\(\int \frac {(1-c^2 x^2)^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx\) [288]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx=-\frac {11 \sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b \sqrt {1-c x}}+\frac {7 \sqrt {-1+c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b \sqrt {1-c x}}+\frac {11 \sqrt {-1+c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b \sqrt {1-c x}}-\frac {7 \sqrt {-1+c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b \sqrt {1-c x}}+\text {Int}\left (\frac {1}{x \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))},x\right ) \]

[Out]

-11/8*Chi((a+b*arccosh(c*x))/b)*cosh(a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)+7/16*Chi(3*(a+b*arccosh(c*x))/b)*cosh
(3*a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)-1/16*Chi(5*(a+b*arccosh(c*x))/b)*cosh(5*a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(
1/2)+11/8*Shi((a+b*arccosh(c*x))/b)*sinh(a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)-7/16*Shi(3*(a+b*arccosh(c*x))/b)*
sinh(3*a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)+1/16*Shi(5*(a+b*arccosh(c*x))/b)*sinh(5*a/b)*(c*x-1)^(1/2)/b/(-c*x+
1)^(1/2)+Unintegrable(1/x/(a+b*arccosh(c*x))/(-c^2*x^2+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.84 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx \]

[In]

Int[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])),x]

[Out]

(-11*Sqrt[-1 + c*x]*Cosh[a/b]*CoshIntegral[(a + b*ArcCosh[c*x])/b])/(8*b*Sqrt[1 - c*x]) + (7*Sqrt[-1 + c*x]*Co
sh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt[1 - c*x]) - (Sqrt[-1 + c*x]*Cosh[(5*a)/b]*Cos
hIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt[1 - c*x]) + (11*Sqrt[-1 + c*x]*Sinh[a/b]*SinhIntegral[(a + b
*ArcCosh[c*x])/b])/(8*b*Sqrt[1 - c*x]) - (7*Sqrt[-1 + c*x]*Sinh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))
/b])/(16*b*Sqrt[1 - c*x]) + (Sqrt[-1 + c*x]*Sinh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b*Sqrt
[1 - c*x]) + Defer[Int][1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}-\frac {3 c^2 x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}+\frac {3 c^4 x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}-\frac {c^6 x^5}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))}\right ) \, dx \\ & = -\left (\left (3 c^2\right ) \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\right )+\left (3 c^4\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx-c^6 \int \frac {x^5}{\sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {\sqrt {-1+c x} \text {Subst}\left (\int \frac {\cosh ^5\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b \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 )}{b \sqrt {1-c x}}+\frac {\left (3 \sqrt {-1+c x}\right ) \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 \sqrt {1-c x}}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {\sqrt {-1+c x} \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}+\frac {5 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {5 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b \sqrt {1-c x}}+\frac {\left (3 \sqrt {-1+c x}\right ) \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 \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 )}{b \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 )}{b \sqrt {1-c x}}+\int \frac {1}{x \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 )}{b \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 )}{b \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b \sqrt {1-c x}}-\frac {\left (5 \sqrt {-1+c x}\right ) \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 )}{16 b \sqrt {1-c x}}-\frac {\left (5 \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 )}{8 b \sqrt {1-c x}}+\frac {\left (3 \sqrt {-1+c x}\right ) \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 \sqrt {1-c x}}+\frac {\left (9 \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 \sqrt {1-c x}}+\int \frac {1}{x \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 )}{b \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 )}{b \sqrt {1-c x}}-\frac {\left (5 \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 )}{8 b \sqrt {1-c x}}+\frac {\left (9 \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 \sqrt {1-c x}}-\frac {\left (5 \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 )}{16 b \sqrt {1-c x}}+\frac {\left (3 \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 \sqrt {1-c x}}-\frac {\left (\sqrt {-1+c x} \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b \sqrt {1-c x}}+\frac {\left (5 \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 )}{8 b \sqrt {1-c x}}-\frac {\left (9 \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 \sqrt {1-c x}}+\frac {\left (5 \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 )}{16 b \sqrt {1-c x}}-\frac {\left (3 \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 \sqrt {1-c x}}+\frac {\left (\sqrt {-1+c x} \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b \sqrt {1-c x}}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ & = -\frac {11 \sqrt {-1+c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b \sqrt {1-c x}}+\frac {7 \sqrt {-1+c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b \sqrt {1-c x}}-\frac {\sqrt {-1+c x} \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b \sqrt {1-c x}}+\frac {11 \sqrt {-1+c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b \sqrt {1-c x}}-\frac {7 \sqrt {-1+c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b \sqrt {1-c x}}+\frac {\sqrt {-1+c x} \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b \sqrt {1-c x}}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.54 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx \]

[In]

Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])),x]

[Out]

Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 1.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}d x\]

[In]

int((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x)),x)

[Out]

int((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx=\text {Timed out} \]

[In]

integrate((-c**2*x**2+1)**(5/2)/x/(a+b*acosh(c*x)),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)/((b*arccosh(c*x) + a)*x), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arccosh(c*x)),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 [N/A]

Not integrable

Time = 2.72 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x (a+b \text {arccosh}(c x))} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]

[In]

int((1 - c^2*x^2)^(5/2)/(x*(a + b*acosh(c*x))),x)

[Out]

int((1 - c^2*x^2)^(5/2)/(x*(a + b*acosh(c*x))), x)