3.7 \(\int (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=30 \[ a x-\frac{b \sqrt{c^2 x^2+1}}{c}+b x \sinh ^{-1}(c x) \]

[Out]

a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]

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Rubi [A]  time = 0.0132673, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5653, 261} \[ a x-\frac{b \sqrt{c^2 x^2+1}}{c}+b x \sinh ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcSinh[c*x],x]

[Out]

a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]

Rule 5653

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=a x+b \int \sinh ^{-1}(c x) \, dx\\ &=a x+b x \sinh ^{-1}(c x)-(b c) \int \frac{x}{\sqrt{1+c^2 x^2}} \, dx\\ &=a x-\frac{b \sqrt{1+c^2 x^2}}{c}+b x \sinh ^{-1}(c x)\\ \end{align*}

Mathematica [A]  time = 0.007988, size = 30, normalized size = 1. \[ a x-\frac{b \sqrt{c^2 x^2+1}}{c}+b x \sinh ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcSinh[c*x],x]

[Out]

a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]

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Maple [A]  time = 0.004, size = 31, normalized size = 1. \begin{align*} ax+{\frac{b}{c} \left ({\it Arcsinh} \left ( cx \right ) cx-\sqrt{{c}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arcsinh(c*x),x)

[Out]

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

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Maxima [A]  time = 1.03064, size = 41, normalized size = 1.37 \begin{align*} a x + \frac{{\left (c x \operatorname{arsinh}\left (c x\right ) - \sqrt{c^{2} x^{2} + 1}\right )} b}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b/c

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Fricas [A]  time = 2.43868, size = 95, normalized size = 3.17 \begin{align*} \frac{b c x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + a c x - \sqrt{c^{2} x^{2} + 1} b}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c*x*log(c*x + sqrt(c^2*x^2 + 1)) + a*c*x - sqrt(c^2*x^2 + 1)*b)/c

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Sympy [A]  time = 0.160008, size = 26, normalized size = 0.87 \begin{align*} a x + b \left (\begin{cases} x \operatorname{asinh}{\left (c x \right )} - \frac{\sqrt{c^{2} x^{2} + 1}}{c} & \text{for}\: c \neq 0 \\0 & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*asinh(c*x),x)

[Out]

a*x + b*Piecewise((x*asinh(c*x) - sqrt(c**2*x**2 + 1)/c, Ne(c, 0)), (0, True))

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Giac [A]  time = 1.28654, size = 55, normalized size = 1.83 \begin{align*}{\left (x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) - \frac{\sqrt{c^{2} x^{2} + 1}}{c}\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*log(c*x + sqrt(c^2*x^2 + 1)) - sqrt(c^2*x^2 + 1)/c)*b + a*x