\(\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx\) [304]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \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 = 2.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int { \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.68 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {1}{x^{2} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]

[In]

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

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.86 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \, dx=\int \frac {1}{x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]

[In]

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

[Out]

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