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

Optimal. Leaf size=195 \[ \frac{2}{125} a^4 c^2 x^5-\frac{76}{675} a^2 c^2 x^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^2-\frac{2 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac{8 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac{16 c^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{15 a}+\frac{298 c^2 x}{225} \]

[Out]

(298*c^2*x)/225 - (76*a^2*c^2*x^3)/675 + (2*a^4*c^2*x^5)/125 - (16*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*
x])/(15*a) + (8*c^2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/(45*a) - (2*c^2*(-1 + a*x)^(5/2)*(1 + a*x)^
(5/2)*ArcCosh[a*x])/(25*a) + (8*c^2*x*ArcCosh[a*x]^2)/15 + (4*c^2*x*(1 - a^2*x^2)*ArcCosh[a*x]^2)/15 + (c^2*x*
(1 - a^2*x^2)^2*ArcCosh[a*x]^2)/5

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Rubi [A]  time = 0.46169, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5681, 5718, 194, 5654, 8} \[ \frac{2}{125} a^4 c^2 x^5-\frac{76}{675} a^2 c^2 x^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^2-\frac{2 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac{8 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac{16 c^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{15 a}+\frac{298 c^2 x}{225} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(298*c^2*x)/225 - (76*a^2*c^2*x^3)/675 + (2*a^4*c^2*x^5)/125 - (16*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*
x])/(15*a) + (8*c^2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x])/(45*a) - (2*c^2*(-1 + a*x)^(5/2)*(1 + a*x)^
(5/2)*ArcCosh[a*x])/(25*a) + (8*c^2*x*ArcCosh[a*x]^2)/15 + (4*c^2*x*(1 - a^2*x^2)*ArcCosh[a*x]^2)/15 + (c^2*x*
(1 - a^2*x^2)^2*ArcCosh[a*x]^2)/5

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 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 )^2 \cosh ^{-1}(a x)^2 \, dx &=\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{1}{5} (4 c) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^2 \, dx-\frac{1}{5} \left (2 a c^2\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx\\ &=-\frac{2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{1}{25} \left (2 c^2\right ) \int \left (-1+a^2 x^2\right )^2 \, dx+\frac{1}{15} \left (8 c^2\right ) \int \cosh ^{-1}(a x)^2 \, dx+\frac{1}{15} \left (8 a c^2\right ) \int x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \, dx\\ &=\frac{8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac{2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{1}{25} \left (2 c^2\right ) \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx-\frac{1}{45} \left (8 c^2\right ) \int \left (-1+a^2 x^2\right ) \, dx-\frac{1}{15} \left (16 a c^2\right ) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{58 c^2 x}{225}-\frac{76}{675} a^2 c^2 x^3+\frac{2}{125} a^4 c^2 x^5-\frac{16 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{15 a}+\frac{8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac{2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2+\frac{1}{15} \left (16 c^2\right ) \int 1 \, dx\\ &=\frac{298 c^2 x}{225}-\frac{76}{675} a^2 c^2 x^3+\frac{2}{125} a^4 c^2 x^5-\frac{16 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{15 a}+\frac{8 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac{2 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^2+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^2+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^2\\ \end{align*}

Mathematica [A]  time = 0.222418, size = 101, normalized size = 0.52 \[ \frac{c^2 \left (54 a^5 x^5-380 a^3 x^3+225 a x \left (3 a^4 x^4-10 a^2 x^2+15\right ) \cosh ^{-1}(a x)^2-30 \sqrt{a x-1} \sqrt{a x+1} \left (9 a^4 x^4-38 a^2 x^2+149\right ) \cosh ^{-1}(a x)+4470 a x\right )}{3375 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(4470*a*x - 380*a^3*x^3 + 54*a^5*x^5 - 30*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(149 - 38*a^2*x^2 + 9*a^4*x^4)*Arc
Cosh[a*x] + 225*a*x*(15 - 10*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*x]^2))/(3375*a)

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Maple [A]  time = 0.047, size = 140, normalized size = 0.7 \begin{align*}{\frac{{c}^{2}}{3375\,a} \left ( 675\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{5}{x}^{5}-270\,{\rm arccosh} \left (ax\right ){a}^{4}{x}^{4}\sqrt{ax-1}\sqrt{ax+1}-2250\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{3}{x}^{3}+1140\,{\rm arccosh} \left (ax\right )\sqrt{ax-1}\sqrt{ax+1}{a}^{2}{x}^{2}+54\,{x}^{5}{a}^{5}+3375\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax-4470\,{\rm arccosh} \left (ax\right )\sqrt{ax-1}\sqrt{ax+1}-380\,{x}^{3}{a}^{3}+4470\,ax \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3375/a*c^2*(675*arccosh(a*x)^2*a^5*x^5-270*arccosh(a*x)*a^4*x^4*(a*x-1)^(1/2)*(a*x+1)^(1/2)-2250*arccosh(a*x
)^2*a^3*x^3+1140*arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^2*x^2+54*x^5*a^5+3375*arccosh(a*x)^2*a*x-4470*arcc
osh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)-380*x^3*a^3+4470*a*x)

