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

Optimal. Leaf size=339 \[ -\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^5 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^5 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^5 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b c^5 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^5 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c^5 \sqrt {c x-1}} \]

[Out]

-1/32*Chi(2*(a+b*arccosh(c*x))/b)*cosh(2*a/b)*(-c*x+1)^(1/2)/b/c^5/(c*x-1)^(1/2)+1/16*Chi(4*(a+b*arccosh(c*x))
/b)*cosh(4*a/b)*(-c*x+1)^(1/2)/b/c^5/(c*x-1)^(1/2)+1/32*Chi(6*(a+b*arccosh(c*x))/b)*cosh(6*a/b)*(-c*x+1)^(1/2)
/b/c^5/(c*x-1)^(1/2)-1/16*ln(a+b*arccosh(c*x))*(-c*x+1)^(1/2)/b/c^5/(c*x-1)^(1/2)+1/32*Shi(2*(a+b*arccosh(c*x)
)/b)*sinh(2*a/b)*(-c*x+1)^(1/2)/b/c^5/(c*x-1)^(1/2)-1/16*Shi(4*(a+b*arccosh(c*x))/b)*sinh(4*a/b)*(-c*x+1)^(1/2
)/b/c^5/(c*x-1)^(1/2)-1/32*Shi(6*(a+b*arccosh(c*x))/b)*sinh(6*a/b)*(-c*x+1)^(1/2)/b/c^5/(c*x-1)^(1/2)

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Rubi [A]  time = 0.88, antiderivative size = 430, normalized size of antiderivative = 1.27, number of steps used = 13, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5798, 5781, 5448, 3303, 3298, 3301} \[ -\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{16 b c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{16 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {\sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c^5 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - c^2*x^2]*Cosh[(2*a)/b]*CoshIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(32*b*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) + (Sqrt[1 - c^2*x^2]*Cosh[(4*a)/b]*CoshIntegral[(4*a)/b + 4*ArcCosh[c*x]])/(16*b*c^5*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) + (Sqrt[1 - c^2*x^2]*Cosh[(6*a)/b]*CoshIntegral[(6*a)/b + 6*ArcCosh[c*x]])/(32*b*c^5*Sqrt[-1 + c*x]*S
qrt[1 + c*x]) - (Sqrt[1 - c^2*x^2]*Log[a + b*ArcCosh[c*x]])/(16*b*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (Sqrt[1
- c^2*x^2]*Sinh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(32*b*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sq
rt[1 - c^2*x^2]*Sinh[(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcCosh[c*x]])/(16*b*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
- (Sqrt[1 - c^2*x^2]*Sinh[(6*a)/b]*SinhIntegral[(6*a)/b + 6*ArcCosh[c*x]])/(32*b*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c
*x])

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 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 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}}{a+b \cosh ^{-1}(c x)} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {x^4 \sqrt {-1+c x} \sqrt {1+c x}}{a+b \cosh ^{-1}(c x)} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \frac {\cosh ^4(x) \sinh ^2(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \left (-\frac {1}{16 (a+b x)}-\frac {\cosh (2 x)}{32 (a+b x)}+\frac {\cosh (4 x)}{16 (a+b x)}+\frac {\cosh (6 x)}{32 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \frac {\cosh (6 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \cosh \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 )}{32 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \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 )}{16 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {6 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \sinh \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 )}{32 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \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 )}{16 c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {6 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{16 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{16 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{16 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 0.51, size = 188, normalized size = 0.55 \[ \frac {\sqrt {1-c^2 x^2} \left (-\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+2 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 \log \left (a+b \cosh ^{-1}(c x)\right )\right )}{32 c^5 \sqrt {\frac {c x-1}{c x+1}} (b c x+b)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*(-(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])]) + 2*Cosh[(4*a)/b]*CoshIntegral[4*(a/
b + ArcCosh[c*x])] + Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcCosh[c*x])] - 2*Log[a + b*ArcCosh[c*x]] + Sinh[(2*
a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 2*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] - Sinh[(6*a)
/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])]))/(32*c^5*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{4}}{b \operatorname {arcosh}\left (c x\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.66, size = 591, normalized size = 1.74 \[ \frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, 6 \,\mathrm {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+6 a}{b}}}{64 \left (c x +1\right ) c^{5} \left (c x -1\right ) b}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, -6 \,\mathrm {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )-6 a}{b}}}{64 \left (c x +1\right ) c^{5} \left (c x -1\right ) b}-\frac {\sqrt {-c^{2} x^{2}+1}\, \ln \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}{16 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5} b}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, 4 \,\mathrm {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+4 a}{b}}}{32 \left (c x +1\right ) c^{5} \left (c x -1\right ) b}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+2 a}{b}}}{64 \left (c x +1\right ) c^{5} \left (c x -1\right ) b}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )-2 a}{b}}}{64 \left (c x +1\right ) c^{5} \left (c x -1\right ) b}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )-4 a}{b}}}{32 \left (c x +1\right ) c^{5} \left (c x -1\right ) b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh
(c*x)+6*a)/b)/(c*x+1)/c^5/(c*x-1)/b+1/64*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,
-6*arccosh(c*x)-6*a/b)*exp((b*arccosh(c*x)-6*a)/b)/(c*x+1)/c^5/(c*x-1)/b-1/16*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)
/(c*x+1)^(1/2)/c^5*ln(a+b*arccosh(c*x))/b+1/32*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)
*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)/(c*x+1)/c^5/(c*x-1)/b-1/64*(-c^2*x^2+1)^(1/2)*(-(c*x+1
)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)/(c*x+1)/c^5/(c*x-1
)/b-1/64*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,-2*arccosh(c*x)-2*a/b)*exp((b*ar
ccosh(c*x)-2*a)/b)/(c*x+1)/c^5/(c*x-1)/b+1/32*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*
Ei(1,-4*arccosh(c*x)-4*a/b)*exp((b*arccosh(c*x)-4*a)/b)/(c*x+1)/c^5/(c*x-1)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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