\(\int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\) [255]

Optimal result

Integrand size = 24, antiderivative size = 188 \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {3 x^2 \sqrt {-1+a x}}{8 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{4 a^2}+\frac {3 \sqrt {-1+a x} \text {arccosh}(a x)^2}{8 a^3 \sqrt {1-a x}}-\frac {3 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5938, 5892, 5883, 5939, 5893, 30} \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {a x-1} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}+\frac {3 \sqrt {a x-1} \text {arccosh}(a x)^2}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}-\frac {3 x \sqrt {1-a x} \sqrt {a x+1} \text {arccosh}(a x)}{4 a^2}-\frac {3 x^2 \sqrt {a x-1} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {3 x^2 \sqrt {a x-1}}{8 a \sqrt {1-a x}} \]

[In]

Int[(x^2*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]

[Out]

(-3*x^2*Sqrt[-1 + a*x])/(8*a*Sqrt[1 - a*x]) - (3*x*Sqrt[1 - a*x]*Sqrt[1 + a*x]*ArcCosh[a*x])/(4*a^2) + (3*Sqrt
[-1 + a*x]*ArcCosh[a*x]^2)/(8*a^3*Sqrt[1 - a*x]) - (3*x^2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2)/(4*a*Sqrt[1 - a*x]) -
(x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3)/(2*a^2) + (Sqrt[-1 + a*x]*ArcCosh[a*x]^4)/(8*a^3*Sqrt[1 - a*x])

Rule 30

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

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 5893

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

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*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, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[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)^3}{2 a^2}+\frac {\int \frac {\text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}-\frac {\left (3 \sqrt {-1+a x}\right ) \int x \text {arccosh}(a x)^2 \, dx}{2 a \sqrt {1-a x}} \\ & = -\frac {3 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}+\frac {\left (3 \sqrt {-1+a x}\right ) \int \frac {x^2 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a x}} \\ & = -\frac {3 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{4 a^2}-\frac {3 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}}+\frac {\left (3 \sqrt {-1+a x}\right ) \int \frac {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a^2 \sqrt {1-a x}}-\frac {\left (3 \sqrt {-1+a x}\right ) \int x \, dx}{4 a \sqrt {1-a x}} \\ & = -\frac {3 x^2 \sqrt {-1+a x}}{8 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a x} \sqrt {1+a x} \text {arccosh}(a x)}{4 a^2}+\frac {3 \sqrt {-1+a x} \text {arccosh}(a x)^2}{8 a^3 \sqrt {1-a x}}-\frac {3 x^2 \sqrt {-1+a x} \text {arccosh}(a x)^2}{4 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \text {arccosh}(a x)^3}{2 a^2}+\frac {\sqrt {-1+a x} \text {arccosh}(a x)^4}{8 a^3 \sqrt {1-a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {-((-1+a x) (1+a x))} \left (-3 \left (1+2 \text {arccosh}(a x)^2\right ) \cosh (2 \text {arccosh}(a x))+2 \text {arccosh}(a x) \left (\text {arccosh}(a x)^3+\left (3+2 \text {arccosh}(a x)^2\right ) \sinh (2 \text {arccosh}(a x))\right )\right )}{16 a^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]

[In]

Integrate[(x^2*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]

[Out]

-1/16*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(-3*(1 + 2*ArcCosh[a*x]^2)*Cosh[2*ArcCosh[a*x]] + 2*ArcCosh[a*x]*(ArcCosh
[a*x]^3 + (3 + 2*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 = 0.70 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.36

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{4}}{8 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-3\right )}{32 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}+6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+3\right )}{32 a^{3} \left (a^{2} x^{2}-1\right )}\)	\(255\)

[In]

int(x^2*arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*x^2*arccosh(a*x)^3/(a^2*x^2 - 1), x)

Sympy [F]

\[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^{2} \operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[In]

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

[Out]

Integral(x**2*acosh(a*x)**3/sqrt(-(a*x - 1)*(a*x + 1)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^2*arccosh(a*x)^3/(-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]

\[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int { \frac {x^{2} \operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

integrate(x^2*arccosh(a*x)^3/sqrt(-a^2*x^2 + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \text {arccosh}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx=\int \frac {x^2\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]

[In]

int((x^2*acosh(a*x)^3)/(1 - a^2*x^2)^(1/2),x)

[Out]

int((x^2*acosh(a*x)^3)/(1 - a^2*x^2)^(1/2), x)