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

Optimal. Leaf size=231 \[ -\frac{3 a x^2 \sqrt{c-a^2 c x^2}}{8 \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3}{4} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x) \]

[Out]

(-3*a*x^2*Sqrt[c - a^2*c*x^2])/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x])/4 + (
3*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*ArcCos
h[a*x]^2)/(4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3)/2 - (Sqrt[c - a^2*c*x^2]*A
rcCosh[a*x]^4)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

________________________________________________________________________________________

Rubi [A]  time = 0.536833, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {5713, 5683, 5676, 5662, 5759, 30} \[ -\frac{3 a x^2 \sqrt{c-a^2 c x^2}}{8 \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3}{4} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3,x]

[Out]

(-3*a*x^2*Sqrt[c - a^2*c*x^2])/(8*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x])/4 + (
3*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*a*x^2*Sqrt[c - a^2*c*x^2]*ArcCos
h[a*x]^2)/(4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3)/2 - (Sqrt[c - a^2*c*x^2]*A
rcCosh[a*x]^4)/(8*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

Rule 5713

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

Rule 5683

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

Rule 5676

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

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3 \, dx &=\frac{\sqrt{c-a^2 c x^2} \int \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3 \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{\sqrt{c-a^2 c x^2} \int \frac{\cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 a \sqrt{c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^2 \, dx}{2 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 a^2 \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3}{4} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 \sqrt{c-a^2 c x^2}\right ) \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{4 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 a \sqrt{c-a^2 c x^2}\right ) \int x \, dx}{4 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{3 a x^2 \sqrt{c-a^2 c x^2}}{8 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3}{4} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.200555, size = 98, normalized size = 0.42 \[ -\frac{\sqrt{-c (a x-1) (a x+1)} \left (2 \cosh ^{-1}(a x)^4+\left (6 \cosh ^{-1}(a x)^2+3\right ) \cosh \left (2 \cosh ^{-1}(a x)\right )-2 \left (2 \cosh ^{-1}(a x)^2+3\right ) \cosh ^{-1}(a x) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{16 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(2*ArcCosh[a*x]^4 + (3 + 6*ArcCosh[a*x]^2)*Cosh[2*ArcCosh[a*x]] - 2*ArcCosh[
a*x]*(3 + 2*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]]))/(16*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))

________________________________________________________________________________________

Maple [A]  time = 0.204, size = 256, normalized size = 1.1 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}}{8\,a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{ax-1}}}{\frac{1}{\sqrt{ax+1}}}}+{\frac{4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}-6\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )-3}{ \left ( 32\,ax-32 \right ) \left ( ax+1 \right ) a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,{x}^{3}{a}^{3}-2\,ax+2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}+6\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )+3}{ \left ( 32\,ax-32 \right ) \left ( ax+1 \right ) a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,{x}^{3}{a}^{3}-2\,ax-2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^3*(-a^2*c*x^2+c)^(1/2),x)

[Out]

-1/8*(-c*(a^2*x^2-1))^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)/a*arccosh(a*x)^4+1/32*(-c*(a^2*x^2-1))^(1/2)*(2*x^3*a^
3-2*a*x+2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a*x)^3-6*arccosh(a*x)^2+
6*arccosh(a*x)-3)/(a*x-1)/(a*x+1)/a+1/32*(-c*(a^2*x^2-1))^(1/2)*(2*x^3*a^3-2*a*x-2*(a*x+1)^(1/2)*(a*x-1)^(1/2)
*x^2*a^2+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a*x)^3+6*arccosh(a*x)^2+6*arccosh(a*x)+3)/(a*x-1)/(a*x+1)/a

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3*(-a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3*(-a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{acosh}^{3}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**3*(-a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(sqrt(-c*(a*x - 1)*(a*x + 1))*acosh(a*x)**3, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^3*(-a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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