Integrand size = 10, antiderivative size = 63 \[ \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^2}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{4 a^2} \] Output:
-1/8*2^(1/2)*Pi^(1/2)*erf(2^(1/2)*arcsinh(a*x)^(1/2))/a^2+1/8*2^(1/2)*Pi^( 1/2)*erfi(2^(1/2)*arcsinh(a*x)^(1/2))/a^2
Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {\text {arcsinh}(a x)}}+\Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )}{4 \sqrt {2} a^2} \] Input:
Integrate[x/Sqrt[ArcSinh[a*x]],x]
Output:
((Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -2*ArcSinh[a*x]])/Sqrt[ArcSinh[a*x]] + Ga mma[1/2, 2*ArcSinh[a*x]])/(4*Sqrt[2]*a^2)
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {6195, 5971, 27, 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.
\(\displaystyle \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx\) |
\(\Big \downarrow \) 6195 |
\(\displaystyle \frac {\int \frac {a x \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \frac {\int \frac {\sinh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -\frac {i \sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {i \int \frac {\sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{2 a^2}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle -\frac {i \left (\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{2 a^2}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle -\frac {i \left (i \int e^{2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{2 a^2}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle -\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{2 a^2}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{2 a^2}\) |
Input:
Int[x/Sqrt[ArcSinh[a*x]],x]
Output:
((-1/2*I)*((-1/2*I)*Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + (I/2)*Sqr t[Pi/2]*Erfi[Sqrt[2]*Sqrt[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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Time = 0.83 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.59
method | result | size |
default | \(-\frac {\sqrt {\pi }\, \sqrt {2}\, \left (\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (x a \right )}\right )-\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (x a \right )}\right )\right )}{8 a^{2}}\) | \(37\) |
Input:
int(x/arcsinh(x*a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/8*Pi^(1/2)*2^(1/2)*(erf(2^(1/2)*arcsinh(x*a)^(1/2))-erfi(2^(1/2)*arcsin h(x*a)^(1/2)))/a^2
Exception generated. \[ \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x/arcsinh(a*x)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {x}{\sqrt {\operatorname {asinh}{\left (a x \right )}}}\, dx \] Input:
integrate(x/asinh(a*x)**(1/2),x)
Output:
Integral(x/sqrt(asinh(a*x)), x)
\[ \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {x}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \] Input:
integrate(x/arcsinh(a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(x/sqrt(arcsinh(a*x)), x)
\[ \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {x}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \] Input:
integrate(x/arcsinh(a*x)^(1/2),x, algorithm="giac")
Output:
integrate(x/sqrt(arcsinh(a*x)), x)
Timed out. \[ \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {x}{\sqrt {\mathrm {asinh}\left (a\,x\right )}} \,d x \] Input:
int(x/asinh(a*x)^(1/2),x)
Output:
int(x/asinh(a*x)^(1/2), x)
\[ \int \frac {x}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {-\frac {4 \sqrt {\mathit {asinh} \left (a x \right )}\, \mathit {asinh} \left (a x \right )}{3}+2 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, a x -4 \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (a x \right )}\, x^{2}}{a^{2} x^{2}+1}d x \right ) a^{3}}{a^{2}} \] Input:
int(x/asinh(a*x)^(1/2),x)
Output:
(2*( - 2*sqrt(asinh(a*x))*asinh(a*x) + 3*sqrt(a**2*x**2 + 1)*sqrt(asinh(a* x))*a*x - 6*int((sqrt(a**2*x**2 + 1)*sqrt(asinh(a*x))*x**2)/(a**2*x**2 + 1 ),x)*a**3))/(3*a**2)