\(\int \frac {(d x)^m}{(a+b \text {sech}^{-1}(c x))^2} \, dx\) [73]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\text {Int}\left (\frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable((d*x)^m/(a+b*arcsech(c*x))^2,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 = {} \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx \]

[In]

Int[(d*x)^m/(a + b*ArcSech[c*x])^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx \]

[In]

Integrate[(d*x)^m/(a + b*ArcSech[c*x])^2,x]

[Out]

Integrate[(d*x)^m/(a + b*ArcSech[c*x])^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.72 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int \frac {\left (d x\right )^{m}}{\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

Integral((d*x)**m/(a + b*asech(c*x))**2, x)

Maxima [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 616, normalized size of antiderivative = 38.50 \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-((c^2*d^m*x^3 - d^m*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^m + (c^2*d^m*x^3 - d^m*x)*x^m)/((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 - sqr
t(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)) + int
egrate(((c^2*d^m*(m + 3)*x^2 - d^m*(m + 1))*(c*x + 1)*(c*x - 1)*x^m + (c^4*d^m*(m + 2)*x^4 - c^2*d^m*(3*m + 5)
*x^2 + 2*d^m*(m + 1))*sqrt(c*x + 1)*sqrt(-c*x + 1)*x^m + (c^4*d^m*(m + 1)*x^4 - 2*c^2*d^m*(m + 1)*x^2 + d^m*(m
+ 1))*x^m)/((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)*sqrt(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 = 18, normalized size of antiderivative = 1.12 \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integrate((d*x)^m/(b*arcsech(c*x) + a)^2, x)

Mupad [N/A]

Not integrable

Time = 3.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {(d x)^m}{\left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \]

[In]

int((d*x)^m/(a + b*acosh(1/(c*x)))^2,x)

[Out]

int((d*x)^m/(a + b*acosh(1/(c*x)))^2, x)