\(\int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx\) [267]

Optimal result

Integrand size = 28, antiderivative size = 339 \[ \int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^5 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^5 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^5 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{16 b c^5 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^5 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^5 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^5 \sqrt {-1+c x}} \]

[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)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5952, 5556, 3384, 3379, 3382} \[ \int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=-\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^5 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^5 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^5 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^5 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{16 b c^5 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{32 b c^5 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \log (a+b \text {arccosh}(c x))}{16 b c^5 \sqrt {c x-1}} \]

[In]

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

[Out]

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

Rule 3379

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 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 3384

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 5556

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

Mathematica [A] (warning: unable to verify)

Time = 0.38 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.55 \[ \int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\frac {\sqrt {1-c^2 x^2} \left (-\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+2 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-2 \log (a+b \text {arccosh}(c x))+\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-2 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-\sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{32 c^5 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \]

[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))

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (4 \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+4 \ln \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c x +\operatorname {Ei}_{1}\left (6 \,\operatorname {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}+\operatorname {Ei}_{1}\left (-6 \,\operatorname {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+6 a}{b}}+2 \,\operatorname {Ei}_{1}\left (4 \,\operatorname {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}-\operatorname {Ei}_{1}\left (2 \,\operatorname {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}-\operatorname {Ei}_{1}\left (-2 \,\operatorname {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+2 a}{b}}+2 \,\operatorname {Ei}_{1}\left (-4 \,\operatorname {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+4 a}{b}}\right )}{64 \left (c x +1\right ) c^{5} \left (c x -1\right ) b}\)	\(297\)

[In]

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

[Out]

1/64*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(a+b*arc
cosh(c*x))+4*ln(a+b*arccosh(c*x))*c*x+Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)+Ei(1,-6*arccosh(c
*x)-6*a/b)*exp(-(-b*arccosh(c*x)+6*a)/b)+2*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)-Ei(1,2*arcco
sh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)-Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-(-b*arccosh(c*x)+2*a)/b)+2*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

Fricas [F]

\[ \int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{4}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^{4} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{a + b \operatorname {acosh}{\left (c x \right )}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{4}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int { \frac {\sqrt {-c^{2} x^{2} + 1} x^{4}}{b \operatorname {arcosh}\left (c x\right ) + a} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \sqrt {1-c^2 x^2}}{a+b \text {arccosh}(c x)} \, dx=\int \frac {x^4\,\sqrt {1-c^2\,x^2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]

[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)