3.242 \(\int (c-a^2 c x^2) \cosh ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=175 \[ -\frac{2}{9} a^2 c x^3 \cosh ^{-1}(a x)+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{2}{27} a c x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{122 c \sqrt{a x-1} \sqrt{a x+1}}{27 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^3+\frac{14}{3} c x \cosh ^{-1}(a x)+\frac{c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{3 a}-\frac{2 c \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{a} \]

[Out]

(-122*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (2*a*c*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/27 + (14*c*x*ArcCosh[a
*x])/3 - (2*a^2*c*x^3*ArcCosh[a*x])/9 - (2*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + (c*(-1 + a*x)^(3
/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(3*a) + (2*c*x*ArcCosh[a*x]^3)/3 + (c*x*(1 - a^2*x^2)*ArcCosh[a*x]^3)/3

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Rubi [A]  time = 0.477469, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {5681, 5718, 5680, 12, 460, 74, 5654} \[ -\frac{2}{9} a^2 c x^3 \cosh ^{-1}(a x)+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{2}{27} a c x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{122 c \sqrt{a x-1} \sqrt{a x+1}}{27 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^3+\frac{14}{3} c x \cosh ^{-1}(a x)+\frac{c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{3 a}-\frac{2 c \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{a} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)*ArcCosh[a*x]^3,x]

[Out]

(-122*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(27*a) + (2*a*c*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/27 + (14*c*x*ArcCosh[a
*x])/3 - (2*a^2*c*x^3*ArcCosh[a*x])/9 - (2*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/a + (c*(-1 + a*x)^(3
/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(3*a) + (2*c*x*ArcCosh[a*x]^3)/3 + (c*x*(1 - a^2*x^2)*ArcCosh[a*x]^3)/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 5680

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] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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]

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^3 \, dx &=\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{3} (2 c) \int \cosh ^{-1}(a x)^3 \, dx+(a c) \int x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac{c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3-\frac{1}{3} (2 c) \int \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x) \, dx-(2 a c) \int \frac{x \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{2}{3} c x \cosh ^{-1}(a x)-\frac{2}{9} a^2 c x^3 \cosh ^{-1}(a x)-\frac{2 c \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a}+\frac{c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+(4 c) \int \cosh ^{-1}(a x) \, dx+\frac{1}{3} (2 a c) \int \frac{x \left (-3+a^2 x^2\right )}{3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{14}{3} c x \cosh ^{-1}(a x)-\frac{2}{9} a^2 c x^3 \cosh ^{-1}(a x)-\frac{2 c \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a}+\frac{c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{9} (2 a c) \int \frac{x \left (-3+a^2 x^2\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-(4 a c) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{4 c \sqrt{-1+a x} \sqrt{1+a x}}{a}+\frac{2}{27} a c x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{14}{3} c x \cosh ^{-1}(a x)-\frac{2}{9} a^2 c x^3 \cosh ^{-1}(a x)-\frac{2 c \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a}+\frac{c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3-\frac{1}{27} (14 a c) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{122 c \sqrt{-1+a x} \sqrt{1+a x}}{27 a}+\frac{2}{27} a c x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{14}{3} c x \cosh ^{-1}(a x)-\frac{2}{9} a^2 c x^3 \cosh ^{-1}(a x)-\frac{2 c \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{a}+\frac{c (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{3 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^3+\frac{1}{3} c x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.116233, size = 109, normalized size = 0.62 \[ \frac{c \left (2 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2-61\right )-9 a x \left (a^2 x^2-3\right ) \cosh ^{-1}(a x)^3+9 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 x^2-7\right ) \cosh ^{-1}(a x)^2-6 a x \left (a^2 x^2-21\right ) \cosh ^{-1}(a x)\right )}{27 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - a^2*c*x^2)*ArcCosh[a*x]^3,x]

[Out]

(c*(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-61 + a^2*x^2) - 6*a*x*(-21 + a^2*x^2)*ArcCosh[a*x] + 9*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]*(-7 + a^2*x^2)*ArcCosh[a*x]^2 - 9*a*x*(-3 + a^2*x^2)*ArcCosh[a*x]^3))/(27*a)

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Maple [A]  time = 0.049, size = 140, normalized size = 0.8 \begin{align*} -{\frac{c}{27\,a} \left ( 9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{3}{x}^{3}-9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}{a}^{2}{x}^{2}-27\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}ax+63\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}+6\,{\rm arccosh} \left (ax\right ){a}^{3}{x}^{3}-2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-126\,ax{\rm arccosh} \left (ax\right )+122\,\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)*arccosh(a*x)^3,x)

