\(\int \frac {1}{(a+b \sec ^{-1}(c x))^3} \, dx\) [46]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\text {Int}\left (\frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3},x\right ) \]

[Out]

Unintegrable(1/(a+b*arcsec(c*x))^3,x)

Rubi [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 10, 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 = {} \[ \int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx \]

[In]

Int[(a + b*ArcSec[c*x])^(-3),x]

[Out]

Defer[Int][(a + b*ArcSec[c*x])^(-3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.57 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx \]

[In]

Integrate[(a + b*ArcSec[c*x])^(-3),x]

[Out]

Integrate[(a + b*ArcSec[c*x])^(-3), x]

Maple [N/A] (verified)

Not integrable

Time = 0.58 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

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

[In]

int(1/(a+b*arcsec(c*x))^3,x)

[Out]

int(1/(a+b*arcsec(c*x))^3,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 4.00 \[ \int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

integral(1/(b^3*arcsec(c*x)^3 + 3*a*b^2*arcsec(c*x)^2 + 3*a^2*b*arcsec(c*x) + a^3), x)

Sympy [N/A]

Not integrable

Time = 2.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{\left (a + b \operatorname {asec}{\left (c x \right )}\right )^{3}}\, dx \]

[In]

integrate(1/(a+b*asec(c*x))**3,x)

[Out]

Integral((a + b*asec(c*x))**(-3), x)

Maxima [N/A]

Not integrable

Time = 28.50 (sec) , antiderivative size = 1744, normalized size of antiderivative = 174.40 \[ \int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

