\(\int \frac {1}{x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx\) [386]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {1}{x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {1}{x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} + 1} x \operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

integral(sqrt(a^2*x^2 + 1)/((a^2*x^3 + x)*arcsinh(a*x)), x)

Sympy [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {1}{x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} + 1} x \operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} + 1} x \operatorname {arsinh}\left (a x\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.69 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)} \, dx=\int \frac {1}{x\,\mathrm {asinh}\left (a\,x\right )\,\sqrt {a^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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