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

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.41, size = 21, normalized size = 0.91 \[ \frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )}{d} \]

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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giac [B]  time = 0.13, size = 56, normalized size = 2.43 \[ \frac {1}{2} \, a {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{2} \, a {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} \]

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)

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maple [A]  time = 0.02, size = 19, normalized size = 0.83 \[ \frac {a \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )}{d} \]

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 = 0.31, size = 23, normalized size = 1.00 \[ \frac {a \cosh \left (d x + c\right )}{d} + \frac {a \sinh \left (d x + c\right )}{d} \]

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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mupad [B]  time = 0.06, size = 11, normalized size = 0.48 \[ \frac {a\,{\mathrm {e}}^{c+d\,x}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*exp(c + d*x))/d

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sympy [A]  time = 0.16, size = 29, normalized size = 1.26 \[ a \left (\begin {cases} \frac {\sinh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cosh {\relax (c )} & \text {otherwise} \end {cases}\right ) + a \left (\begin {cases} \frac {\cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sinh {\relax (c )} & \text {otherwise} \end {cases}\right ) \]

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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