\(\int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx\) [177]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx=\text {Int}\left (\frac {\sqrt {a+b \arccos (c x)}}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx=\int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx \]

[In]

Int[Sqrt[a + b*ArcCos[c*x]]/x^2,x]

[Out]

Defer[Int][Sqrt[a + b*ArcCos[c*x]]/x^2, x]

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

Mathematica [N/A]

Not integrable

Time = 7.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx=\int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx \]

[In]

Integrate[Sqrt[a + b*ArcCos[c*x]]/x^2,x]

[Out]

Integrate[Sqrt[a + b*ArcCos[c*x]]/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 1.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx=\int \frac {\sqrt {a + b \operatorname {acos}{\left (c x \right )}}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx=\int { \frac {\sqrt {b \arccos \left (c x\right ) + a}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.97 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx=\int { \frac {\sqrt {b \arccos \left (c x\right ) + a}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \arccos (c x)}}{x^2} \, dx=\int \frac {\sqrt {a+b\,\mathrm {acos}\left (c\,x\right )}}{x^2} \,d x \]

[In]

int((a + b*acos(c*x))^(1/2)/x^2,x)

[Out]

int((a + b*acos(c*x))^(1/2)/x^2, x)