\(\int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [227]

Optimal result

Integrand size = 24, antiderivative size = 151 \[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {x \sqrt {1-a x} \sqrt {1+a x}}{4 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)}{4 a^3 \sqrt {1-a x}}-\frac {x^2 \sqrt {-1+a x} \text {arccosh}(a x)}{2 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^3}{6 a^3 \sqrt {1-a x}} \]

[Out]

1/4*arccosh(a*x)*(a*x-1)^(1/2)/a^3/(-a*x+1)^(1/2)-1/2*x^2*arccosh(a*x)*(a*x-1)^(1/2)/a/(-a*x+1)^(1/2)+1/6*arcc
osh(a*x)^3*(a*x-1)^(1/2)/a^3/(-a*x+1)^(1/2)-1/4*x*(-a*x+1)^(1/2)*(a*x+1)^(1/2)/a^2-1/2*x*arccosh(a*x)^2*(-a^2*
x^2+1)^(1/2)/a^2

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5938, 5892, 5883, 92, 54} \[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {a x-1} \text {arccosh}(a x)^3}{6 a^3 \sqrt {1-a x}}+\frac {\sqrt {a x-1} \text {arccosh}(a x)}{4 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{2 a^2}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{4 a^2}-\frac {x^2 \sqrt {a x-1} \text {arccosh}(a x)}{2 a \sqrt {1-a x}} \]

[In]

Int[(x^2*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]

[Out]

-1/4*(x*Sqrt[1 - a*x]*Sqrt[1 + a*x])/a^2 + (Sqrt[-1 + a*x]*ArcCosh[a*x])/(4*a^3*Sqrt[1 - a*x]) - (x^2*Sqrt[-1
+ a*x]*ArcCosh[a*x])/(2*a*Sqrt[1 - a*x]) - (x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)/(2*a^2) + (Sqrt[-1 + a*x]*ArcC
osh[a*x]^3)/(6*a^3*Sqrt[1 - a*x])

Rule 54

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5892

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])]*(a + b*ArcCosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e,
n}, x] && EqQ[c^2*d + e, 0] && NeQ[n, -1]

Rule 5938

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

Rubi steps \begin{align*} \text {integral}& = -\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{2 a^2}+\frac {\int \frac {\text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}-\frac {\sqrt {-1+a x} \int x \text {arccosh}(a x) \, dx}{a \sqrt {1-a x}} \\ & = -\frac {x^2 \sqrt {-1+a x} \text {arccosh}(a x)}{2 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^3}{6 a^3 \sqrt {1-a x}}+\frac {\sqrt {-1+a x} \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a x}} \\ & = -\frac {x \sqrt {1-a x} \sqrt {1+a x}}{4 a^2}-\frac {x^2 \sqrt {-1+a x} \text {arccosh}(a x)}{2 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^3}{6 a^3 \sqrt {1-a x}}+\frac {\sqrt {-1+a x} \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a^2 \sqrt {1-a x}} \\ & = -\frac {x \sqrt {1-a x} \sqrt {1+a x}}{4 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)}{4 a^3 \sqrt {1-a x}}-\frac {x^2 \sqrt {-1+a x} \text {arccosh}(a x)}{2 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^3}{6 a^3 \sqrt {1-a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.58 \[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-((-1+a x) (1+a x))} \left (4 \text {arccosh}(a x)^3-6 \text {arccosh}(a x) \cosh (2 \text {arccosh}(a x))+\left (3+6 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))\right )}{24 a^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]

[In]

Integrate[(x^2*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]

[Out]

-1/24*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(4*ArcCosh[a*x]^3 - 6*ArcCosh[a*x]*Cosh[2*ArcCosh[a*x]] + (3 + 6*ArcCosh[
a*x]^2)*Sinh[2*ArcCosh[a*x]]))/(a^3*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.58

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{3}}{6 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \operatorname {arccosh}\left (a x \right )^{2}-2 \,\operatorname {arccosh}\left (a x \right )+1\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \operatorname {arccosh}\left (a x \right )^{2}+2 \,\operatorname {arccosh}\left (a x \right )+1\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}\)	\(239\)

[In]

int(x^2*arccosh(a*x)^2/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^3/(a^2*x^2-1)*arccosh(a*x)^3-1/16*(-a^2*x^2+1)^(1/2)*(2*
a^3*x^3-2*a*x+2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(2*arccosh(a*x)^2-2*arccosh(a
*x)+1)/a^3/(a^2*x^2-1)-1/16*(-a^2*x^2+1)^(1/2)*(2*a^3*x^3-2*a*x-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+(a*x-1)^
(1/2)*(a*x+1)^(1/2))*(2*arccosh(a*x)^2+2*arccosh(a*x)+1)/a^3/(a^2*x^2-1)

Fricas [F]

\[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*x^2*arccosh(a*x)^2/(a^2*x^2 - 1), x)

Sympy [F]

\[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{2} \operatorname {acosh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[In]

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

[Out]

Integral(x**2*acosh(a*x)**2/sqrt(-(a*x - 1)*(a*x + 1)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

\[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

integrate(x^2*arccosh(a*x)^2/sqrt(-a^2*x^2 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \text {arccosh}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^2\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]

[In]

int((x^2*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2),x)

[Out]

int((x^2*acosh(a*x)^2)/(1 - a^2*x^2)^(1/2), x)