Integrand size = 12, antiderivative size = 102 \[ \int \sqrt {a+b \text {arcsinh}(c x)} \, dx=x \sqrt {a+b \text {arcsinh}(c x)}+\frac {\sqrt {b} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c}-\frac {\sqrt {b} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 c} \]
1/4*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c-1/4* erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*b^(1/2)*Pi^(1/2)/c/exp(a/b)+x*(a+b* arcsinh(c*x))^(1/2)
Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.99 \[ \int \sqrt {a+b \text {arcsinh}(c x)} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {a+b \text {arcsinh}(c x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )}{\sqrt {\frac {a}{b}+\text {arcsinh}(c x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}}\right )}{2 c} \]
(Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]])/ Sqrt[a/b + ArcSinh[c*x]]) + Gamma[3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[-(( a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b))
Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6187, 6234, 25, 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.
\(\displaystyle \int \sqrt {a+b \text {arcsinh}(c x)} \, dx\) |
\(\Big \downarrow \) 6187 |
\(\displaystyle x \sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} b c \int \frac {x}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx\) |
\(\Big \downarrow \) 6234 |
\(\displaystyle x \sqrt {a+b \text {arcsinh}(c x)}-\frac {\int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}+x \sqrt {a+b \text {arcsinh}(c x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \sqrt {a+b \text {arcsinh}(c x)}+\frac {\int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \int \frac {e^{-\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i \int \frac {e^{\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))\right )}{2 c}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-i \int e^{\frac {a+b \text {arcsinh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 c}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\) |
x*Sqrt[a + b*ArcSinh[c*x]] - ((I/2)*((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sq rt[a + b*ArcSinh[c*x]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b* ArcSinh[c*x]]/Sqrt[b]])/E^(a/b)))/c
3.2.38.3.1 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_.) + (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[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \sqrt {a +b \,\operatorname {arcsinh}\left (c x \right )}d x\]
Exception generated. \[ \int \sqrt {a+b \text {arcsinh}(c x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {a+b \text {arcsinh}(c x)} \, dx=\int \sqrt {a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
\[ \int \sqrt {a+b \text {arcsinh}(c x)} \, dx=\int { \sqrt {b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
\[ \int \sqrt {a+b \text {arcsinh}(c x)} \, dx=\int { \sqrt {b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
Timed out. \[ \int \sqrt {a+b \text {arcsinh}(c x)} \, dx=\int \sqrt {a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]