\(\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx\) [349]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.12 (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 = {} \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.42 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.99 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.07 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 21.61 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{x^{2} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 491, normalized size of antiderivative = 17.54 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]

[In]

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

[Out]

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