3.343 \(\int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \cosh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=236 \[ -\frac {\sqrt {c x-1} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c^5 \sqrt {1-c x}}-\frac {\sqrt {c x-1} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^5 \sqrt {1-c x}}+\frac {\sqrt {c x-1} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c^5 \sqrt {1-c x}}+\frac {\sqrt {c x-1} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^5 \sqrt {1-c x}}-\frac {x^4 \sqrt {c x-1}}{b c \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

-x^4*(c*x-1)^(1/2)/b/c/(a+b*arccosh(c*x))/(-c*x+1)^(1/2)+cosh(2*a/b)*Shi(2*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)
/b^2/c^5/(-c*x+1)^(1/2)+1/2*cosh(4*a/b)*Shi(4*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)/b^2/c^5/(-c*x+1)^(1/2)-Chi(2
*(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(c*x-1)^(1/2)/b^2/c^5/(-c*x+1)^(1/2)-1/2*Chi(4*(a+b*arccosh(c*x))/b)*sinh(4
*a/b)*(c*x-1)^(1/2)/b^2/c^5/(-c*x+1)^(1/2)

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Rubi [A]  time = 0.79, antiderivative size = 301, normalized size of antiderivative = 1.28, number of steps used = 11, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5798, 5775, 5670, 5448, 3303, 3298, 3301} \[ -\frac {\sqrt {c x-1} \sqrt {c x+1} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^5 \sqrt {1-c^2 x^2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c^5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c^5 \sqrt {1-c^2 x^2}}-\frac {x^4 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))) - (Sqrt[-1 + c*x]*Sqrt[1 +
c*x]*CoshIntegral[(2*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(b^2*c^5*Sqrt[1 - c^2*x^2]) - (Sqrt[-1 + c*x]*Sqrt[
1 + c*x]*CoshIntegral[(4*a)/b + 4*ArcCosh[c*x]]*Sinh[(4*a)/b])/(2*b^2*c^5*Sqrt[1 - c^2*x^2]) + (Sqrt[-1 + c*x]
*Sqrt[1 + c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(b^2*c^5*Sqrt[1 - c^2*x^2]) + (Sqrt[-1 +
c*x]*Sqrt[1 + c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcCosh[c*x]])/(2*b^2*c^5*Sqrt[1 - c^2*x^2])

Rule 3298

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

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]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5670

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Cosh[x]^m*Sinh[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

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]

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {x^4 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt {1-c^2 x^2}}\\ &=-\frac {x^4 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^5 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^4 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 (a+b x)}+\frac {\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^5 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^4 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^5 \sqrt {1-c^2 x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^5 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^4 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^5 \sqrt {1-c^2 x^2}}+\frac {\left (\sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^5 \sqrt {1-c^2 x^2}}-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x} \sinh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^5 \sqrt {1-c^2 x^2}}-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x} \sinh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^5 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^4 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c^5 \sqrt {1-c^2 x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c^5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^5 \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{2 b^2 c^5 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.51, size = 149, normalized size = 0.63 \[ \frac {\sqrt {1-c^2 x^2} \left (\frac {2 b c^4 x^4}{a+b \cosh ^{-1}(c x)}+2 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+\sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{2 b^2 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*((2*b*c^4*x^4)/(a + b*ArcCosh[c*x]) + 2*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b]
+ CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] - 2*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] -
Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(2*b^2*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1} x^{4}}{a^{2} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} - a^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b*arccosh(c*x))^2/(-c^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*x^2 + 1)*x^4/(a^2*c^2*x^2 + (b^2*c^2*x^2 - b^2)*arccosh(c*x)^2 - a^2 + 2*(a*b*c^2*x^2 - a*
b)*arccosh(c*x)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b*arccosh(c*x))^2/(-c^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.87, size = 758, normalized size = 3.21 \[ -\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right )}{16 \left (c^{2} x^{2}-1\right ) c^{5} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}-\frac {\left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, \Ei \left (1, 4 \,\mathrm {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {b \,\mathrm {arccosh}\left (c x \right )-4 a}{b}}}{4 b^{2} \left (c^{2} x^{2}-1\right ) c^{5}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (8 \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3} b \,c^{3}+8 x^{4} b \,c^{4}-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x b c -8 x^{2} b \,c^{2}+4 \,\mathrm {arccosh}\left (c x \right ) \Ei \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {4 a}{b}} b +4 \Ei \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {4 a}{b}} a +b \right )}{16 \left (c^{2} x^{2}-1\right ) c^{5} b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {3 \sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}}{8 \left (c^{2} x^{2}-1\right ) c^{5} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right )}{4 \left (c^{2} x^{2}-1\right ) c^{5} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}-\frac {\left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, \Ei \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\mathrm {arccosh}\left (c x \right )-2 a}{b}}}{2 b^{2} \left (c^{2} x^{2}-1\right ) c^{5}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \sqrt {c x +1}\, \sqrt {c x -1}\, x b c +2 x^{2} b \,c^{2}+2 \,\mathrm {arccosh}\left (c x \right ) \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}} b +2 \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}} a -b \right )}{4 \left (c^{2} x^{2}-1\right ) c^{5} b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a+b*arccosh(c*x))^2/(-c^2*x^2+1)^(1/2),x)

