Integrand size = 14, antiderivative size = 92 \[ \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{2 \sqrt {b} d} \]
1/2*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/b^(1/2)+1/ 2*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/d/exp(a/b)/b^(1/2)
Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\frac {e^{-\frac {a}{b}} \left (-e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )}{2 d \sqrt {a+b \text {arcsinh}(c+d x)}} \]
(-(E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d *x]]) + Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/(2*d*E^(a/b)*Sqrt[a + b*ArcSinh[c + d*x]])
Time = 0.45 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6273, 6189, 3042, 3788, 26, 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.
\(\displaystyle \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx\) |
\(\Big \downarrow \) 6273 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6189 |
\(\displaystyle \frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle \frac {\frac {1}{2} i \int -\frac {i e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int \frac {i e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{2} \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b d}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}+\int e^{\frac {a+b \text {arcsinh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}}{b d}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{b d}\) |
((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/2 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(2*E^(a/b))) /(b*d)
3.3.8.3.1 Defintions of rubi rules used
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_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
\[\int \frac {1}{\sqrt {a +b \,\operatorname {arcsinh}\left (d x +c \right )}}d x\]
Exception generated. \[ \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}\, dx \]
\[ \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}} \,d x } \]
\[ \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )}} \,d x \]