\(\int x^m \text {arcsinh}(a x^n) \, dx\) [307]

Optimal result

Integrand size = 10, antiderivative size = 77 \[ \int x^m \text {arcsinh}\left (a x^n\right ) \, dx=\frac {x^{1+m} \text {arcsinh}\left (a x^n\right )}{1+m}-\frac {a n x^{1+m+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m+n}{2 n},\frac {1+m+3 n}{2 n},-a^2 x^{2 n}\right )}{(1+m) (1+m+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5875, 12, 371} \[ \int x^m \text {arcsinh}\left (a x^n\right ) \, dx=\frac {x^{m+1} \text {arcsinh}\left (a x^n\right )}{m+1}-\frac {a n x^{m+n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+n+1}{2 n},\frac {m+3 n+1}{2 n},-a^2 x^{2 n}\right )}{(m+1) (m+n+1)} \]

[In]

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

[Out]

(x^(1 + m)*ArcSinh[a*x^n])/(1 + m) - (a*n*x^(1 + m + n)*Hypergeometric2F1[1/2, (1 + m + n)/(2*n), (1 + m + 3*n
)/(2*n), -(a^2*x^(2*n))])/((1 + m)*(1 + m + 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 {x^{1+m} \text {arcsinh}\left (a x^n\right )}{1+m}-\frac {\int \frac {a n x^{m+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx}{1+m} \\ & = \frac {x^{1+m} \text {arcsinh}\left (a x^n\right )}{1+m}-\frac {(a n) \int \frac {x^{m+n}}{\sqrt {1+a^2 x^{2 n}}} \, dx}{1+m} \\ & = \frac {x^{1+m} \text {arcsinh}\left (a x^n\right )}{1+m}-\frac {a n x^{1+m+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m+n}{2 n},\frac {1+m+3 n}{2 n},-a^2 x^{2 n}\right )}{(1+m) (1+m+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.96 \[ \int x^m \text {arcsinh}\left (a x^n\right ) \, dx=\frac {x^{1+m} \left ((1+m+n) \text {arcsinh}\left (a x^n\right )-a n x^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m+n}{2 n},\frac {1+m+3 n}{2 n},-a^2 x^{2 n}\right )\right )}{(1+m) (1+m+n)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x**m*asinh(a*x**n), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate(x^m*arcsinh(a*x^n), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^m*asinh(a*x^n), x)