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Maxima [A]  time = 1.24127, size = 181, normalized size = 0.93 \begin{align*} \frac{2}{125} \, a^{4} c^{2} x^{5} - \frac{76}{675} \, a^{2} c^{2} x^{3} + \frac{298}{225} \, c^{2} x - \frac{2}{225} \,{\left (9 \, \sqrt{a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 38 \, \sqrt{a^{2} x^{2} - 1} c^{2} x^{2} + \frac{149 \, \sqrt{a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \operatorname{arcosh}\left (a x\right ) + \frac{1}{15} \,{\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname{arcosh}\left (a x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/125*a^4*c^2*x^5 - 76/675*a^2*c^2*x^3 + 298/225*c^2*x - 2/225*(9*sqrt(a^2*x^2 - 1)*a^2*c^2*x^4 - 38*sqrt(a^2*
x^2 - 1)*c^2*x^2 + 149*sqrt(a^2*x^2 - 1)*c^2/a^2)*a*arccosh(a*x) + 1/15*(3*a^4*c^2*x^5 - 10*a^2*c^2*x^3 + 15*c
^2*x)*arccosh(a*x)^2

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Fricas [A]  time = 2.02822, size = 321, normalized size = 1.65 \begin{align*} \frac{54 \, a^{5} c^{2} x^{5} - 380 \, a^{3} c^{2} x^{3} + 4470 \, a c^{2} x + 225 \,{\left (3 \, a^{5} c^{2} x^{5} - 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} - 30 \,{\left (9 \, a^{4} c^{2} x^{4} - 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{3375 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3375*(54*a^5*c^2*x^5 - 380*a^3*c^2*x^3 + 4470*a*c^2*x + 225*(3*a^5*c^2*x^5 - 10*a^3*c^2*x^3 + 15*a*c^2*x)*lo
g(a*x + sqrt(a^2*x^2 - 1))^2 - 30*(9*a^4*c^2*x^4 - 38*a^2*c^2*x^2 + 149*c^2)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(
a^2*x^2 - 1)))/a

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Sympy [A]  time = 4.98755, size = 182, normalized size = 0.93 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{5} \operatorname{acosh}^{2}{\left (a x \right )}}{5} + \frac{2 a^{4} c^{2} x^{5}}{125} - \frac{2 a^{3} c^{2} x^{4} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{25} - \frac{2 a^{2} c^{2} x^{3} \operatorname{acosh}^{2}{\left (a x \right )}}{3} - \frac{76 a^{2} c^{2} x^{3}}{675} + \frac{76 a c^{2} x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{225} + c^{2} x \operatorname{acosh}^{2}{\left (a x \right )} + \frac{298 c^{2} x}{225} - \frac{298 c^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{225 a} & \text{for}\: a \neq 0 \\- \frac{\pi ^{2} c^{2} 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)**2*acosh(a*x)**2,x)

[Out]

Piecewise((a**4*c**2*x**5*acosh(a*x)**2/5 + 2*a**4*c**2*x**5/125 - 2*a**3*c**2*x**4*sqrt(a**2*x**2 - 1)*acosh(
a*x)/25 - 2*a**2*c**2*x**3*acosh(a*x)**2/3 - 76*a**2*c**2*x**3/675 + 76*a*c**2*x**2*sqrt(a**2*x**2 - 1)*acosh(
a*x)/225 + c**2*x*acosh(a*x)**2 + 298*c**2*x/225 - 298*c**2*sqrt(a**2*x**2 - 1)*acosh(a*x)/(225*a), Ne(a, 0)),
(-pi**2*c**2*x/4, True))

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Giac [A]  time = 1.19807, size = 184, normalized size = 0.94 \begin{align*} \frac{2}{3375} \,{\left (27 \, a^{4} x^{5} - 190 \, a^{2} x^{3} + 2235 \, x - \frac{15 \,{\left (9 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} - 20 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 120 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )}{a}\right )} c^{2} + \frac{1}{15} \,{\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3375*(27*a^4*x^5 - 190*a^2*x^3 + 2235*x - 15*(9*(a^2*x^2 - 1)^(5/2) - 20*(a^2*x^2 - 1)^(3/2) + 120*sqrt(a^2*
x^2 - 1))*log(a*x + sqrt(a^2*x^2 - 1))/a)*c^2 + 1/15*(3*a^4*c^2*x^5 - 10*a^2*c^2*x^3 + 15*c^2*x)*log(a*x + sqr
t(a^2*x^2 - 1))^2