### 3.741 $$\int (a+b \cosh (x)+c \sinh (x)) \, dx$$

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

[Out]

a*x + c*Cosh[x] + b*Sinh[x]

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Rubi [A]  time = 0.0087491, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {2637, 2638} $a x+b \sinh (x)+c \cosh (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[a + b*Cosh[x] + c*Sinh[x],x]

[Out]

a*x + c*Cosh[x] + b*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+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*}

Mathematica [A]  time = 0.001935, size = 12, normalized size = 1. $a x+b \sinh (x)+c \cosh (x)$

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 = 0.002, size = 13, normalized size = 1.1 \begin{align*} ax+c\cosh \left ( x \right ) +b\sinh \left ( x \right ) \end{align*}

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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Maxima [A]  time = 1.04637, size = 16, normalized size = 1.33 \begin{align*} a x + c \cosh \left (x\right ) + b \sinh \left (x\right ) \end{align*}

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 = 1.96078, size = 39, normalized size = 3.25 \begin{align*} a x + c \cosh \left (x\right ) + b \sinh \left (x\right ) \end{align*}

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 = 0.164501, size = 12, normalized size = 1. \begin{align*} a x + b \sinh{\left (x \right )} + c \cosh{\left (x \right )} \end{align*}

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]  time = 1.15804, size = 35, normalized size = 2.92 \begin{align*} 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{align*}

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)