Integrand size = 21, antiderivative size = 49 \[ \int \frac {x \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {(-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))}{2 a^2}+\frac {\Gamma (1+n,\text {arcsinh}(a x))}{2 a^2} \] Output:
1/2*arcsinh(a*x)^n*GAMMA(1+n,-arcsinh(a*x))/a^2/((-arcsinh(a*x))^n)+1/2*GA MMA(1+n,arcsinh(a*x))/a^2
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {x \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\frac {(-\text {arcsinh}(a x))^{-n} \text {arcsinh}(a x)^n \Gamma (1+n,-\text {arcsinh}(a x))+\Gamma (1+n,\text {arcsinh}(a x))}{2 a^2} \] Input:
Integrate[(x*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2],x]
Output:
((ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(-ArcSinh[a*x])^n + Gamma[1 + n, ArcSinh[a*x]])/(2*a^2)
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6234, 3042, 26, 3789, 2612}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x \text {arcsinh}(a x)^n}{\sqrt {a^2 x^2+1}} \, dx\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \frac {\int a x \text {arcsinh}(a x)^nd\text {arcsinh}(a x)}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -i \text {arcsinh}(a x)^n \sin (i \text {arcsinh}(a x))d\text {arcsinh}(a x)}{a^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {i \int \text {arcsinh}(a x)^n \sin (i \text {arcsinh}(a x))d\text {arcsinh}(a x)}{a^2}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle -\frac {i \left (\frac {1}{2} i \int e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^nd\text {arcsinh}(a x)-\frac {1}{2} i \int e^{-\text {arcsinh}(a x)} \text {arcsinh}(a x)^nd\text {arcsinh}(a x)\right )}{a^2}\) |
\(\Big \downarrow \) 2612 |
\(\displaystyle -\frac {i \left (\frac {1}{2} i \text {arcsinh}(a x)^n (-\text {arcsinh}(a x))^{-n} \Gamma (n+1,-\text {arcsinh}(a x))+\frac {1}{2} i \Gamma (n+1,\text {arcsinh}(a x))\right )}{a^2}\) |
Input:
Int[(x*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2],x]
Output:
((-I)*(((I/2)*ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(-ArcSinh[a*x])^ n + (I/2)*Gamma[1 + n, ArcSinh[a*x]]))/a^2
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) )^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && !IntegerQ[m]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \frac {x \operatorname {arcsinh}\left (x a \right )^{n}}{\sqrt {a^{2} x^{2}+1}}d x\]
Input:
int(x*arcsinh(x*a)^n/(a^2*x^2+1)^(1/2),x)
Output:
int(x*arcsinh(x*a)^n/(a^2*x^2+1)^(1/2),x)
\[ \int \frac {x \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x*arcsinh(a*x)^n/(a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
integral(x*arcsinh(a*x)^n/sqrt(a^2*x^2 + 1), x)
\[ \int \frac {x \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x \operatorname {asinh}^{n}{\left (a x \right )}}{\sqrt {a^{2} x^{2} + 1}}\, dx \] Input:
integrate(x*asinh(a*x)**n/(a**2*x**2+1)**(1/2),x)
Output:
Integral(x*asinh(a*x)**n/sqrt(a**2*x**2 + 1), x)
\[ \int \frac {x \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x*arcsinh(a*x)^n/(a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate(x*arcsinh(a*x)^n/sqrt(a^2*x^2 + 1), x)
\[ \int \frac {x \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {x \operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1}} \,d x } \] Input:
integrate(x*arcsinh(a*x)^n/(a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate(x*arcsinh(a*x)^n/sqrt(a^2*x^2 + 1), x)
Timed out. \[ \int \frac {x \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {x\,{\mathrm {asinh}\left (a\,x\right )}^n}{\sqrt {a^2\,x^2+1}} \,d x \] Input:
int((x*asinh(a*x)^n)/(a^2*x^2 + 1)^(1/2),x)
Output:
int((x*asinh(a*x)^n)/(a^2*x^2 + 1)^(1/2), x)
\[ \int \frac {x \text {arcsinh}(a x)^n}{\sqrt {1+a^2 x^2}} \, dx=\int \frac {\mathit {asinh} \left (a x \right )^{n} x}{\sqrt {a^{2} x^{2}+1}}d x \] Input:
int(x*asinh(a*x)^n/(a^2*x^2+1)^(1/2),x)
Output:
int((asinh(a*x)**n*x)/sqrt(a**2*x**2 + 1),x)