3.86 \(\int \cosh ^{-1}(a+b x) \, dx\)

Optimal. Leaf size=41 \[ \frac{(a+b x) \cosh ^{-1}(a+b x)}{b}-\frac{\sqrt{a+b x-1} \sqrt{a+b x+1}}{b} \]

[Out]

-((Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/b) + ((a + b*x)*ArcCosh[a + b*x])/b

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Rubi [A]  time = 0.0158476, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5864, 5654, 74} \[ \frac{(a+b x) \cosh ^{-1}(a+b x)}{b}-\frac{\sqrt{a+b x-1} \sqrt{a+b x+1}}{b} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a + b*x],x]

[Out]

-((Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/b) + ((a + b*x)*ArcCosh[a + b*x])/b

Rule 5864

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCosh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rule 5654

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

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

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

Mathematica [A]  time = 0.0362489, size = 56, normalized size = 1.37 \[ x \cosh ^{-1}(a+b x)-\frac{\sqrt{a+b x-1} \sqrt{a+b x+1}-2 a \sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCosh[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.001, size = 36, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ){\rm arccosh} \left (bx+a\right )-\sqrt{bx+a-1}\sqrt{bx+a+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17374, size = 41, normalized size = 1. \begin{align*} \frac{{\left (b x + a\right )} \operatorname{arcosh}\left (b x + a\right ) - \sqrt{{\left (b x + a\right )}^{2} - 1}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.31516, size = 135, normalized size = 3.29 \begin{align*} \frac{{\left (b x + a\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.22608, size = 46, normalized size = 1.12 \begin{align*} \begin{cases} \frac{a \operatorname{acosh}{\left (a + b x \right )}}{b} + x \operatorname{acosh}{\left (a + b x \right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2} - 1}}{b} & \text{for}\: b \neq 0 \\x \operatorname{acosh}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.17564, size = 126, normalized size = 3.07 \begin{align*} -b{\left (\frac{a \log \left ({\left | -a b -{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{\left | b \right |} \right |}\right )}{b{\left | b \right |}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{b^{2}}\right )} + x \log \left (b x + a + \sqrt{{\left (b x + a\right )}^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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