3.1 \(\int \cosh (a+b x) \, dx\)

Optimal. Leaf size=10 \[ \frac {\sinh (a+b x)}{b} \]

[Out]

sinh(b*x+a)/b

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2637} \[ \frac {\sinh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x],x]

[Out]

Sinh[a + b*x]/b

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

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Mathematica [B]  time = 0.01, size = 21, normalized size = 2.10 \[ \frac {\sinh (a) \cosh (b x)}{b}+\frac {\cosh (a) \sinh (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x],x]

[Out]

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

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fricas [A]  time = 0.59, size = 10, normalized size = 1.00 \[ \frac {\sinh \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sinh(b*x + a)/b

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giac [B]  time = 0.13, size = 26, normalized size = 2.60 \[ \frac {e^{\left (b x + a\right )}}{2 \, b} - \frac {e^{\left (-b x - a\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 11, normalized size = 1.10 \[ \frac {\sinh \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sinh(b*x+a)/b

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maxima [A]  time = 0.32, size = 10, normalized size = 1.00 \[ \frac {\sinh \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sinh(b*x + a)/b

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mupad [B]  time = 0.05, size = 10, normalized size = 1.00 \[ \frac {\mathrm {sinh}\left (a+b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x),x)

[Out]

sinh(a + b*x)/b

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sympy [A]  time = 0.13, size = 12, normalized size = 1.20 \[ \begin {cases} \frac {\sinh {\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cosh {\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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