3.44 \(\int (c+d x^2) \cosh ^{-1}(a x) \, dx\)

Optimal. Leaf size=84 \[ -\frac{\sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 c+2 d\right )}{9 a^3}+c x \cosh ^{-1}(a x)-\frac{d x^2 \sqrt{a x-1} \sqrt{a x+1}}{9 a}+\frac{1}{3} d x^3 \cosh ^{-1}(a x) \]

[Out]

-((9*a^2*c + 2*d)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3) - (d*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a) + c*x*Arc
Cosh[a*x] + (d*x^3*ArcCosh[a*x])/3

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Rubi [A]  time = 0.0710306, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5705, 460, 74} \[ -\frac{\sqrt{a x-1} \sqrt{a x+1} \left (9 a^2 c+2 d\right )}{9 a^3}+c x \cosh ^{-1}(a x)-\frac{d x^2 \sqrt{a x-1} \sqrt{a x+1}}{9 a}+\frac{1}{3} d x^3 \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)*ArcCosh[a*x],x]

[Out]

-((9*a^2*c + 2*d)*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a^3) - (d*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(9*a) + c*x*Arc
Cosh[a*x] + (d*x^3*ArcCosh[a*x])/3

Rule 5705

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 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 (c+d x^2\right ) \cosh ^{-1}(a x) \, dx &=c x \cosh ^{-1}(a x)+\frac{1}{3} d x^3 \cosh ^{-1}(a x)-a \int \frac{x \left (c+\frac{d x^2}{3}\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{d x^2 \sqrt{-1+a x} \sqrt{1+a x}}{9 a}+c x \cosh ^{-1}(a x)+\frac{1}{3} d x^3 \cosh ^{-1}(a x)+\frac{1}{9} \left (a \left (-9 c-\frac{2 d}{a^2}\right )\right ) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{\left (9 a^2 c+2 d\right ) \sqrt{-1+a x} \sqrt{1+a x}}{9 a^3}-\frac{d x^2 \sqrt{-1+a x} \sqrt{1+a x}}{9 a}+c x \cosh ^{-1}(a x)+\frac{1}{3} d x^3 \cosh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0596275, size = 60, normalized size = 0.71 \[ \cosh ^{-1}(a x) \left (c x+\frac{d x^3}{3}\right )-\frac{\sqrt{a x-1} \sqrt{a x+1} \left (a^2 \left (9 c+d x^2\right )+2 d\right )}{9 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)*ArcCosh[a*x],x]

[Out]

-(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2*d + a^2*(9*c + d*x^2)))/(9*a^3) + (c*x + (d*x^3)/3)*ArcCosh[a*x]

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Maple [A]  time = 0.009, size = 62, normalized size = 0.7 \begin{align*}{\frac{1}{a} \left ({\frac{a{\rm arccosh} \left (ax\right )d{x}^{3}}{3}}+{\rm arccosh} \left (ax\right )cax-{\frac{{a}^{2}d{x}^{2}+9\,{a}^{2}c+2\,d}{9\,{a}^{2}}\sqrt{ax-1}\sqrt{ax+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)*arccosh(a*x),x)

[Out]

1/a*(1/3*a*arccosh(a*x)*d*x^3+arccosh(a*x)*c*a*x-1/9/a^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(a^2*d*x^2+9*a^2*c+2*d))

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Maxima [A]  time = 1.14981, size = 100, normalized size = 1.19 \begin{align*} -\frac{1}{9} \,{\left (\frac{\sqrt{a^{2} x^{2} - 1} d x^{2}}{a^{2}} + \frac{9 \, \sqrt{a^{2} x^{2} - 1} c}{a^{2}} + \frac{2 \, \sqrt{a^{2} x^{2} - 1} d}{a^{4}}\right )} a + \frac{1}{3} \,{\left (d x^{3} + 3 \, c x\right )} \operatorname{arcosh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)*arccosh(a*x),x, algorithm="maxima")

[Out]

-1/9*(sqrt(a^2*x^2 - 1)*d*x^2/a^2 + 9*sqrt(a^2*x^2 - 1)*c/a^2 + 2*sqrt(a^2*x^2 - 1)*d/a^4)*a + 1/3*(d*x^3 + 3*
c*x)*arccosh(a*x)

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Fricas [A]  time = 2.35689, size = 154, normalized size = 1.83 \begin{align*} \frac{3 \,{\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (a^{2} d x^{2} + 9 \, a^{2} c + 2 \, d\right )} \sqrt{a^{2} x^{2} - 1}}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)*arccosh(a*x),x, algorithm="fricas")

[Out]

1/9*(3*(a^3*d*x^3 + 3*a^3*c*x)*log(a*x + sqrt(a^2*x^2 - 1)) - (a^2*d*x^2 + 9*a^2*c + 2*d)*sqrt(a^2*x^2 - 1))/a
^3

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Sympy [A]  time = 0.742982, size = 90, normalized size = 1.07 \begin{align*} \begin{cases} c x \operatorname{acosh}{\left (a x \right )} + \frac{d x^{3} \operatorname{acosh}{\left (a x \right )}}{3} - \frac{c \sqrt{a^{2} x^{2} - 1}}{a} - \frac{d x^{2} \sqrt{a^{2} x^{2} - 1}}{9 a} - \frac{2 d \sqrt{a^{2} x^{2} - 1}}{9 a^{3}} & \text{for}\: a \neq 0 \\\frac{i \pi \left (c x + \frac{d x^{3}}{3}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)*acosh(a*x),x)

[Out]

Piecewise((c*x*acosh(a*x) + d*x**3*acosh(a*x)/3 - c*sqrt(a**2*x**2 - 1)/a - d*x**2*sqrt(a**2*x**2 - 1)/(9*a) -
2*d*sqrt(a**2*x**2 - 1)/(9*a**3), Ne(a, 0)), (I*pi*(c*x + d*x**3/3)/2, True))

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Giac [A]  time = 1.13231, size = 107, normalized size = 1.27 \begin{align*} \frac{1}{3} \,{\left (d x^{3} + 3 \, c x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{9 \, \sqrt{a^{2} x^{2} - 1} a^{2} c +{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} d + 3 \, \sqrt{a^{2} x^{2} - 1} d}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)*arccosh(a*x),x, algorithm="giac")

[Out]

1/3*(d*x^3 + 3*c*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 1/9*(9*sqrt(a^2*x^2 - 1)*a^2*c + (a^2*x^2 - 1)^(3/2)*d + 3*
sqrt(a^2*x^2 - 1)*d)/a^3