\(\int \frac {x}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx\) [106]

Optimal result

Integrand size = 16, antiderivative size = 204 \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=-\frac {c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^2}-\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d^2}-\frac {c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5859, 5830, 6873, 6874, 5407, 2236, 2235, 5406} \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=-\frac {\sqrt {\pi } c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^2}-\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d^2}-\frac {\sqrt {\pi } c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d^2} \]

[In]

Int[x/Sqrt[a + b*ArcSinh[c + d*x]],x]

[Out]

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

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 5406

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5407

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] + Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5830

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst
[Int[(a + b*x)^n*Cosh[x]*(c*d + e*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ
[m, 0]

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-\frac {c}{d}+\frac {x}{d}}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {\cosh (x) \left (-\frac {c}{d}+\frac {\sinh (x)}{d}\right )}{\sqrt {a+b x}} \, dx,x,\text {arcsinh}(c+d x)\right )}{d} \\ & = -\frac {2 \text {Subst}\left (\int \cosh \left (\frac {a-x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {2 \text {Subst}\left (\int \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {2 \text {Subst}\left (\int \left (c \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+\frac {1}{2} \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {\text {Subst}\left (\int \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2}-\frac {(2 c) \text {Subst}\left (\int \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {\text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 b d^2}+\frac {\text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 b d^2}-\frac {c \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2}-\frac {c \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^2}-\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d^2}-\frac {c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{4 \sqrt {b} d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.06 \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {\frac {e^{-\frac {a}{b}} \left (4 c e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )-4 c \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\sqrt {2 \pi } \left (\text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (-\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )+\text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )\right )}{\sqrt {b}}}{8 d^2} \]

[In]

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

[Out]

((4*c*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]] - 4*c*Sqrt[-((a + b*ArcSinh[
c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/(E^(a/b)*Sqrt[a + b*ArcSinh[c + d*x]]) - (Sqrt[2*Pi]*
(Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(-Cosh[(2*a)/b] + Sinh[(2*a)/b]) + Erf[(Sqrt[2]*Sqrt[a +
b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b])))/Sqrt[b])/(8*d^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x/sqrt(a + b*asinh(c + d*x)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x/(a + b*asinh(c + d*x))^(1/2),x)

[Out]

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