\(\int \frac {1}{x (a+b \sec ^{-1}(c x))} \, dx\) [35]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx=\text {Int}\left (\frac {1}{x \left (a+b \sec ^{-1}(c x)\right )},x\right ) \]

[Out]

Unintegrable(1/x/(a+b*arcsec(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 14, 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 = {} \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx=\int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx \]

[In]

Int[1/(x*(a + b*ArcSec[c*x])),x]

[Out]

Defer[Int][1/(x*(a + b*ArcSec[c*x])), x]

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

Mathematica [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx=\int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx \]

[In]

Integrate[1/(x*(a + b*ArcSec[c*x])),x]

[Out]

Integrate[1/(x*(a + b*ArcSec[c*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

\[\int \frac {1}{x \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )}d x\]

[In]

int(1/x/(a+b*arcsec(c*x)),x)

[Out]

int(1/x/(a+b*arcsec(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(a+b*arcsec(c*x)),x, algorithm="fricas")

[Out]

integral(1/(b*x*arcsec(c*x) + a*x), x)

Sympy [N/A]

Not integrable

Time = 0.90 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx=\int \frac {1}{x \left (a + b \operatorname {asec}{\left (c x \right )}\right )}\, dx \]

[In]

integrate(1/x/(a+b*asec(c*x)),x)

[Out]

Integral(1/(x*(a + b*asec(c*x))), x)

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(a+b*arcsec(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((b*arcsec(c*x) + a)*x), x)

Giac [N/A]

Not integrable

Time = 1.69 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x} \,d x } \]

[In]

integrate(1/x/(a+b*arcsec(c*x)),x, algorithm="giac")

[Out]

integrate(1/((b*arcsec(c*x) + a)*x), x)

Mupad [N/A]

Not integrable

Time = 0.79 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )} \, dx=\int \frac {1}{x\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )} \,d x \]

[In]

int(1/(x*(a + b*acos(1/(c*x)))),x)

[Out]

int(1/(x*(a + b*acos(1/(c*x)))), x)