3.292 \(\int \frac{x^4}{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=98 \[ \frac{\sqrt{a x-1} \text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^5 \sqrt{1-a x}}+\frac{\sqrt{a x-1} \text{Chi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^5 \sqrt{1-a x}}+\frac{3 \sqrt{a x-1} \log \left (\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt{1-a x}} \]

[Out]

(Sqrt[-1 + a*x]*CoshIntegral[2*ArcCosh[a*x]])/(2*a^5*Sqrt[1 - a*x]) + (Sqrt[-1 + a*x]*CoshIntegral[4*ArcCosh[a
*x]])/(8*a^5*Sqrt[1 - a*x]) + (3*Sqrt[-1 + a*x]*Log[ArcCosh[a*x]])/(8*a^5*Sqrt[1 - a*x])

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Rubi [A]  time = 0.465873, antiderivative size = 137, 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} \[ \frac{\sqrt{a x-1} \sqrt{a x+1} \text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^5 \sqrt{1-a^2 x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \text{Chi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^5 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \log \left (\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]

[Out]

(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*CoshIntegral[2*ArcCosh[a*x]])/(2*a^5*Sqrt[1 - a^2*x^2]) + (Sqrt[-1 + a*x]*Sqrt[1
+ a*x]*CoshIntegral[4*ArcCosh[a*x]])/(8*a^5*Sqrt[1 - a^2*x^2]) + (3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[ArcCosh[
a*x]])/(8*a^5*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^4}{\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^4}{\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 ^4(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5 \sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 x}+\frac{\cosh (2 x)}{2 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5 \sqrt{1-a^2 x^2}}\\ &=\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \log \left (\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt{1-a^2 x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt{1-a^2 x^2}}+\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5 \sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{-1+a x} \sqrt{1+a x} \text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^5 \sqrt{1-a^2 x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \text{Chi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^5 \sqrt{1-a^2 x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \log \left (\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.112654, size = 69, normalized size = 0.7 \[ \frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left (4 \text{Chi}\left (2 \cosh ^{-1}(a x)\right )+\text{Chi}\left (4 \cosh ^{-1}(a x)\right )+3 \log \left (\cosh ^{-1}(a x)\right )\right )}{8 a^5 \sqrt{-(a x-1) (a x+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]

[Out]

(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(4*CoshIntegral[2*ArcCosh[a*x]] + CoshIntegral[4*ArcCosh[a*x]] + 3*Log[A
rcCosh[a*x]]))/(8*a^5*Sqrt[-((-1 + a*x)*(1 + a*x))])

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Maple [B]  time = 0.263, size = 249, normalized size = 2.5 \begin{align*}{\frac{{\it Ei} \left ( 1,4\,{\rm arccosh} \left (ax\right ) \right ) }{16\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\it Ei} \left ( 1,-4\,{\rm arccosh} \left (ax\right ) \right ) }{16\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3\,\ln \left ({\rm arccosh} \left (ax\right ) \right ) }{8\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\it Ei} \left ( 1,2\,{\rm arccosh} \left (ax\right ) \right ) }{4\,{a}^{5} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\it Ei} \left ( 1,-2\,{\rm arccosh} \left (ax\right ) \right ) }{4\,{a}^{5} \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^4/arccosh(a*x)/(-a^2*x^2+1)^(1/2),x)

[Out]

1/16*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5/(a^2*x^2-1)*Ei(1,4*arccosh(a*x))+1/16*(-a^2*x^2+1)^(1/
2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5/(a^2*x^2-1)*Ei(1,-4*arccosh(a*x))-3/8*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x
+1)^(1/2)/a^5/(a^2*x^2-1)*ln(arccosh(a*x))+1/4*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5/(a^2*x^2-1)*
Ei(1,2*arccosh(a*x))+1/4*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a^5/(a^2*x^2-1)*Ei(1,-2*arccosh(a*x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\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^4/arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^4/(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^{4}}{{\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^4/arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*x^4/((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^{4}}{\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**4/acosh(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(x**4/(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^{4}}{\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^4/arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/(sqrt(-a^2*x^2 + 1)*arccosh(a*x)), x)