[Out]

-1/16*(-c^2*x^2+1)^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4+8*c^5*x^5+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c
^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)/(c^2*x^2-1)/c^5/(a+b*arccosh(c*x))/b-1/4*((c*x+1)^(1/2)*(c*x-
1)^(1/2)*x*c+c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*Ei(1,4*arccosh(c*x)+4*a/b)*exp(-(b*arccosh(c*x)-4*a)/b)/b^2/(c^2*x^
2-1)/c^5+1/16*(-c^2*x^2+1)^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/c^5*(8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^
3*b*c^3+8*x^4*b*c^4-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*b*c-8*x^2*b*c^2+4*arccosh(c*x)*Ei(1,-4*arccosh(c*x)-4*a/b)
*exp(-4*a/b)*b+4*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(-4*a/b)*a+b)/b^2/(a+b*arccosh(c*x))+3/8*(-c^2*x^2+1)^(1/2)*(c
*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/c^5/(a+b*arccosh(c*x))/b-1/4*(-c^2*x^2+1)^(1/2)*(-2*(c*x+1)^(1/2)*(c*x-1
)^(1/2)*x^2*c^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)/(c^2*x^2-1)/c^5/(a+b*arccosh(c*x))/b-1/2*((c*x+1)
^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*Ei(1,2*arccosh(c*x)+2*a/b)*exp(-(b*arccosh(c*x)-2*a)/b)
/b^2/(c^2*x^2-1)/c^5+1/4*(-c^2*x^2+1)^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/c^5*(2*(c*x+1)^(1/2)*(c*x-
1)^(1/2)*x*b*c+2*x^2*b*c^2+2*arccosh(c*x)*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-2*a/b)*b+2*Ei(1,-2*arccosh(c*x)-2*a
/b)*exp(-2*a/b)*a-b)/b^2/(a+b*arccosh(c*x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {c^{3} x^{7} - c x^{5} + {\left (c^{2} x^{6} - x^{4}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{{\left ({\left (c x + 1\right )} \sqrt {c x - 1} b^{2} c^{2} x + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )} \sqrt {c x - 1} a b c^{2} x + {\left (a b c^{3} x^{2} - a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}} + \int \frac {4 \, c^{5} x^{8} - 9 \, c^{3} x^{6} + 5 \, c x^{4} + {\left (4 \, c^{3} x^{6} - 3 \, c x^{4}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 4 \, {\left (2 \, c^{4} x^{7} - 3 \, c^{2} x^{5} + x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{{\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} b^{2} c^{3} x^{2} + 2 \, {\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{4} - 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} a b c^{3} x^{2} + 2 \, {\left (a b c^{4} x^{3} - a b c^{2} x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b*arccosh(c*x))^2/(-c^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-(c^3*x^7 - c*x^5 + (c^2*x^6 - x^4)*sqrt(c*x + 1)*sqrt(c*x - 1))/(((c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x + (b^2*c^
3*x^2 - b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)*sqrt(c*x - 1)
*a*b*c^2*x + (a*b*c^3*x^2 - a*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)) + integrate((4*c^5*x^8 - 9*c^3*x^6 + 5*c*x^4
+ (4*c^3*x^6 - 3*c*x^4)*(c*x + 1)*(c*x - 1) + 4*(2*c^4*x^7 - 3*c^2*x^5 + x^3)*sqrt(c*x + 1)*sqrt(c*x - 1))/((
(c*x + 1)^(3/2)*(c*x - 1)*b^2*c^3*x^2 + 2*(b^2*c^4*x^3 - b^2*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 - 2
*b^2*c^3*x^2 + b^2*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + ((c*x + 1)^(3/2)*
(c*x - 1)*a*b*c^3*x^2 + 2*(a*b*c^4*x^3 - a*b*c^2*x)*(c*x + 1)*sqrt(c*x - 1) + (a*b*c^5*x^4 - 2*a*b*c^3*x^2 + a
*b*c)*sqrt(c*x + 1))*sqrt(-c*x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*acosh(c*x))^2*(1 - c^2*x^2)^(1/2)),x)

[Out]

int(x^4/((a + b*acosh(c*x))^2*(1 - c^2*x^2)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(a+b*acosh(c*x))**2/(-c**2*x**2+1)**(1/2),x)

[Out]

Integral(x**4/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))**2), x)

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