### 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]

________________________________________________________________________________________

Rubi [A]  time = 0.0087967, antiderivative size = 9, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0037957, size = 9, normalized size = 1. $a \sinh (x)+b \cosh (x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[a*Cosh[x] + b*Sinh[x],x]

[Out]

b*Cosh[x] + a*Sinh[x]

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 10, normalized size = 1.1 \begin{align*} b\cosh \left ( x \right ) +a\sinh \left ( x \right ) \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.02031, size = 12, normalized size = 1.33 \begin{align*} b \cosh \left (x\right ) + a \sinh \left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 2.43486, size = 31, normalized size = 3.44 \begin{align*} b \cosh \left (x\right ) + a \sinh \left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.129453, size = 8, normalized size = 0.89 \begin{align*} a \sinh{\left (x \right )} + b \cosh{\left (x \right )} \end{align*}

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)

________________________________________________________________________________________

Giac [B]  time = 1.12386, size = 31, normalized size = 3.44 \begin{align*} \frac{1}{2} \, b{\left (e^{\left (-x\right )} + e^{x}\right )} - \frac{1}{2} \, a{\left (e^{\left (-x\right )} - e^{x}\right )} \end{align*}

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)