3.783 \(\int \frac {\sinh (x)}{1+\cosh (x)+\sinh (x)} \, dx\)

Optimal. Leaf size=18 \[ \frac {x}{2}-\frac {\sinh (x)}{2}+\frac {\cosh (x)}{2} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3131} \[ \frac {x}{2}-\frac {\sinh (x)}{2}+\frac {\cosh (x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(1 + Cosh[x] + Sinh[x]),x]

[Out]

x/2 + Cosh[x]/2 - Sinh[x]/2

Rule 3131

Int[((A_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x
_)]), x_Symbol] :> Simp[((2*a*A - c*C)*x)/(2*a^2), x] + (-Simp[(C*Cos[d + e*x])/(2*a*e), x] + Simp[(c*C*Sin[d
+ e*x])/(2*a*b*e), x] + Simp[((-(a^2*C) + 2*a*c*A + b^2*C)*Log[RemoveContent[a + b*Cos[d + e*x] + c*Sin[d + e*
x], x]])/(2*a^2*b*e), x]) /; FreeQ[{a, b, c, d, e, A, C}, x] && EqQ[b^2 + c^2, 0]

Rubi steps

\begin {align*} \int \frac {\sinh (x)}{1+\cosh (x)+\sinh (x)} \, dx &=\frac {x}{2}+\frac {\cosh (x)}{2}-\frac {\sinh (x)}{2}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 18, normalized size = 1.00 \[ \frac {x}{2}-\frac {\sinh (x)}{2}+\frac {\cosh (x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(1 + Cosh[x] + Sinh[x]),x]

[Out]

x/2 + Cosh[x]/2 - Sinh[x]/2

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fricas [A]  time = 0.41, size = 19, normalized size = 1.06 \[ \frac {x \cosh \relax (x) + x \sinh \relax (x) + 1}{2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 10, normalized size = 0.56 \[ \frac {1}{2} \, x + \frac {1}{2} \, e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x + 1/2*e^(-x)

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maple [B]  time = 0.17, size = 28, normalized size = 1.56 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*ln(tanh(1/2*x)-1)+1/(tanh(1/2*x)+1)+1/2*ln(tanh(1/2*x)+1)

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maxima [A]  time = 0.74, size = 10, normalized size = 0.56 \[ \frac {1}{2} \, x + \frac {1}{2} \, e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x + 1/2*e^(-x)

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mupad [B]  time = 0.05, size = 10, normalized size = 0.56 \[ \frac {x}{2}+\frac {{\mathrm {e}}^{-x}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/2 + exp(-x)/2

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sympy [B]  time = 0.56, size = 34, normalized size = 1.89 \[ \frac {x \tanh {\left (\frac {x}{2} \right )}}{2 \tanh {\left (\frac {x}{2} \right )} + 2} + \frac {x}{2 \tanh {\left (\frac {x}{2} \right )} + 2} + \frac {2}{2 \tanh {\left (\frac {x}{2} \right )} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*tanh(x/2)/(2*tanh(x/2) + 2) + x/(2*tanh(x/2) + 2) + 2/(2*tanh(x/2) + 2)

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