3.580 \(\int (a \cosh (x)+b \sinh (x)) \, dx\)

Optimal. Leaf size=9 \[ a \sinh (x)+b \cosh (x) \]

[Out]

b*cosh(x)+a*sinh(x)

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2637, 2638} \[ a \sinh (x)+b \cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[a*Cosh[x] + b*Sinh[x],x]

[Out]

b*Cosh[x] + a*Sinh[x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int (a \cosh (x)+b \sinh (x)) \, dx &=a \int \cosh (x) \, dx+b \int \sinh (x) \, dx\\ &=b \cosh (x)+a \sinh (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ a \sinh (x)+b \cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[a*Cosh[x] + b*Sinh[x],x]

[Out]

b*Cosh[x] + a*Sinh[x]

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fricas [A]  time = 0.40, size = 9, normalized size = 1.00 \[ b \cosh \relax (x) + a \sinh \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*cosh(x) + a*sinh(x)

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giac [B]  time = 0.13, size = 23, normalized size = 2.56 \[ \frac {1}{2} \, b {\left (e^{\left (-x\right )} + e^{x}\right )} - \frac {1}{2} \, a {\left (e^{\left (-x\right )} - e^{x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*(e^(-x) + e^x) - 1/2*a*(e^(-x) - e^x)

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maple [A]  time = 0.02, size = 10, normalized size = 1.11 \[ b \cosh \relax (x )+a \sinh \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*cosh(x)+a*sinh(x)

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maxima [A]  time = 0.35, size = 9, normalized size = 1.00 \[ b \cosh \relax (x) + a \sinh \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*cosh(x) + a*sinh(x)

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mupad [B]  time = 1.50, size = 9, normalized size = 1.00 \[ b\,\mathrm {cosh}\relax (x)+a\,\mathrm {sinh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a*cosh(x) + b*sinh(x),x)

[Out]

b*cosh(x) + a*sinh(x)

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sympy [A]  time = 0.11, size = 8, normalized size = 0.89 \[ a \sinh {\relax (x )} + b \cosh {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*sinh(x) + b*cosh(x)

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