\(\int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx\) [527]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{m} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.80 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^{m} \operatorname {asinh}^{n}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{m} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x^{m} \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.62 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^m \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x^m\,{\mathrm {asinh}\left (a\,x\right )}^n}{\sqrt {a^2\,x^2+1}} \,d x \]

[In]

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

[Out]

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