\(\int \frac {1}{x (a+b \arccos (c x))^2} \, dx\) [166]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{x (a+b \arccos (c x))^2} \, dx=\text {Int}\left (\frac {1}{x (a+b \arccos (c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 14, 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 = {} \[ \int \frac {1}{x (a+b \arccos (c x))^2} \, dx=\int \frac {1}{x (a+b \arccos (c x))^2} \, dx \]

[In]

Int[1/(x*(a + b*ArcCos[c*x])^2),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x (a+b \arccos (c x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x (a+b \arccos (c x))^2} \, dx=\int \frac {1}{x (a+b \arccos (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

\[\int \frac {1}{x \left (a +b \arccos \left (c x \right )\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.14 \[ \int \frac {1}{x (a+b \arccos (c x))^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

integral(1/(b^2*x*arccos(c*x)^2 + 2*a*b*x*arccos(c*x) + a^2*x), x)

Sympy [N/A]

Not integrable

Time = 1.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+b \arccos (c x))^2} \, dx=\int \frac {1}{x \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral(1/(x*(a + b*acos(c*x))**2), x)

Maxima [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 166, normalized size of antiderivative = 11.86 \[ \int \frac {1}{x (a+b \arccos (c x))^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

-((b^2*c*x*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*x)*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)/(a*b*c
^3*x^4 - a*b*c*x^2 + (b^2*c^3*x^4 - b^2*c*x^2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)), x) - sqrt(c*x + 1)
*sqrt(-c*x + 1))/(b^2*c*x*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*x)

Giac [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x (a+b \arccos (c x))^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

integrate(1/((b*arccos(c*x) + a)^2*x), x)

Mupad [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x (a+b \arccos (c x))^2} \, dx=\int \frac {1}{x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \]

[In]

int(1/(x*(a + b*acos(c*x))^2),x)

[Out]

int(1/(x*(a + b*acos(c*x))^2), x)