-(16*(a*b^2*c^2*log(c)^2 + a^3*c^2)*x^3 + 8*(2*b^3*c^2*x^3 - b^3*x)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^3 + 24
*(2*a*b^2*c^2*x^3 - a*b^2*x)*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 2*(2*a*b^2*c^2*x^3 - a*b^2*x)*log(c^2*x^2
)^2 + 8*(2*a*b^2*c^2*x^3 - a*b^2*x)*log(x)^2 + 2*(4*b^3*x*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 - b^3*x*log(c^
2*x^2)^2 - 8*b^3*x*log(c)*log(x) - 4*b^3*x*log(x)^2 + 8*a*b^2*x*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - 4*(b^3*l
og(c)^2 - a^2*b)*x + 4*(b^3*x*log(c) + b^3*x*log(x))*log(c^2*x^2))*sqrt(c*x + 1)*sqrt(c*x - 1) - 8*(a*b^2*log(
c)^2 + a^3)*x + 2*(8*(b^3*c^2*log(c)^2 + 3*a^2*b*c^2)*x^3 + (2*b^3*c^2*x^3 - b^3*x)*log(c^2*x^2)^2 + 4*(2*b^3*
c^2*x^3 - b^3*x)*log(x)^2 - 4*(b^3*log(c)^2 + 3*a^2*b)*x - 4*(2*b^3*c^2*x^3*log(c) - b^3*x*log(c) + (2*b^3*c^2
*x^3 - b^3*x)*log(x))*log(c^2*x^2) + 8*(2*b^3*c^2*x^3*log(c) - b^3*x*log(c))*log(x))*arctan(sqrt(c*x + 1)*sqrt
(c*x - 1)) - (16*b^6*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^4 + b^6*log(c^2*x^2)^4 + 16*b^6*log(c)^4 + 64*b^6*log
(c)*log(x)^3 + 16*b^6*log(x)^4 + 64*a*b^5*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^3 + 32*a^2*b^4*log(c)^2 + 16*a^4
*b^2 - 8*(b^6*log(c) + b^6*log(x))*log(c^2*x^2)^3 + 8*(b^6*log(c^2*x^2)^2 + 4*b^6*log(c)^2 + 8*b^6*log(c)*log(
x) + 4*b^6*log(x)^2 + 12*a^2*b^4 - 4*(b^6*log(c) + b^6*log(x))*log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1
))^2 + 8*(3*b^6*log(c)^2 + 6*b^6*log(c)*log(x) + 3*b^6*log(x)^2 + a^2*b^4)*log(c^2*x^2)^2 + 32*(3*b^6*log(c)^2
+ a^2*b^4)*log(x)^2 + 16*(a*b^5*log(c^2*x^2)^2 + 4*a*b^5*log(c)^2 + 8*a*b^5*log(c)*log(x) + 4*a*b^5*log(x)^2
+ 4*a^3*b^3 - 4*(a*b^5*log(c) + a*b^5*log(x))*log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - 32*(b^6*log(
c)^3 + 3*b^6*log(c)*log(x)^2 + b^6*log(x)^3 + a^2*b^4*log(c) + (3*b^6*log(c)^2 + a^2*b^4)*log(x))*log(c^2*x^2)
+ 64*(b^6*log(c)^3 + a^2*b^4*log(c))*log(x))*integrate(2*(6*a*c^2*x^2 + (6*b*c^2*x^2 - b)*arctan(sqrt(c*x + 1
)*sqrt(c*x - 1)) - a)/(4*b^4*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + b^4*log(c^2*x^2)^2 + 4*b^4*log(c)^2 + 8*b
^4*log(c)*log(x) + 4*b^4*log(x)^2 + 8*a*b^3*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*a^2*b^2 - 4*(b^4*log(c) +
b^4*log(x))*log(c^2*x^2)), x) - 8*(2*a*b^2*c^2*x^3*log(c) - a*b^2*x*log(c) + (2*a*b^2*c^2*x^3 - a*b^2*x)*log(x
))*log(c^2*x^2) + 16*(2*a*b^2*c^2*x^3*log(c) - a*b^2*x*log(c))*log(x))/(16*b^6*arctan(sqrt(c*x + 1)*sqrt(c*x -
1))^4 + b^6*log(c^2*x^2)^4 + 16*b^6*log(c)^4 + 64*b^6*log(c)*log(x)^3 + 16*b^6*log(x)^4 + 64*a*b^5*arctan(sqr
t(c*x + 1)*sqrt(c*x - 1))^3 + 32*a^2*b^4*log(c)^2 + 16*a^4*b^2 - 8*(b^6*log(c) + b^6*log(x))*log(c^2*x^2)^3 +
8*(b^6*log(c^2*x^2)^2 + 4*b^6*log(c)^2 + 8*b^6*log(c)*log(x) + 4*b^6*log(x)^2 + 12*a^2*b^4 - 4*(b^6*log(c) + b
^6*log(x))*log(c^2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1))^2 + 8*(3*b^6*log(c)^2 + 6*b^6*log(c)*log(x) + 3*b
^6*log(x)^2 + a^2*b^4)*log(c^2*x^2)^2 + 32*(3*b^6*log(c)^2 + a^2*b^4)*log(x)^2 + 16*(a*b^5*log(c^2*x^2)^2 + 4*
a*b^5*log(c)^2 + 8*a*b^5*log(c)*log(x) + 4*a*b^5*log(x)^2 + 4*a^3*b^3 - 4*(a*b^5*log(c) + a*b^5*log(x))*log(c^
2*x^2))*arctan(sqrt(c*x + 1)*sqrt(c*x - 1)) - 32*(b^6*log(c)^3 + 3*b^6*log(c)*log(x)^2 + b^6*log(x)^3 + a^2*b^
4*log(c) + (3*b^6*log(c)^2 + a^2*b^4)*log(x))*log(c^2*x^2) + 64*(b^6*log(c)^3 + a^2*b^4*log(c))*log(x))

Giac [N/A]

Not integrable

Time = 52.48 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int { \frac {1}{{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

integrate((b*arcsec(c*x) + a)^(-3), x)

Mupad [N/A]

Not integrable

Time = 0.75 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\left (a+b \sec ^{-1}(c x)\right )^3} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3} \,d x \]

[In]

int(1/(a + b*acos(1/(c*x)))^3,x)

[Out]

int(1/(a + b*acos(1/(c*x)))^3, x)