\(\int x^m \text {arcsinh}(a x)^3 \, dx\) [118]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\frac {x^{1+m} \text {arcsinh}(a x)^3}{1+m}-\frac {3 a \text {Int}\left (\frac {x^{1+m} \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}},x\right )}{1+m} \]

[Out]

x^(1+m)*arcsinh(a*x)^3/(1+m)-3*a*Unintegrable(x^(1+m)*arcsinh(a*x)^2/(a^2*x^2+1)^(1/2),x)/(1+m)

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int x^m \text {arcsinh}(a x)^3 \, dx \]

[In]

Int[x^m*ArcSinh[a*x]^3,x]

[Out]

(x^(1 + m)*ArcSinh[a*x]^3)/(1 + m) - (3*a*Defer[Int][(x^(1 + m)*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2], x])/(1 + m)

Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \text {arcsinh}(a x)^3}{1+m}-\frac {(3 a) \int \frac {x^{1+m} \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{1+m} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int x^m \text {arcsinh}(a x)^3 \, dx \]

[In]

Integrate[x^m*ArcSinh[a*x]^3,x]

[Out]

Integrate[x^m*ArcSinh[a*x]^3, x]

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int x^{m} \operatorname {arcsinh}\left (a x \right )^{3}d x\]

[In]

int(x^m*arcsinh(a*x)^3,x)

[Out]

int(x^m*arcsinh(a*x)^3,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int { x^{m} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]

[In]

integrate(x^m*arcsinh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^m*arcsinh(a*x)^3, x)

Sympy [N/A]

Not integrable

Time = 3.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int x^{m} \operatorname {asinh}^{3}{\left (a x \right )}\, dx \]

[In]

integrate(x**m*asinh(a*x)**3,x)

[Out]

Integral(x**m*asinh(a*x)**3, x)

Maxima [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 134, normalized size of antiderivative = 13.40 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int { x^{m} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]

[In]

integrate(x^m*arcsinh(a*x)^3,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int { x^{m} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]

[In]

integrate(x^m*arcsinh(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^m*arcsinh(a*x)^3, x)

Mupad [N/A]

Not integrable

Time = 2.88 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int x^m\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]

[In]

int(x^m*asinh(a*x)^3,x)

[Out]

int(x^m*asinh(a*x)^3, x)