### 3.596 $$\int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx$$

Optimal. Leaf size=23 $\frac{a \sinh (c+d x)}{d}+\frac{a \cosh (c+d x)}{d}$

[Out]

(a*Cosh[c + d*x])/d + (a*Sinh[c + d*x])/d

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Rubi [A]  time = 0.0152554, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {2637, 2638} $\frac{a \sinh (c+d x)}{d}+\frac{a \cosh (c+d x)}{d}$

Antiderivative was successfully veriﬁed.

[In]

Int[a*Cosh[c + d*x] + a*Sinh[c + d*x],x]

[Out]

(a*Cosh[c + d*x])/d + (a*Sinh[c + d*x])/d

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

Mathematica [A]  time = 0.0169584, size = 45, normalized size = 1.96 $\frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}+\frac{a \sinh (c) \cosh (d x)}{d}+\frac{a \cosh (c) \sinh (d x)}{d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a*Cosh[c]*Cosh[d*x])/d + (a*Cosh[d*x]*Sinh[c])/d + (a*Cosh[c]*Sinh[d*x])/d + (a*Sinh[c]*Sinh[d*x])/d

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Maple [A]  time = 0.009, size = 19, normalized size = 0.8 \begin{align*}{\frac{a \left ( \cosh \left ( dx+c \right ) +\sinh \left ( dx+c \right ) \right ) }{d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(a*cosh(d*x+c)+a*sinh(d*x+c),x)

[Out]

a*(cosh(d*x+c)+sinh(d*x+c))/d

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Maxima [A]  time = 1.06774, size = 31, normalized size = 1.35 \begin{align*} \frac{a \cosh \left (d x + c\right )}{d} + \frac{a \sinh \left (d x + c\right )}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*cosh(d*x + c)/d + a*sinh(d*x + c)/d

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Fricas [A]  time = 1.97798, size = 53, normalized size = 2.3 \begin{align*} \frac{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*cosh(d*x + c) + a*sinh(d*x + c))/d

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Sympy [A]  time = 0.254128, size = 29, normalized size = 1.26 \begin{align*} a \left (\begin{cases} \frac{\sinh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \cosh{\left (c \right )} & \text{otherwise} \end{cases}\right ) + a \left (\begin{cases} \frac{\cosh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \sinh{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*Piecewise((sinh(c + d*x)/d, Ne(d, 0)), (x*cosh(c), True)) + a*Piecewise((cosh(c + d*x)/d, Ne(d, 0)), (x*sinh
(c), True))

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Giac [B]  time = 1.14229, size = 76, normalized size = 3.3 \begin{align*} \frac{1}{2} \, a{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{1}{2} \, a{\left (\frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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