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

Optimal. Leaf size=112 \[ -\frac{2}{27} a^2 c x^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{2}{3} c x \cosh ^{-1}(a x)^2+\frac{2 c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{9 a}-\frac{4 c \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{3 a}+\frac{14 c x}{9} \]

[Out]

(14*c*x)/9 - (2*a^2*c*x^3)/27 - (4*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*a) + (2*c*(-1 + a*x)^(3/2)*
(1 + a*x)^(3/2)*ArcCosh[a*x])/(9*a) + (2*c*x*ArcCosh[a*x]^2)/3 + (c*x*(1 - a^2*x^2)*ArcCosh[a*x]^2)/3

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Rubi [A]  time = 0.263092, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5681, 5718, 5654, 8} \[ -\frac{2}{27} a^2 c x^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{2}{3} c x \cosh ^{-1}(a x)^2+\frac{2 c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{9 a}-\frac{4 c \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{3 a}+\frac{14 c x}{9} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)*ArcCosh[a*x]^2,x]

[Out]

(14*c*x)/9 - (2*a^2*c*x^3)/27 - (4*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(3*a) + (2*c*(-1 + a*x)^(3/2)*
(1 + a*x)^(3/2)*ArcCosh[a*x])/(9*a) + (2*c*x*ArcCosh[a*x]^2)/3 + (c*x*(1 - a^2*x^2)*ArcCosh[a*x]^2)/3

Rule 5681

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

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2 \, dx &=\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{1}{3} (2 c) \int \cosh ^{-1}(a x)^2 \, dx+\frac{1}{3} (2 a c) \int x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \, dx\\ &=\frac{2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^2+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2-\frac{1}{9} (2 c) \int \left (-1+a^2 x^2\right ) \, dx-\frac{1}{3} (4 a c) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{2 c x}{9}-\frac{2}{27} a^2 c x^3-\frac{4 c \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 a}+\frac{2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^2+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{1}{3} (4 c) \int 1 \, dx\\ &=\frac{14 c x}{9}-\frac{2}{27} a^2 c x^3-\frac{4 c \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{3 a}+\frac{2 c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{9 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^2+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2\\ \end{align*}

Mathematica [A]  time = 0.10888, size = 73, normalized size = 0.65 \[ \frac{c \left (-2 a^3 x^3-9 a x \left (a^2 x^2-3\right ) \cosh ^{-1}(a x)^2+6 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2-7\right ) \cosh ^{-1}(a x)+42 a x\right )}{27 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - a^2*c*x^2)*ArcCosh[a*x]^2,x]

[Out]

(c*(42*a*x - 2*a^3*x^3 + 6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-7 + a^2*x^2)*ArcCosh[a*x] - 9*a*x*(-3 + a^2*x^2)*Arc
Cosh[a*x]^2))/(27*a)

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Maple [A]  time = 0.04, size = 90, normalized size = 0.8 \begin{align*} -{\frac{c}{27\,a} \left ( 9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{3}{x}^{3}-6\,{\rm arccosh} \left (ax\right )\sqrt{ax-1}\sqrt{ax+1}{a}^{2}{x}^{2}-27\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax+42\,{\rm arccosh} \left (ax\right )\sqrt{ax-1}\sqrt{ax+1}+2\,{x}^{3}{a}^{3}-42\,ax \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27/a*c*(9*arccosh(a*x)^2*a^3*x^3-6*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^2*x^2-27*arccosh(a*x)^2*a*x+4
2*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+2*x^3*a^3-42*a*x)

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Maxima [A]  time = 1.15525, size = 103, normalized size = 0.92 \begin{align*} -\frac{2}{27} \, a^{2} c x^{3} + \frac{2}{9} \,{\left (\sqrt{a^{2} x^{2} - 1} c x^{2} - \frac{7 \, \sqrt{a^{2} x^{2} - 1} c}{a^{2}}\right )} a \operatorname{arcosh}\left (a x\right ) - \frac{1}{3} \,{\left (a^{2} c x^{3} - 3 \, c x\right )} \operatorname{arcosh}\left (a x\right )^{2} + \frac{14}{9} \, c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/27*a^2*c*x^3 + 2/9*(sqrt(a^2*x^2 - 1)*c*x^2 - 7*sqrt(a^2*x^2 - 1)*c/a^2)*a*arccosh(a*x) - 1/3*(a^2*c*x^3 -
3*c*x)*arccosh(a*x)^2 + 14/9*c*x

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Fricas [A]  time = 2.08833, size = 216, normalized size = 1.93 \begin{align*} -\frac{2 \, a^{3} c x^{3} - 42 \, a c x + 9 \,{\left (a^{3} c x^{3} - 3 \, a c x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 6 \,{\left (a^{2} c x^{2} - 7 \, c\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{27 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(2*a^3*c*x^3 - 42*a*c*x + 9*(a^3*c*x^3 - 3*a*c*x)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 6*(a^2*c*x^2 - 7*c)*s
qrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1)))/a

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Sympy [A]  time = 1.25717, size = 105, normalized size = 0.94 \begin{align*} \begin{cases} - \frac{a^{2} c x^{3} \operatorname{acosh}^{2}{\left (a x \right )}}{3} - \frac{2 a^{2} c x^{3}}{27} + \frac{2 a c x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{9} + c x \operatorname{acosh}^{2}{\left (a x \right )} + \frac{14 c x}{9} - \frac{14 c \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{9 a} & \text{for}\: a \neq 0 \\- \frac{\pi ^{2} c x}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c*x**3*acosh(a*x)**2/3 - 2*a**2*c*x**3/27 + 2*a*c*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)/9 + c*x
*acosh(a*x)**2 + 14*c*x/9 - 14*c*sqrt(a**2*x**2 - 1)*acosh(a*x)/(9*a), Ne(a, 0)), (-pi**2*c*x/4, True))

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Giac [A]  time = 1.19174, size = 127, normalized size = 1.13 \begin{align*} -\frac{1}{3} \,{\left (a^{2} c x^{3} - 3 \, c x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - \frac{2}{27} \,{\left (a^{2} x^{3} - 21 \, x - \frac{3 \,{\left ({\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} - 6 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a}\right )} c \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a^2*c*x^3 - 3*c*x)*log(a*x + sqrt(a^2*x^2 - 1))^2 - 2/27*(a^2*x^3 - 21*x - 3*((a^2*x^2 - 1)^(3/2) - 6*sq
rt(a^2*x^2 - 1))*log(a*x + sqrt(a^2*x^2 - 1))/a)*c