\(\int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx\) [59]

Optimal result

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

[Out]

Unintegrable(1/x^2/arcsinh(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 = {} \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx \]

[In]

Int[1/(x^2*ArcSinh[a*x]^2),x]

[Out]

Defer[Int][1/(x^2*ArcSinh[a*x]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx \]

[In]

Integrate[1/(x^2*ArcSinh[a*x]^2),x]

[Out]

Integrate[1/(x^2*ArcSinh[a*x]^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{2} \operatorname {arcsinh}\left (a x \right )^{2}}d x\]

[In]

int(1/x^2/arcsinh(a*x)^2,x)

[Out]

int(1/x^2/arcsinh(a*x)^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx=\int { \frac {1}{x^{2} \operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^2/arcsinh(a*x)^2,x, algorithm="fricas")

[Out]

integral(1/(x^2*arcsinh(a*x)^2), x)

Sympy [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(1/x**2/asinh(a*x)**2,x)

[Out]

Integral(1/(x**2*asinh(a*x)**2), x)

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 237, normalized size of antiderivative = 23.70 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx=\int { \frac {1}{x^{2} \operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^2/arcsinh(a*x)^2,x, algorithm="maxima")

[Out]

-(a^3*x^3 + a*x + (a^2*x^2 + 1)^(3/2))/((a^3*x^4 + sqrt(a^2*x^2 + 1)*a^2*x^3 + a*x^2)*log(a*x + sqrt(a^2*x^2 +
1))) - integrate((a^5*x^5 + 2*a^3*x^3 + (a^3*x^3 + 3*a*x)*(a^2*x^2 + 1) + a*x + (2*a^4*x^4 + 5*a^2*x^2 + 2)*s
qrt(a^2*x^2 + 1))/((a^5*x^7 + (a^2*x^2 + 1)*a^3*x^5 + 2*a^3*x^5 + a*x^3 + 2*(a^4*x^6 + a^2*x^4)*sqrt(a^2*x^2 +
1))*log(a*x + sqrt(a^2*x^2 + 1))), x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx=\int { \frac {1}{x^{2} \operatorname {arsinh}\left (a x\right )^{2}} \,d x } \]

[In]

integrate(1/x^2/arcsinh(a*x)^2,x, algorithm="giac")

[Out]

integrate(1/(x^2*arcsinh(a*x)^2), x)

Mupad [N/A]

Not integrable

Time = 2.67 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arcsinh}(a x)^2} \, dx=\int \frac {1}{x^2\,{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]

[In]

int(1/(x^2*asinh(a*x)^2),x)

[Out]

int(1/(x^2*asinh(a*x)^2), x)