\(\int \frac {1}{(a+b \text {sech}^{-1}(c x))^2} \, dx\) [58]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\text {Int}\left (\frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable(1/(a+b*arcsech(c*x))^2,x)

Rubi [N/A]

Not integrable

Time = 0.01 (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 = {} \[ \int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx \]

[In]

Int[(a + b*ArcSech[c*x])^(-2),x]

[Out]

Defer[Int][(a + b*ArcSech[c*x])^(-2), x]

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

Mathematica [N/A]

Not integrable

Time = 76.82 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx \]

[In]

Integrate[(a + b*ArcSech[c*x])^(-2),x]

[Out]

Integrate[(a + b*ArcSech[c*x])^(-2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.60 \[ \int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral((a + b*asech(c*x))**(-2), x)

Maxima [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 535, normalized size of antiderivative = 53.50 \[ \int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(c^2*x^3 + (c^2*x^3 - x)*sqrt(c*x + 1)*sqrt(-c*x + 1) - x)/((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) - (b^
2*log(c) + b^2*log(x) - a*b)*sqrt(c*x + 1)*sqrt(-c*x + 1) + a*b - (b^2*c^2*x^2 - sqrt(c*x + 1)*sqrt(-c*x + 1)*
b^2 - b^2)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + (b^2*c^2*x^2 - b^2)*log(x)) + integrate((c^4*x^4 - 2*c^2*x^
2 + (3*c^2*x^2 - 1)*(c*x + 1)*(c*x - 1) + (2*c^4*x^4 - 5*c^2*x^2 + 2)*sqrt(c*x + 1)*sqrt(-c*x + 1) + 1)/((b^2*
c^4*log(c) - a*b*c^4)*x^4 - (b^2*log(c) + b^2*log(x) - a*b)*(c*x + 1)*(c*x - 1) - 2*(b^2*c^2*log(c) - a*b*c^2)
*x^2 + b^2*log(c) - 2*((b^2*c^2*log(c) - a*b*c^2)*x^2 - b^2*log(c) + a*b + (b^2*c^2*x^2 - b^2)*log(x))*sqrt(c*
x + 1)*sqrt(-c*x + 1) - a*b - (b^2*c^4*x^4 - 2*b^2*c^2*x^2 - (c*x + 1)*(c*x - 1)*b^2 - 2*(b^2*c^2*x^2 - b^2)*s
qrt(c*x + 1)*sqrt(-c*x + 1) + b^2)*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) + (b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)
*log(x)), x)

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate((b*arcsech(c*x) + a)^(-2), x)

Mupad [N/A]

Not integrable

Time = 3.96 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]

[In]

int(1/(a + b*acosh(1/(c*x)))^2,x)

[Out]

int(1/(a + b*acosh(1/(c*x)))^2, x)