\(\int x \text {arcsinh}(a x^n) \, dx\) [309]

Optimal result

Integrand size = 8, antiderivative size = 65 \[ \int x \text {arcsinh}\left (a x^n\right ) \, dx=\frac {1}{2} x^2 \text {arcsinh}\left (a x^n\right )-\frac {a n x^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2 n},\frac {1}{2} \left (3+\frac {2}{n}\right ),-a^2 x^{2 n}\right )}{2 (2+n)} \]

[Out]

1/2*x^2*arcsinh(a*x^n)-1/2*a*n*x^(2+n)*hypergeom([1/2, 1/2*(2+n)/n],[3/2+1/n],-a^2*x^(2*n))/(2+n)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5875, 12, 371} \[ \int x \text {arcsinh}\left (a x^n\right ) \, dx=\frac {1}{2} x^2 \text {arcsinh}\left (a x^n\right )-\frac {a n x^{n+2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+2}{2 n},\frac {1}{2} \left (3+\frac {2}{n}\right ),-a^2 x^{2 n}\right )}{2 (n+2)} \]

[In]

Int[x*ArcSinh[a*x^n],x]

[Out]

(x^2*ArcSinh[a*x^n])/2 - (a*n*x^(2 + n)*Hypergeometric2F1[1/2, (2 + n)/(2*n), (3 + 2/n)/2, -(a^2*x^(2*n))])/(2
*(2 + n))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 5875

Int[((a_.) + ArcSinh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcSin
h[u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/Sqrt[1 + u^2]),
x], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)
^(m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \text {arcsinh}\left (a x^n\right )-\frac {1}{2} \int \frac {a n x^{1+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx \\ & = \frac {1}{2} x^2 \text {arcsinh}\left (a x^n\right )-\frac {1}{2} (a n) \int \frac {x^{1+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx \\ & = \frac {1}{2} x^2 \text {arcsinh}\left (a x^n\right )-\frac {a n x^{2+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+n}{2 n},\frac {1}{2} \left (3+\frac {2}{n}\right ),-a^2 x^{2 n}\right )}{2 (2+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int x \text {arcsinh}\left (a x^n\right ) \, dx=\frac {x^2 \left ((2+n) \text {arcsinh}\left (a x^n\right )-a n x^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}+\frac {1}{n},\frac {3}{2}+\frac {1}{n},-a^2 x^{2 n}\right )\right )}{2 (2+n)} \]

[In]

Integrate[x*ArcSinh[a*x^n],x]

[Out]

(x^2*((2 + n)*ArcSinh[a*x^n] - a*n*x^n*Hypergeometric2F1[1/2, 1/2 + n^(-1), 3/2 + n^(-1), -(a^2*x^(2*n))]))/(2
*(2 + n))

Maple [F]

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

[In]

int(x*arcsinh(a*x^n),x)

[Out]

int(x*arcsinh(a*x^n),x)

Fricas [F(-2)]

Exception generated. \[ \int x \text {arcsinh}\left (a x^n\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int x \text {arcsinh}\left (a x^n\right ) \, dx=\int x \operatorname {asinh}{\left (a x^{n} \right )}\, dx \]

[In]

integrate(x*asinh(a*x**n),x)

[Out]

Integral(x*asinh(a*x**n), x)

Maxima [F]

\[ \int x \text {arcsinh}\left (a x^n\right ) \, dx=\int { x \operatorname {arsinh}\left (a x^{n}\right ) \,d x } \]

[In]

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

[Out]

-1/4*n*x^2 - a*n*integrate(1/2*x*x^n/(a^3*x^(3*n) + a*x^n + (a^2*x^(2*n) + 1)^(3/2)), x) + 1/2*x^2*log(a*x^n +
sqrt(a^2*x^(2*n) + 1)) + n*integrate(1/2*x/(a^2*x^(2*n) + 1), x)

Giac [F]

\[ \int x \text {arcsinh}\left (a x^n\right ) \, dx=\int { x \operatorname {arsinh}\left (a x^{n}\right ) \,d x } \]

[In]

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

[Out]

integrate(x*arcsinh(a*x^n), x)

Mupad [F(-1)]

Timed out. \[ \int x \text {arcsinh}\left (a x^n\right ) \, dx=\int x\,\mathrm {asinh}\left (a\,x^n\right ) \,d x \]

[In]

int(x*asinh(a*x^n),x)

[Out]

int(x*asinh(a*x^n), x)