\(\int \text {arcsinh}(a x^n) \, dx\) [310]

Optimal result

Integrand size = 6, antiderivative size = 56 \[ \int \text {arcsinh}\left (a x^n\right ) \, dx=x \text {arcsinh}\left (a x^n\right )-\frac {a n x^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-a^2 x^{2 n}\right )}{1+n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5874, 12, 371} \[ \int \text {arcsinh}\left (a x^n\right ) \, dx=x \text {arcsinh}\left (a x^n\right )-\frac {a n x^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-a^2 x^{2 n}\right )}{n+1} \]

[In]

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

[Out]

x*ArcSinh[a*x^n] - (a*n*x^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/(2*n), (3 + n^(-1))/2, -(a^2*x^(2*n))])/(1 +
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 5874

Int[ArcSinh[u_], x_Symbol] :> Simp[x*ArcSinh[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/Sqrt[1 + u^2]), x], x]
/; InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \text {arcsinh}\left (a x^n\right ) \, dx=x \text {arcsinh}\left (a x^n\right )-\frac {a n x^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2 n},\frac {1}{2} \left (3+\frac {1}{n}\right ),-a^2 x^{2 n}\right )}{1+n} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(asinh(a*x**n), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(arcsinh(a*x^n), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(asinh(a*x^n), x)