\(\int \frac {(b x)^m}{\arccos (a x)^2} \, dx\) [124]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {(b x)^m}{\arccos (a x)^2} \, dx=\text {Int}\left (\frac {(b x)^m}{\arccos (a x)^2},x\right ) \]

[Out]

Unintegrable((b*x)^m/arccos(a*x)^2,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 12, 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 = {} \[ \int \frac {(b x)^m}{\arccos (a x)^2} \, dx=\int \frac {(b x)^m}{\arccos (a x)^2} \, dx \]

[In]

Int[(b*x)^m/ArcCos[a*x]^2,x]

[Out]

Defer[Int][(b*x)^m/ArcCos[a*x]^2, x]

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

Mathematica [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {(b x)^m}{\arccos (a x)^2} \, dx=\int \frac {(b x)^m}{\arccos (a x)^2} \, dx \]

[In]

Integrate[(b*x)^m/ArcCos[a*x]^2,x]

[Out]

Integrate[(b*x)^m/ArcCos[a*x]^2, x]

Maple [N/A] (verified)

Not integrable

Time = 2.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

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

[In]

int((b*x)^m/arccos(a*x)^2,x)

[Out]

int((b*x)^m/arccos(a*x)^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {(b x)^m}{\arccos (a x)^2} \, dx=\int { \frac {\left (b x\right )^{m}}{\arccos \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

integral((b*x)^m/arccos(a*x)^2, x)

Sympy [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {(b x)^m}{\arccos (a x)^2} \, dx=\int \frac {\left (b x\right )^{m}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate((b*x)**m/acos(a*x)**2,x)

[Out]

Integral((b*x)**m/acos(a*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.95 (sec) , antiderivative size = 156, normalized size of antiderivative = 13.00 \[ \int \frac {(b x)^m}{\arccos (a x)^2} \, dx=\int { \frac {\left (b x\right )^{m}}{\arccos \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {(b x)^m}{\arccos (a x)^2} \, dx=\int { \frac {\left (b x\right )^{m}}{\arccos \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

integrate((b*x)^m/arccos(a*x)^2, x)

Mupad [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {(b x)^m}{\arccos (a x)^2} \, dx=\int \frac {{\left (b\,x\right )}^m}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]

[In]

int((b*x)^m/acos(a*x)^2,x)

[Out]

int((b*x)^m/acos(a*x)^2, x)