Integrand size = 8, antiderivative size = 84 \[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {4 x}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {2 \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a}+\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{3 a} \]
2/3*erf(arcsinh(a*x)^(1/2))*Pi^(1/2)/a+2/3*erfi(arcsinh(a*x)^(1/2))*Pi^(1/ 2)/a-2/3*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(3/2)-4/3*x/arcsinh(a*x)^(1/2)
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {e^{-\text {arcsinh}(a x)} \left (1+e^{2 \text {arcsinh}(a x)}-2 \text {arcsinh}(a x)+2 e^{2 \text {arcsinh}(a x)} \text {arcsinh}(a x)+2 e^{\text {arcsinh}(a x)} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )+2 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )}{3 a \text {arcsinh}(a x)^{3/2}} \]
-1/3*(1 + E^(2*ArcSinh[a*x]) - 2*ArcSinh[a*x] + 2*E^(2*ArcSinh[a*x])*ArcSi nh[a*x] + 2*E^ArcSinh[a*x]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -ArcSinh[a*x]] + 2*E^ArcSinh[a*x]*ArcSinh[a*x]^(3/2)*Gamma[1/2, ArcSinh[a*x]])/(a*E^ArcS inh[a*x]*ArcSinh[a*x]^(3/2))
Time = 0.53 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {6188, 6233, 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.
\(\displaystyle \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 6188 |
\(\displaystyle \frac {2}{3} a \int \frac {x}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}dx-\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 6233 |
\(\displaystyle \frac {2}{3} a \left (\frac {2 \int \frac {1}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 6189 |
\(\displaystyle \frac {2}{3} a \left (\frac {2 \int \frac {\sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2}{3} a \left (-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}\right )\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle -\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2}{3} a \left (-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {i e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^2}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {2}{3} a \left (\frac {2 \left (\frac {1}{2} \int \frac {e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)+\frac {1}{2} \int \frac {e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {2}{3} a \left (\frac {2 \left (\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\int e^{\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {2}{3} a \left (\frac {2 \left (\int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {2}{3} a \left (\frac {2 \left (\frac {1}{2} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
(-2*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) + (2*a*((-2*x)/(a*Sqrt[Arc Sinh[a*x]]) + (2*((Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/2 + (Sqrt[Pi]*Erfi[Sq rt[ArcSinh[a*x]]])/2))/a^2))/3
3.2.9.3.1 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[Sqrt[1 + c^ 2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1) ) Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
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_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* (n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e , c^2*d] && LtQ[n, -1]
Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {-\frac {4 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, a x}{3}+\frac {2 \operatorname {arcsinh}\left (a x \right )^{2} \pi \,\operatorname {erf}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )}{3}+\frac {2 \operatorname {arcsinh}\left (a x \right )^{2} \pi \,\operatorname {erfi}\left (\sqrt {\operatorname {arcsinh}\left (a x \right )}\right )}{3}-\frac {2 \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}}{3}}{\sqrt {\pi }\, a \operatorname {arcsinh}\left (a x \right )^{2}}\) | \(81\) |
2/3*(-2*arcsinh(a*x)^(3/2)*Pi^(1/2)*a*x+arcsinh(a*x)^2*Pi*erf(arcsinh(a*x) ^(1/2))+arcsinh(a*x)^2*Pi*erfi(arcsinh(a*x)^(1/2))-arcsinh(a*x)^(1/2)*Pi^( 1/2)*(a^2*x^2+1)^(1/2))/Pi^(1/2)/a/arcsinh(a*x)^2
Exception generated. \[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {1}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {1}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]