\(\int \frac {x^m}{\text {arccosh}(a x)^2} \, dx\) [120]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {x^m}{\text {arccosh}(a x)^2} \, dx=\text {Int}\left (\frac {x^m}{\text {arccosh}(a x)^2},x\right ) \]

[Out]

Unintegrable(x^m/arccosh(a*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 = {} \[ \int \frac {x^m}{\text {arccosh}(a x)^2} \, dx=\int \frac {x^m}{\text {arccosh}(a x)^2} \, dx \]

[In]

Int[x^m/ArcCosh[a*x]^2,x]

[Out]

Defer[Int][x^m/ArcCosh[a*x]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m}{\text {arccosh}(a x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {x^m}{\text {arccosh}(a x)^2} \, dx=\int \frac {x^m}{\text {arccosh}(a x)^2} \, dx \]

[In]

Integrate[x^m/ArcCosh[a*x]^2,x]

[Out]

Integrate[x^m/ArcCosh[a*x]^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.73 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {x^{m}}{\operatorname {arccosh}\left (a x \right )^{2}}d x\]

[In]

int(x^m/arccosh(a*x)^2,x)

[Out]

int(x^m/arccosh(a*x)^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {x^m}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x^{m}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^m/arccosh(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^m/arccosh(a*x)^2, x)

Sympy [N/A]

Not integrable

Time = 1.59 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\text {arccosh}(a x)^2} \, dx=\int \frac {x^{m}}{\operatorname {acosh}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(x**m/acosh(a*x)**2,x)

[Out]

Integral(x**m/acosh(a*x)**2, x)

Maxima [N/A]

Not integrable

Time = 0.72 (sec) , antiderivative size = 305, normalized size of antiderivative = 30.50 \[ \int \frac {x^m}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x^{m}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^m/arccosh(a*x)^2,x, algorithm="maxima")

[Out]

-((a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1)*x^m + (a^3*x^3 - a*x)*x^m)/((a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)
*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) + integrate(((a^3*(m + 1)*x^3 - a*(m - 1)*x)*(a*x + 1)*(a*
x - 1)*x^m + (2*a^4*(m + 1)*x^4 - a^2*(3*m + 1)*x^2 + m)*sqrt(a*x + 1)*sqrt(a*x - 1)*x^m + (a^5*(m + 1)*x^5 -
2*a^3*(m + 1)*x^3 + a*(m + 1)*x)*x^m)/((a^5*x^5 + (a*x + 1)*(a*x - 1)*a^3*x^3 - 2*a^3*x^3 + 2*(a^4*x^4 - a^2*x
^2)*sqrt(a*x + 1)*sqrt(a*x - 1) + a*x)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {x^m}{\text {arccosh}(a x)^2} \, dx=\int { \frac {x^{m}}{\operatorname {arcosh}\left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^m/arccosh(a*x)^2,x, algorithm="giac")

[Out]

integrate(x^m/arccosh(a*x)^2, x)

Mupad [N/A]

Not integrable

Time = 2.52 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {x^m}{\text {arccosh}(a x)^2} \, dx=\int \frac {x^m}{{\mathrm {acosh}\left (a\,x\right )}^2} \,d x \]

[In]

int(x^m/acosh(a*x)^2,x)

[Out]

int(x^m/acosh(a*x)^2, x)