Integrand size = 17, antiderivative size = 33 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=-\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(x)}\right ) \] Output:
-1/2*Pi^(1/2)*erf(arcsinh(x)^(1/2))+1/2*Pi^(1/2)*erfi(arcsinh(x)^(1/2))
Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\frac {1}{2} \left (\frac {\sqrt {-\text {arcsinh}(x)} \Gamma \left (\frac {1}{2},-\text {arcsinh}(x)\right )}{\sqrt {\text {arcsinh}(x)}}+\Gamma \left (\frac {1}{2},\text {arcsinh}(x)\right )\right ) \] Input:
Integrate[x/(Sqrt[1 + x^2]*Sqrt[ArcSinh[x]]),x]
Output:
((Sqrt[-ArcSinh[x]]*Gamma[1/2, -ArcSinh[x]])/Sqrt[ArcSinh[x]] + Gamma[1/2, ArcSinh[x]])/2
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6234, 3042, 26, 3789, 2611, 2633, 2634}
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.
\(\displaystyle \int \frac {x}{\sqrt {x^2+1} \sqrt {\text {arcsinh}(x)}} \, dx\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle \int \frac {x}{\sqrt {\text {arcsinh}(x)}}d\text {arcsinh}(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sin (i \text {arcsinh}(x))}{\sqrt {\text {arcsinh}(x)}}d\text {arcsinh}(x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sin (i \text {arcsinh}(x))}{\sqrt {\text {arcsinh}(x)}}d\text {arcsinh}(x)\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle -i \left (\frac {1}{2} i \int \frac {e^{\text {arcsinh}(x)}}{\sqrt {\text {arcsinh}(x)}}d\text {arcsinh}(x)-\frac {1}{2} i \int \frac {e^{-\text {arcsinh}(x)}}{\sqrt {\text {arcsinh}(x)}}d\text {arcsinh}(x)\right )\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle -i \left (i \int e^{\text {arcsinh}(x)}d\sqrt {\text {arcsinh}(x)}-i \int e^{-\text {arcsinh}(x)}d\sqrt {\text {arcsinh}(x)}\right )\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle -i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(x)}\right )-i \int e^{-\text {arcsinh}(x)}d\sqrt {\text {arcsinh}(x)}\right )\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(x)}\right )\right )\) |
Input:
Int[x/(Sqrt[1 + x^2]*Sqrt[ArcSinh[x]]),x]
Output:
(-I)*((-1/2*I)*Sqrt[Pi]*Erf[Sqrt[ArcSinh[x]]] + (I/2)*Sqrt[Pi]*Erfi[Sqrt[A rcSinh[x]]])
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_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
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]
\[\int \frac {x}{\sqrt {x^{2}+1}\, \sqrt {\operatorname {arcsinh}\left (x \right )}}d x\]
Input:
int(x/(x^2+1)^(1/2)/arcsinh(x)^(1/2),x)
Output:
int(x/(x^2+1)^(1/2)/arcsinh(x)^(1/2),x)
Exception generated. \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x/(x^2+1)^(1/2)/arcsinh(x)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\int \frac {x}{\sqrt {x^{2} + 1} \sqrt {\operatorname {asinh}{\left (x \right )}}}\, dx \] Input:
integrate(x/(x**2+1)**(1/2)/asinh(x)**(1/2),x)
Output:
Integral(x/(sqrt(x**2 + 1)*sqrt(asinh(x))), x)
\[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\int { \frac {x}{\sqrt {x^{2} + 1} \sqrt {\operatorname {arsinh}\left (x\right )}} \,d x } \] Input:
integrate(x/(x^2+1)^(1/2)/arcsinh(x)^(1/2),x, algorithm="maxima")
Output:
integrate(x/(sqrt(x^2 + 1)*sqrt(arcsinh(x))), x)
\[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\int { \frac {x}{\sqrt {x^{2} + 1} \sqrt {\operatorname {arsinh}\left (x\right )}} \,d x } \] Input:
integrate(x/(x^2+1)^(1/2)/arcsinh(x)^(1/2),x, algorithm="giac")
Output:
integrate(x/(sqrt(x^2 + 1)*sqrt(arcsinh(x))), x)
Timed out. \[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\int \frac {x}{\sqrt {\mathrm {asinh}\left (x\right )}\,\sqrt {x^2+1}} \,d x \] Input:
int(x/(asinh(x)^(1/2)*(x^2 + 1)^(1/2)),x)
Output:
int(x/(asinh(x)^(1/2)*(x^2 + 1)^(1/2)), x)
\[ \int \frac {x}{\sqrt {1+x^2} \sqrt {\text {arcsinh}(x)}} \, dx=\int \frac {\sqrt {x^{2}+1}\, \sqrt {\mathit {asinh} \left (x \right )}\, x}{\mathit {asinh} \left (x \right ) x^{2}+\mathit {asinh} \left (x \right )}d x \] Input:
int(x/(x^2+1)^(1/2)/asinh(x)^(1/2),x)
Output:
int((sqrt(x**2 + 1)*sqrt(asinh(x))*x)/(asinh(x)*x**2 + asinh(x)),x)