\(\int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx\) [226]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx=\text {Int}\left (\frac {1}{\sqrt {d x} (a+b \arccos (c x))^2},x\right ) \]

[Out]

Unintegrable(1/(a+b*arccos(c*x))^2/(d*x)^(1/2),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 = {} \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx=\int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 40.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx=\int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.98 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.17 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx=\int \frac {1}{\sqrt {d x} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.68 (sec) , antiderivative size = 196, normalized size of antiderivative = 10.89 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,\sqrt {d\,x}} \,d x \]

[In]

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

[Out]

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