3.8.41 \(\int (a+b \cosh (x)+c \sinh (x)) \, dx\) [741]

Optimal. Leaf size=12 \[ a x+c \cosh (x)+b \sinh (x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2717, 2718} \begin {gather*} a x+b \sinh (x)+c \cosh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[a + b*Cosh[x] + c*Sinh[x],x]

[Out]

a*x + c*Cosh[x] + b*Sinh[x]

Rule 2717

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

Rule 2718

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} a x+c \cosh (x)+b \sinh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*Cosh[x] + c*Sinh[x],x]

[Out]

a*x + c*Cosh[x] + b*Sinh[x]

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Maple [A]
time = 0.54, size = 13, normalized size = 1.08

method	result	size
default	\(a x +c \cosh \left (x \right )+b \sinh \left (x \right )\)	\(13\)
risch	\(\frac {\left (b \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} c +2 a x \,{\mathrm e}^{x}-b +c \right ) {\mathrm e}^{-x}}{2}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*cosh(x)+c*sinh(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 12, normalized size = 1.00 \begin {gather*} a x + c \cosh \left (x\right ) + b \sinh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + c*cosh(x) + b*sinh(x)

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Fricas [A]
time = 0.34, size = 12, normalized size = 1.00 \begin {gather*} a x + c \cosh \left (x\right ) + b \sinh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + c*cosh(x) + b*sinh(x)

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Sympy [A]
time = 0.05, size = 12, normalized size = 1.00 \begin {gather*} a x + b \sinh {\left (x \right )} + c \cosh {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*sinh(x) + c*cosh(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
time = 0.43, size = 26, normalized size = 2.17 \begin {gather*} a x + \frac {1}{2} \, c {\left (e^{\left (-x\right )} + e^{x}\right )} - \frac {1}{2} \, b {\left (e^{\left (-x\right )} - e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.05, size = 12, normalized size = 1.00 \begin {gather*} a\,x+c\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*cosh(x) + c*sinh(x),x)

[Out]

a*x + c*cosh(x) + b*sinh(x)

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