\(\int \frac {(1-c^2 x^2)^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx\) [281]

   Optimal result
   Rubi [N/A]
   Mathematica [N/A]
   Maple [N/A] (verified)
   Fricas [N/A]
   Sympy [N/A]
   Maxima [N/A]
   Giac [N/A]
   Mupad [N/A]

Optimal result

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

[Out]

1/2*c*Chi(2*(a+b*arccosh(c*x))/b)*cosh(2*a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)-3/2*c*ln(a+b*arccosh(c*x))*(c*x-1
)^(1/2)/b/(-c*x+1)^(1/2)-1/2*c*Shi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(c*x-1)^(1/2)/b/(-c*x+1)^(1/2)+Unintegr
able(1/x^2/(a+b*arccosh(c*x))/(-c^2*x^2+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.43 (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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

(c*Sqrt[-1 + c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(2*b*Sqrt[1 - c*x]) - (3*c*Sqrt[-1 +
 c*x]*Log[a + b*ArcCosh[c*x]])/(2*b*Sqrt[1 - c*x]) - (c*Sqrt[-1 + c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*Ar
cCosh[c*x]))/b])/(2*b*Sqrt[1 - c*x]) + Defer[Int][1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]

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

Mathematica [N/A]

Not integrable

Time = 2.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.90 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 11.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {arccosh}(c x))} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \]

[In]

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

[Out]

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