Integrand size = 14, antiderivative size = 107 \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=-\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 \sqrt {b} c^2} \]
-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)
Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.01 \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\frac {e^{-\frac {2 a}{b}} \left (\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{4 \sqrt {2} c^2 \sqrt {a+b \text {arcsinh}(c x)}} \]
(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]])
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6195, 25, 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.
\(\displaystyle \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx\) |
\(\Big \downarrow \) 6195 |
\(\displaystyle \frac {\int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b c^2}\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle -\frac {\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b c^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 b c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int -\frac {i \sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 b c^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 b c^2}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle \frac {i \left (\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))\right )}{2 b c^2}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-i \int e^{\frac {2 (a+b \text {arcsinh}(c x))}{b}-\frac {2 a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b c^2}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {i \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 b c^2}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 b c^2}\) |
((I/2)*((I/2)*Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSi nh[c*x]])/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*Ar cSinh[c*x]])/Sqrt[b]])/E^((2*a)/b)))/(b*c^2)
3.2.46.3.1 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[(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]
\[\int \frac {x}{\sqrt {a +b \,\operatorname {arcsinh}\left (c x \right )}}d x\]
Exception generated. \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {x}{\sqrt {a + b \operatorname {asinh}{\left (c x \right )}}}\, dx \]
\[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {x}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]
\[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int { \frac {x}{\sqrt {b \operatorname {arsinh}\left (c x\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx=\int \frac {x}{\sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )}} \,d x \]