3.66 \(\int (a+b \cosh (c+d x)) \, dx\)

Optimal. Leaf size=15 \[ a x+\frac {b \sinh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2637} \[ a x+\frac {b \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Cosh[c + d*x],x]

[Out]

a*x + (b*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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.73 \[ a x+\frac {b \sinh (c) \cosh (d x)}{d}+\frac {b \cosh (c) \sinh (d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Cosh[c + d*x],x]

[Out]

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

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fricas [A]  time = 0.39, size = 17, normalized size = 1.13 \[ \frac {a d x + b \sinh \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*d*x + b*sinh(d*x + c))/d

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giac [B]  time = 0.11, size = 32, normalized size = 2.13 \[ a x + \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 16, normalized size = 1.07 \[ a x +\frac {b \sinh \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.29, size = 15, normalized size = 1.00 \[ a x + \frac {b \sinh \left (d x + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*sinh(d*x + c)/d

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mupad [B]  time = 0.06, size = 15, normalized size = 1.00 \[ a\,x+\frac {b\,\mathrm {sinh}\left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*cosh(c + d*x),x)

[Out]

a*x + (b*sinh(c + d*x))/d

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sympy [A]  time = 0.13, size = 17, normalized size = 1.13 \[ a x + b \left (\begin {cases} \frac {\sinh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cosh {\relax (c )} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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