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

Optimal. Leaf size=65 \[ \frac{3 \sqrt{a x-1} \text{Chi}\left (\cosh ^{-1}(a x)\right )}{4 a^4 \sqrt{1-a x}}+\frac{\sqrt{a x-1} \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a^4 \sqrt{1-a x}} \]

[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])

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Rubi [A]  time = 0.450474, antiderivative size = 91, normalized size of antiderivative = 1.4, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5798, 5781, 3312, 3301} \[ \frac{3 \sqrt{a x-1} \sqrt{a x+1} \text{Chi}\left (\cosh ^{-1}(a x)\right )}{4 a^4 \sqrt{1-a^2 x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a^4 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((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[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rule 3312

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 3301

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]

Rubi steps

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

Mathematica [A]  time = 0.0873484, size = 60, normalized size = 0.92 \[ \frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left (3 \text{Chi}\left (\cosh ^{-1}(a x)\right )+\text{Chi}\left (3 \cosh ^{-1}(a x)\right )\right )}{4 a^4 \sqrt{-(a x-1) (a x+1)}} \]

Warning: Unable to verify antiderivative.

[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))])

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Maple [B]  time = 0.21, size = 200, normalized size = 3.1 \begin{align*}{\frac{{\it Ei} \left ( 1,3\,{\rm arccosh} \left (ax\right ) \right ) }{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\it Ei} \left ( 1,-3\,{\rm arccosh} \left (ax\right ) \right ) }{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{3\,{\it Ei} \left ( 1,{\rm arccosh} \left (ax\right ) \right ) }{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{3\,{\it Ei} \left ( 1,-{\rm arccosh} \left (ax\right ) \right ) }{8\,{a}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} x^{3}}{{\left (a^{2} x^{2} - 1\right )} \operatorname{arcosh}\left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{-a^{2} x^{2} + 1} \operatorname{arcosh}\left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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