\(\int \frac {1}{\sqrt {d x} (a+b \arccos (c x))} \, dx\) [222]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.47 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))} \, dx=\int \frac {1}{\sqrt {d x} (a+b \arccos (c x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.45 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))} \, dx=\int \frac {1}{\sqrt {d x} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d x} (a+b \arccos (c x))} \, dx=\int \frac {1}{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d\,x}} \,d x \]

[In]

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

[Out]

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