[Out]

-1/27/a*c*(9*arccosh(a*x)^3*a^3*x^3-9*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^2*x^2-27*arccosh(a*x)^3*a*x
+63*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+6*arccosh(a*x)*a^3*x^3-2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2-12
6*a*x*arccosh(a*x)+122*(a*x-1)^(1/2)*(a*x+1)^(1/2))

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Maxima [A]  time = 1.22694, size = 167, normalized size = 0.95 \begin{align*} \frac{1}{3} \,{\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 )^{2} - \frac{1}{3} \,{\left (a^{2} c x^{3} - 3 \, c x\right )} \operatorname{arcosh}\left (a x\right )^{3} + \frac{2}{27} \,{\left (\sqrt{a^{2} x^{2} - 1} c x^{2} - \frac{3 \,{\left (a^{2} c x^{3} - 21 \, c x\right )} \operatorname{arcosh}\left (a x\right )}{a} - \frac{61 \, \sqrt{a^{2} x^{2} - 1} c}{a^{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)*arccosh(a*x)^3,x, algorithm="maxima")

[Out]

1/3*(sqrt(a^2*x^2 - 1)*c*x^2 - 7*sqrt(a^2*x^2 - 1)*c/a^2)*a*arccosh(a*x)^2 - 1/3*(a^2*c*x^3 - 3*c*x)*arccosh(a
*x)^3 + 2/27*(sqrt(a^2*x^2 - 1)*c*x^2 - 3*(a^2*c*x^3 - 21*c*x)*arccosh(a*x)/a - 61*sqrt(a^2*x^2 - 1)*c/a^2)*a

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Fricas [A]  time = 2.17679, size = 316, normalized size = 1.81 \begin{align*} -\frac{9 \,{\left (a^{3} c x^{3} - 3 \, a c x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 9 \,{\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 )^{2} + 6 \,{\left (a^{3} c x^{3} - 21 \, a c x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 2 \,{\left (a^{2} c x^{2} - 61 \, c\right )} \sqrt{a^{2} x^{2} - 1}}{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)^3,x, algorithm="fricas")

[Out]

-1/27*(9*(a^3*c*x^3 - 3*a*c*x)*log(a*x + sqrt(a^2*x^2 - 1))^3 - 9*(a^2*c*x^2 - 7*c)*sqrt(a^2*x^2 - 1)*log(a*x
+ sqrt(a^2*x^2 - 1))^2 + 6*(a^3*c*x^3 - 21*a*c*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(a^2*c*x^2 - 61*c)*sqrt(a^2
*x^2 - 1))/a

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Sympy [A]  time = 2.48989, size = 160, normalized size = 0.91 \begin{align*} \begin{cases} - \frac{a^{2} c x^{3} \operatorname{acosh}^{3}{\left (a x \right )}}{3} - \frac{2 a^{2} c x^{3} \operatorname{acosh}{\left (a x \right )}}{9} + \frac{a c x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{3} + \frac{2 a c x^{2} \sqrt{a^{2} x^{2} - 1}}{27} + c x \operatorname{acosh}^{3}{\left (a x \right )} + \frac{14 c x \operatorname{acosh}{\left (a x \right )}}{3} - \frac{7 c \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{3 a} - \frac{122 c \sqrt{a^{2} x^{2} - 1}}{27 a} & \text{for}\: a \neq 0 \\- \frac{i \pi ^{3} c x}{8} & \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)**3,x)

[Out]

Piecewise((-a**2*c*x**3*acosh(a*x)**3/3 - 2*a**2*c*x**3*acosh(a*x)/9 + a*c*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)
**2/3 + 2*a*c*x**2*sqrt(a**2*x**2 - 1)/27 + c*x*acosh(a*x)**3 + 14*c*x*acosh(a*x)/3 - 7*c*sqrt(a**2*x**2 - 1)*
acosh(a*x)**2/(3*a) - 122*c*sqrt(a**2*x**2 - 1)/(27*a), Ne(a, 0)), (-I*pi**3*c*x/8, True))

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Giac [A]  time = 1.31812, size = 196, normalized size = 1.12 \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 )^{3} - \frac{1}{27} \,{\left (6 \,{\left (a^{2} x^{3} - 21 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{9 \,{\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 )^{2}}{a} - \frac{2 \,{\left ({\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} - 60 \, \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)^3,x, algorithm="giac")

[Out]

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