\(\int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx\) [225]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx=\text {Int}\left (\frac {\sqrt {d x}}{(a+b \arccos (c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx=\int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 16.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx=\int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.97 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx=\int { \frac {\sqrt {d x}}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx=\int \frac {\sqrt {d x}}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.88 (sec) , antiderivative size = 181, normalized size of antiderivative = 10.06 \[ \int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx=\int { \frac {\sqrt {d x}}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx=\int { \frac {\sqrt {d x}}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d x}}{(a+b \arccos (c x))^2} \, dx=\int \frac {\sqrt {d\,x}}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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