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

Optimal. Leaf size=46 \[ a x-\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1}}{d}+\frac{b (c+d x) \cosh ^{-1}(c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0235829, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5864, 5654, 74} \[ a x-\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1}}{d}+\frac{b (c+d x) \cosh ^{-1}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcCosh[c + d*x],x]

[Out]

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

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

Mathematica [A]  time = 0.0492835, size = 61, normalized size = 1.33 \[ a x-\frac{b \left (\sqrt{c+d x-1} \sqrt{c+d x+1}-2 c \sinh ^{-1}\left (\frac{\sqrt{c+d x-1}}{\sqrt{2}}\right )\right )}{d}+b x \cosh ^{-1}(c+d x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcCosh[c + d*x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 41, normalized size = 0.9 \begin{align*} ax+{\frac{b}{d} \left ( \left ( dx+c \right ){\rm arccosh} \left (dx+c\right )-\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.22302, size = 47, normalized size = 1.02 \begin{align*} a x + \frac{{\left ({\left (d x + c\right )} \operatorname{arcosh}\left (d x + c\right ) - \sqrt{{\left (d x + c\right )}^{2} - 1}\right )} b}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.16048, size = 154, normalized size = 3.35 \begin{align*} \frac{a d x +{\left (b d x + b c\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.1967, size = 51, normalized size = 1.11 \begin{align*} a x + b \left (\begin{cases} \frac{c \operatorname{acosh}{\left (c + d x \right )}}{d} + x \operatorname{acosh}{\left (c + d x \right )} - \frac{\sqrt{c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} & \text{for}\: d \neq 0 \\x \operatorname{acosh}{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.14232, size = 135, normalized size = 2.93 \begin{align*} -{\left (d{\left (\frac{c \log \left ({\left | -c d -{\left (x{\left | d \right |} - \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{\left | d \right |} \right |}\right )}{d{\left | d \right |}} + \frac{\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt{{\left (d x + c\right )}^{2} - 1}\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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