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

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

[Out]

Cosh[a + b*x]/b

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Rubi [A]  time = 0.0056658, antiderivative size = 10, normalized size of antiderivative = 1., 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 = {2638} \[ \frac{\cosh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x],x]

[Out]

Cosh[a + b*x]/b

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

Mathematica [B]  time = 0.0109847, size = 21, normalized size = 2.1 \[ \frac{\sinh (a) \sinh (b x)}{b}+\frac{\cosh (a) \cosh (b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.002, size = 11, normalized size = 1.1 \begin{align*}{\frac{\cosh \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(b*x+a)/b

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Maxima [A]  time = 1.09713, size = 14, normalized size = 1.4 \begin{align*} \frac{\cosh \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(b*x + a)/b

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Fricas [A]  time = 1.9584, size = 23, normalized size = 2.3 \begin{align*} \frac{\cosh \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(b*x + a)/b

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Sympy [A]  time = 0.166357, size = 12, normalized size = 1.2 \begin{align*} \begin{cases} \frac{\cosh{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \sinh{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.43722, size = 35, normalized size = 3.5 \begin{align*} \frac{e^{\left (b x + a\right )}}{2 \, b} + \frac{e^{\left (-b x - a\right )}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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