Integrand size = 12, antiderivative size = 223 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=-\frac {2 x^4 \sqrt {1+a^2 x^2}}{3 a \text {arcsinh}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\text {arcsinh}(a x)}}-\frac {20 x^5}{3 \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{24 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}-\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{8 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{24 a^5} \]
1/12*erf(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^5+1/12*erfi(arcsinh(a*x)^(1/2))*Pi ^(1/2)/a^5-3/8*erf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5-3/8*er fi(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5+5/24*erf(5^(1/2)*arcsi nh(a*x)^(1/2))*5^(1/2)*Pi^(1/2)/a^5+5/24*erfi(5^(1/2)*arcsinh(a*x)^(1/2))* 5^(1/2)*Pi^(1/2)/a^5-2/3*x^4*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(3/2)-16/3*x ^3/a^2/arcsinh(a*x)^(1/2)-20/3*x^5/arcsinh(a*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.52 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=\frac {-\frac {e^{5 \text {arcsinh}(a x)} (1+10 \text {arcsinh}(a x))+10 \sqrt {5} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-5 \text {arcsinh}(a x)\right )}{48 \text {arcsinh}(a x)^{3/2}}+\frac {e^{3 \text {arcsinh}(a x)} (1+6 \text {arcsinh}(a x))+6 \sqrt {3} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )}{16 \text {arcsinh}(a x)^{3/2}}-\frac {e^{\text {arcsinh}(a x)} (1+2 \text {arcsinh}(a x))+2 (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )}{24 \text {arcsinh}(a x)^{3/2}}-\frac {e^{-\text {arcsinh}(a x)} \left (1-2 \text {arcsinh}(a x)+2 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )}{24 \text {arcsinh}(a x)^{3/2}}+\frac {1}{16} \left (\frac {e^{-3 \text {arcsinh}(a x)}}{\text {arcsinh}(a x)^{3/2}}-\frac {6 e^{-3 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}+6 \sqrt {3} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )\right )-\frac {e^{-5 \text {arcsinh}(a x)} \left (1-10 \text {arcsinh}(a x)+10 \sqrt {5} e^{5 \text {arcsinh}(a x)} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},5 \text {arcsinh}(a x)\right )\right )}{48 \text {arcsinh}(a x)^{3/2}}}{a^5} \]
(-1/48*(E^(5*ArcSinh[a*x])*(1 + 10*ArcSinh[a*x]) + 10*Sqrt[5]*(-ArcSinh[a* x])^(3/2)*Gamma[1/2, -5*ArcSinh[a*x]])/ArcSinh[a*x]^(3/2) + (E^(3*ArcSinh[ a*x])*(1 + 6*ArcSinh[a*x]) + 6*Sqrt[3]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -3 *ArcSinh[a*x]])/(16*ArcSinh[a*x]^(3/2)) - (E^ArcSinh[a*x]*(1 + 2*ArcSinh[a *x]) + 2*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -ArcSinh[a*x]])/(24*ArcSinh[a*x] ^(3/2)) - (1 - 2*ArcSinh[a*x] + 2*E^ArcSinh[a*x]*ArcSinh[a*x]^(3/2)*Gamma[ 1/2, ArcSinh[a*x]])/(24*E^ArcSinh[a*x]*ArcSinh[a*x]^(3/2)) + (1/(E^(3*ArcS inh[a*x])*ArcSinh[a*x]^(3/2)) - 6/(E^(3*ArcSinh[a*x])*Sqrt[ArcSinh[a*x]]) + 6*Sqrt[3]*Gamma[1/2, 3*ArcSinh[a*x]])/16 - (1 - 10*ArcSinh[a*x] + 10*Sqr t[5]*E^(5*ArcSinh[a*x])*ArcSinh[a*x]^(3/2)*Gamma[1/2, 5*ArcSinh[a*x]])/(48 *E^(5*ArcSinh[a*x])*ArcSinh[a*x]^(3/2)))/a^5
Time = 0.94 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.46, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6194, 6233, 6195, 5971, 2009}
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.
| \(\displaystyle \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {10}{3} a \int \frac {x^5}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}dx+\frac {8 \int \frac {x^3}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{3/2}}dx}{3 a}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {10}{3} a \left (\frac {10 \int \frac {x^4}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^5}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {8 \left (\frac {6 \int \frac {x^2}{\sqrt {\text {arcsinh}(a x)}}dx}{a}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {8 \left (\frac {6 \int \frac {a^2 x^2 \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}+\frac {10}{3} a \left (\frac {10 \int \frac {a^4 x^4 \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^6}-\frac {2 x^5}{a \sqrt {\text {arcsinh}(a x)}}\right )-\frac {2 x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {10}{3} a \left (\frac {10 \int \left (-\frac {3 \cosh (3 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}+\frac {\cosh (5 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {a^2 x^2+1}}{8 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^6}-\frac {2 x^5}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {8 \left (\frac {6 \int \left (\frac {\cosh (3 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {a^2 x^2+1}}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^4}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {10}{3} a \left (\frac {10 \left (\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^6}-\frac {2 x^5}{a \sqrt {\text {arcsinh}(a x)}}\right )+\frac {8 \left (\frac {6 \left (-\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^4}-\frac {2 x^3}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{3 a \text {arcsinh}(a x)^{3/2}}\) |
(-2*x^4*Sqrt[1 + a^2*x^2])/(3*a*ArcSinh[a*x]^(3/2)) + (8*((-2*x^3)/(a*Sqrt [ArcSinh[a*x]]) + (6*(-1/8*(Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]]) + (Sqrt[Pi/3 ]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/8 - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]]) /8 + (Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/8))/a^4))/(3*a) + (10*a *((-2*x^5)/(a*Sqrt[ArcSinh[a*x]]) + (10*((Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]] )/16 - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/32 + (Sqrt[Pi/5]*Erf[S qrt[5]*Sqrt[ArcSinh[a*x]]])/32 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/16 - (Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/32 + (Sqrt[Pi/5]*Erfi[Sqrt[5 ]*Sqrt[ArcSinh[a*x]]])/32))/a^6))/3
3.2.5.3.1 Defintions of rubi rules used
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[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (- Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n + 1)/ Sqrt[1 + c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
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[(((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]
\[\int \frac {x^{4}}{\operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x^{4}}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\text {arcsinh}(a x)^{5/2}} \, dx=\int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \]