\(\int \frac {x^3}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx\) [293]

Optimal result

Integrand size = 24, antiderivative size = 65 \[ \int \frac {x^3}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {3 \sqrt {-1+a x} \text {Chi}(\text {arccosh}(a x))}{4 a^4 \sqrt {1-a x}}+\frac {\sqrt {-1+a x} \text {Chi}(3 \text {arccosh}(a x))}{4 a^4 \sqrt {1-a x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5952, 3393, 3382} \[ \int \frac {x^3}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {3 \sqrt {a x-1} \text {Chi}(\text {arccosh}(a x))}{4 a^4 \sqrt {1-a x}}+\frac {\sqrt {a x-1} \text {Chi}(3 \text {arccosh}(a x))}{4 a^4 \sqrt {1-a x}} \]

[In]

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

[Out]

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

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5952

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {x^3}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) (3 \text {Chi}(\text {arccosh}(a x))+\text {Chi}(3 \text {arccosh}(a x)))}{4 a^4 \sqrt {-((-1+a x) (1+a x))}} \]

[In]

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

[Out]

(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(3*CoshIntegral[ArcCosh[a*x]] + CoshIntegral[3*ArcCosh[a*x]]))/(4*a^4*Sq
rt[-((-1 + a*x)*(1 + a*x))])

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \left (\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )+\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (a x \right )\right )+3 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (a x \right )\right )+3 \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (a x \right )\right )\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}\)	\(78\)

[In]

int(x^3/arccosh(a*x)/(-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)*(Ei(1,3*arccosh(a*x))+Ei(1,-3*arccosh(a*x))+3*Ei(1,arccosh(
a*x))+3*Ei(1,-arccosh(a*x)))/a^4/(a^2*x^2-1)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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