\(\int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx\) [227]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx=\text {Int}\left (\frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2},x\right ) \]

[Out]

Unintegrable(1/(d*x)^(3/2)/(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 = {} \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx=\int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx \]

[In]

Int[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2),x]

[Out]

Defer[Int][1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 22.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx=\int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx \]

[In]

Integrate[1/((d*x)^(3/2)*(a + b*ArcCos[c*x])^2),x]

[Out]

Integrate[1/((d*x)^(3/2)*(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

\[\int \frac {1}{\left (d x \right )^{\frac {3}{2}} \left (a +b \arccos \left (c x \right )\right )^{2}}d x\]

[In]

int(1/(d*x)^(3/2)/(a+b*arccos(c*x))^2,x)

[Out]

int(1/(d*x)^(3/2)/(a+b*arccos(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.83 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x)^(3/2)/(a+b*arccos(c*x))^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 11.73 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx=\int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

integrate(1/(d*x)**(3/2)/(a+b*acos(c*x))**2,x)

[Out]

Integral(1/((d*x)**(3/2)*(a + b*acos(c*x))**2), x)

Maxima [N/A]

Not integrable

Time = 1.93 (sec) , antiderivative size = 218, normalized size of antiderivative = 12.11 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x)^(3/2)/(a+b*arccos(c*x))^2,x, algorithm="maxima")

[Out]

((b^2*c*d^2*x^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*d^2*x^2)*sqrt(d)*integrate(1/2*(c^2*x^2 - 3
)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(x)/(a*b*c^3*d^2*x^5 - a*b*c*d^2*x^3 + (b^2*c^3*d^2*x^5 - b^2*c*d^2*x^3)*ar
ctan2(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^2*x^2*a
rctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c*d^2*x^2)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx=\int { \frac {1}{\left (d x\right )^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(d*x)^(3/2)/(a+b*arccos(c*x))^2,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(d x)^{3/2} (a+b \arccos (c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d\,x\right )}^{3/2}} \,d x \]

[In]

int(1/((a + b*acos(c*x))^2*(d*x)^(3/2)),x)

[Out]

int(1/((a + b*acos(c*x))^2*(d*x)^(3/2)), x)