\(\int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx\) [387]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\text {Int}\left (\frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)},x\right ) \]

[Out]

Unintegrable(1/x^2/arcsinh(a*x)/(a^2*x^2+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} + 1} x^{2} \operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

integral(sqrt(a^2*x^2 + 1)/((a^2*x^4 + x^2)*arcsinh(a*x)), x)

Sympy [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {1}{x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral(1/(x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)), x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} + 1} x^{2} \operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(a^2*x^2 + 1)*x^2*arcsinh(a*x)), x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} + 1} x^{2} \operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

integrate(1/(sqrt(a^2*x^2 + 1)*x^2*arcsinh(a*x)), x)

Mupad [N/A]

Not integrable

Time = 2.79 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {1}{x^2\,\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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