\(\int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\) [299]

Optimal result

Integrand size = 28, antiderivative size = 197 \[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {3 \sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b c^4 \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b c^4 \sqrt {1-c x}}-\frac {3 \sqrt {-1+c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b c^4 \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b c^4 \sqrt {1-c x}} \]

[Out]

3/4*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)*(c*x-1)^(1/2)/b/c^4/(-c*x+1)^(1/2)+1/4*Chi(3*(a+b*arccosh(c*x))/b)*cos
h(3*a/b)*(c*x-1)^(1/2)/b/c^4/(-c*x+1)^(1/2)-3/4*Shi((a+b*arccosh(c*x))/b)*sinh(a/b)*(c*x-1)^(1/2)/b/c^4/(-c*x+
1)^(1/2)-1/4*Shi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)*(c*x-1)^(1/2)/b/c^4/(-c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5952, 3393, 3384, 3379, 3382} \[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {3 \sqrt {c x-1} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b c^4 \sqrt {1-c x}}+\frac {\sqrt {c x-1} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b c^4 \sqrt {1-c x}}-\frac {3 \sqrt {c x-1} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b c^4 \sqrt {1-c x}}-\frac {\sqrt {c x-1} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b c^4 \sqrt {1-c x}} \]

[In]

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

[Out]

(3*Sqrt[-1 + c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*c^4*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Co
sh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b*c^4*Sqrt[1 - c*x]) - (3*Sqrt[-1 + c*x]*Sinh[a/b]*Si
nhIntegral[(a + b*ArcCosh[c*x])/b])/(4*b*c^4*Sqrt[1 - c*x]) - (Sqrt[-1 + c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a
+ b*ArcCosh[c*x]))/b])/(4*b*c^4*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 3393

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.66 \[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{4 b c^4 \sqrt {-((-1+c x) (1+c x))}} \]

[In]

Integrate[x^3/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]

[Out]

(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(3*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + Cosh[(3*a)/b]*CoshIntegr
al[3*(a/b + ArcCosh[c*x])] - 3*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b
+ ArcCosh[c*x])]))/(4*b*c^4*Sqrt[-((-1 + c*x)*(1 + c*x))])

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}+3 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}+3 \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}}\right )}{8 b \left (c^{2} x^{2}-1\right ) c^{4}}\)	\(182\)

[In]

int(x^3/(a+b*arccosh(c*x))/(-c^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x^{3}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

integral(-sqrt(-c^2*x^2 + 1)*x^3/(a*c^2*x^2 + (b*c^2*x^2 - b)*arccosh(c*x) - a), x)

Sympy [F]

\[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x^{3}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

[In]

integrate(x**3/(a+b*acosh(c*x))/(-c**2*x**2+1)**(1/2),x)

[Out]

Integral(x**3/(sqrt(-(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))), x)

Maxima [F]

\[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {x^{3}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {x^3}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]

[In]

int(x^3/((a + b*acosh(c*x))*(1 - c^2*x^2)^(1/2)),x)

[Out]

int(x^3/((a + b*acosh(c*x))*(1 - c^2*x^2)^(1/2)), x)