3.100 \(\int (a+b \sinh (c+d x)) \, dx\)

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

[Out]

a*x + (b*Cosh[c + d*x])/d

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Rubi [A]  time = 0.0097391, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2638} \[ a x+\frac{b \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sinh[c + d*x],x]

[Out]

a*x + (b*Cosh[c + d*x])/d

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

Mathematica [A]  time = 0.0094447, size = 26, normalized size = 1.73 \[ a x+\frac{b \sinh (c) \sinh (d x)}{d}+\frac{b \cosh (c) \cosh (d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sinh[c + d*x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 16, normalized size = 1.1 \begin{align*} ax+{\frac{b\cosh \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03804, size = 20, normalized size = 1.33 \begin{align*} a x + \frac{b \cosh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*cosh(d*x + c)/d

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Fricas [A]  time = 2.00046, size = 39, normalized size = 2.6 \begin{align*} \frac{a d x + b \cosh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*d*x + b*cosh(d*x + c))/d

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Sympy [A]  time = 0.167423, size = 17, normalized size = 1.13 \begin{align*} a x + b \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+b*sinh(d*x+c),x)

[Out]

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

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Giac [B]  time = 1.24204, size = 42, normalized size = 2.8 \begin{align*} a x + \frac{1}{2} \, b{\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+b*sinh(d*x+c),x, algorithm="giac")

[Out]

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