Integrand size = 10, antiderivative size = 84 \[ \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 x \sqrt {1+a^2 x^2}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a^2}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{a^2} \]
1/2*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2+1/2*erfi(2^(1/2)* arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2-2*x*(a^2*x^2+1)^(1/2)/a/arcsinh(a *x)^(1/2)
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.87 \[ \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx=\frac {\frac {\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )}{\sqrt {2} \sqrt {\text {arcsinh}(a x)}}-\frac {\Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )}{\sqrt {2}}-\frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}}{a^2} \]
((Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -2*ArcSinh[a*x]])/(Sqrt[2]*Sqrt[ArcSinh[a *x]]) - Gamma[1/2, 2*ArcSinh[a*x]]/Sqrt[2] - Sinh[2*ArcSinh[a*x]]/Sqrt[Arc Sinh[a*x]])/a^2
Time = 0.37 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6193, 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.
\(\displaystyle \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 6193 |
\(\displaystyle \frac {2 \int \frac {\cosh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \int \frac {\sin \left (2 i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle -\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {i e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)+\frac {1}{2} \int \frac {e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {2 \left (\int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\int e^{2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {2 \left (\int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\) |
(-2*x*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) + (2*((Sqrt[Pi/2]*Erf[Sqrt [2]*Sqrt[ArcSinh[a*x]]])/2 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]]) /2))/a^2
3.2.2.3.1 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_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (2 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x -\operatorname {arcsinh}\left (a x \right ) \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )-\operatorname {arcsinh}\left (a x \right ) \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )\right )}{2 \sqrt {\pi }\, a^{2} \operatorname {arcsinh}\left (a x \right )}\) | \(82\) |
-1/2*2^(1/2)*(2*2^(1/2)*arcsinh(a*x)^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*a*x- arcsinh(a*x)*Pi*erf(2^(1/2)*arcsinh(a*x)^(1/2))-arcsinh(a*x)*Pi*erfi(2^(1/ 2)*arcsinh(a*x)^(1/2)))/Pi^(1/2)/a^2/arcsinh(a*x)
Exception generated. \[ \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x}{\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {x}{\operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {x}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}} \,d x \]