \relax (x)}{4 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*(cosh(x)+sinh(x)),x, algorithm="fricas")

[Out]

-1/4*((2*x - 1)*cosh(x) - (2*x + 1)*sinh(x))/(cosh(x) - sinh(x))

________________________________________________________________________________________

giac [A]  time = 0.13, size = 10, normalized size = 0.45 \[ -\frac {1}{2} \, x + \frac {1}{4} \, e^{\left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*(cosh(x)+sinh(x)),x, algorithm="giac")

[Out]

-1/2*x + 1/4*e^(2*x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 17, normalized size = 0.77 \[ \frac {\cosh \relax (x ) \sinh \relax (x )}{2}-\frac {x}{2}+\frac {\left (\cosh ^{2}\relax (x )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)*(cosh(x)+sinh(x)),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 22, normalized size = 1.00 \[ \frac {1}{2} \, \cosh \relax (x)^{2} - \frac {1}{2} \, x + \frac {1}{8} \, e^{\left (2 \, x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*(cosh(x)+sinh(x)),x, algorithm="maxima")

[Out]

1/2*cosh(x)^2 - 1/2*x + 1/8*e^(2*x) - 1/8*e^(-2*x)

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 10, normalized size = 0.45 \[ \frac {{\mathrm {e}}^{2\,x}}{4}-\frac {x}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)*(cosh(x) + sinh(x)),x)

[Out]

exp(2*x)/4 - x/2

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 31, normalized size = 1.41 \[ \frac {x \sinh ^{2}{\relax (x )}}{2} - \frac {x \cosh ^{2}{\relax (x )}}{2} + \frac {\sinh {\relax (x )} \cosh {\relax (x )}}{2} + \frac {\cosh ^{2}{\relax (x )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*(cosh(x)+sinh(x)),x)

[Out]

x*sinh(x)**2/2 - x*cosh(x)**2/2 + sinh(x)*cosh(x)/2 + cosh(x)**2/2

________________________________________________________________________________________