\(\int \frac {1}{x (a+b \sec ^{-1}(c x))^2} \, dx\) [41]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\text {Int}\left (\frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable(1/x/(a+b*arcsec(c*x))^2,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 = {} \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

[In]

Int[1/(x*(a + b*ArcSec[c*x])^2),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.40 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.14 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

integral(1/(b^2*x*arcsec(c*x)^2 + 2*a*b*x*arcsec(c*x) + a^2*x), x)

Sympy [N/A]

Not integrable

Time = 1.84 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int \frac {1}{x \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral(1/(x*(a + b*asec(c*x))**2), x)

Maxima [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 560, normalized size of antiderivative = 40.00 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

-(4*sqrt(c*x + 1)*sqrt(c*x - 1)*(b*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + a) - (4*b^3*arctan(sqrt(c*x + 1)*sqrt
(c*x - 1))^2 + b^3*log(c^2*x^2)^2 + 4*b^3*log(c)^2 + 8*b^3*log(c)*log(x) + 4*b^3*log(x)^2 + 8*a*b^2*arctan(sqr
t(c*x + 1)*sqrt(c*x - 1)) + 4*a^2*b - 4*(b^3*log(c) + b^3*log(x))*log(c^2*x^2))*integrate(-4*(b*c^2*x*arctan(s
qrt(c*x + 1)*sqrt(c*x - 1)) + a*c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1)/(4*b^3*log(c)^2 + 4*a^2*b - 4*(b^3*c^2*log(
c)^2 + a^2*b*c^2)*x^2 - 4*(b^3*c^2*x^2 - b^3)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 - (b^3*c^2*x^2 - b^3)*log(
c^2*x^2)^2 - 4*(b^3*c^2*x^2 - b^3)*log(x)^2 - 8*(a*b^2*c^2*x^2 - a*b^2)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) +
4*(b^3*c^2*x^2*log(c) - b^3*log(c) + (b^3*c^2*x^2 - b^3)*log(x))*log(c^2*x^2) - 8*(b^3*c^2*x^2*log(c) - b^3*lo
g(c))*log(x)), x))/(4*b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + b^3*log(c^2*x^2)^2 + 4*b^3*log(c)^2 + 8*b^3*
log(c)*log(x) + 4*b^3*log(x)^2 + 8*a*b^2*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*a^2*b - 4*(b^3*log(c) + b^3*l
og(x))*log(c^2*x^2))

Giac [N/A]

Not integrable

Time = 4.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

integrate(1/((b*arcsec(c*x) + a)^2*x), x)

Mupad [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x \left (a+b \sec ^{-1}(c x)\right )^2} \, dx=\int \frac {1}{x\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]

[In]

int(1/(x*(a + b*acos(1/(c*x)))^2),x)

[Out]

int(1/(x*(a + b*acos(1/(c*x)))^2), x)