\(\int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx\) [54]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx=\text {Int}\left (\frac {(d x)^m}{a+b \sec ^{-1}(c x)},x\right ) \]

[Out]

Unintegrable((d*x)^m/(a+b*arcsec(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx=\int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx \]

[In]

Int[(d*x)^m/(a + b*ArcSec[c*x]),x]

[Out]

Defer[Int][(d*x)^m/(a + b*ArcSec[c*x]), x]

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

Mathematica [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx=\int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx \]

[In]

Integrate[(d*x)^m/(a + b*ArcSec[c*x]),x]

[Out]

Integrate[(d*x)^m/(a + b*ArcSec[c*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 1.94 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \frac {\left (d x \right )^{m}}{a +b \,\operatorname {arcsec}\left (c x \right )}d x\]

[In]

int((d*x)^m/(a+b*arcsec(c*x)),x)

[Out]

int((d*x)^m/(a+b*arcsec(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx=\int { \frac {\left (d x\right )^{m}}{b \operatorname {arcsec}\left (c x\right ) + a} \,d x } \]

[In]

integrate((d*x)^m/(a+b*arcsec(c*x)),x, algorithm="fricas")

[Out]

integral((d*x)^m/(b*arcsec(c*x) + a), x)

Sympy [N/A]

Not integrable

Time = 1.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx=\int \frac {\left (d x\right )^{m}}{a + b \operatorname {asec}{\left (c x \right )}}\, dx \]

[In]

integrate((d*x)**m/(a+b*asec(c*x)),x)

[Out]

Integral((d*x)**m/(a + b*asec(c*x)), x)

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx=\int { \frac {\left (d x\right )^{m}}{b \operatorname {arcsec}\left (c x\right ) + a} \,d x } \]

[In]

integrate((d*x)^m/(a+b*arcsec(c*x)),x, algorithm="maxima")

[Out]

integrate((d*x)^m/(b*arcsec(c*x) + a), x)

Giac [N/A]

Not integrable

Time = 0.94 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx=\int { \frac {\left (d x\right )^{m}}{b \operatorname {arcsec}\left (c x\right ) + a} \,d x } \]

[In]

integrate((d*x)^m/(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

integrate((d*x)^m/(b*arcsec(c*x) + a), x)

Mupad [N/A]

Not integrable

Time = 0.72 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {(d x)^m}{a+b \sec ^{-1}(c x)} \, dx=\int \frac {{\left (d\,x\right )}^m}{a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )} \,d x \]

[In]

int((d*x)^m/(a + b*acos(1/(c*x))),x)

[Out]

int((d*x)^m/(a + b*acos(1/(c*x))), x)