\(\int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx\) [146]

Optimal result

Integrand size = 14, antiderivative size = 107 \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=-\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2} \]

[Out]

-1/8*exp(2*a/b)*erf(2^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/c^2/b^(1/2)+1/8*erfi(2^(1/2)*(a
+b*arcsinh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/c^2/exp(2*a/b)/b^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5780, 5556, 12, 3389, 2211, 2236, 2235} \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}-\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2} \]

[In]

Int[x/Sqrt[a + b*ArcSinh[c*x]],x]

[Out]

-1/4*(E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(Sqrt[b]*c^2) + (Sqrt[Pi/2]*Erfi
[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^2*E^((2*a)/b))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 2235

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]

Rule 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5556

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]

Rule 5780

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b c^2} \\ & = -\frac {\text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b c^2}+\frac {\text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b c^2} \\ & = -\frac {\text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b c^2}+\frac {\text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b c^2} \\ & = -\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.01 \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {e^{-\frac {2 a}{b}} \left (\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{4 \sqrt {2} c^2 \sqrt {a+b \text {arcsinh}(c x)}} \]

[In]

Integrate[x/Sqrt[a + b*ArcSinh[c*x]],x]

[Out]

(Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c*x]))/b] + E^((4*a)/b)*Sqrt[a/b + ArcSinh[c*x]
]*Gamma[1/2, (2*(a + b*ArcSinh[c*x]))/b])/(4*Sqrt[2]*c^2*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c*x]])

Maple [F]

\[\int \frac {x}{\sqrt {a +b \,\operatorname {arcsinh}\left (c x \right )}}d x\]

[In]

int(x/(a+b*arcsinh(c*x))^(1/2),x)

[Out]

int(x/(a+b*arcsinh(c*x))^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x/(a+b*arcsinh(c*x))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {x}{\sqrt {a + b \operatorname {asinh}{\left (c x \right )}}}\, dx \]

[In]

integrate(x/(a+b*asinh(c*x))**(1/2),x)

[Out]

Integral(x/sqrt(a + b*asinh(c*x)), x)

Maxima [F]

\[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {x}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]

[In]

integrate(x/(a+b*arcsinh(c*x))^(1/2),x, algorithm="maxima")

[Out]

integrate(x/sqrt(b*arcsinh(c*x) + a), x)

Giac [F]

\[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {x}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]

[In]

integrate(x/(a+b*arcsinh(c*x))^(1/2),x, algorithm="giac")

[Out]

integrate(x/sqrt(b*arcsinh(c*x) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {x}{\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}} \,d x \]

[In]

int(x/(a + b*asinh(c*x))^(1/2),x)

[Out]

int(x/(a + b*asinh(c*x))^(1/2), x)