\(\int \frac {(1-c^2 x^2)^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx\) [331]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c x (a+b \text {arccosh}(c x))}-\frac {9 \sqrt {1-c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{4 b^2 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{4 b^2 \sqrt {-1+c x}}+\frac {9 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 b^2 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 b^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Int}\left (\frac {-1+c^2 x^2}{x^2 (a+b \text {arccosh}(c x))},x\right )}{b c \sqrt {-1+c x}} \]

[Out]

9/4*cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2)-3/4*cosh(3*a/b)*Shi(3*(a+b*arccosh(c*
x))/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2)-9/4*Chi((a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/
2)+3/4*Chi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)*(-c*x+1)^(1/2)/b^2/(c*x-1)^(1/2)-(-c^2*x^2+1)^(3/2)*(c*x-1)^(1/
2)*(c*x+1)^(1/2)/b/c/x/(a+b*arccosh(c*x))-(-c*x+1)^(1/2)*Unintegrable((c^2*x^2-1)/x^2/(a+b*arccosh(c*x)),x)/b/
c/(c*x-1)^(1/2)

Rubi [N/A]

Not integrable

Time = 0.38 (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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*x*(a + b*ArcCosh[c*x]))) - (9*Sqrt[1 - c*x]*CoshInte
gral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(4*b^2*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b*ArcCo
sh[c*x]))/b]*Sinh[(3*a)/b])/(4*b^2*Sqrt[-1 + c*x]) + (9*Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*
x])/b])/(4*b^2*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(4*b
^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Defer[Int][(-1 + c^2*x^2)/(x^2*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c
*x])

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

Mathematica [N/A]

Not integrable

Time = 18.83 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 20.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 476, normalized size of antiderivative = 17.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((-c^2*x^2+1)^(3/2)/x/(a+b*arccosh(c*x))^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 [N/A]

Not integrable

Time = 3.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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