Integrand size = 12, antiderivative size = 285 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 x^4 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {4 x^5}{3 \text {arcsinh}(a x)^{3/2}}-\frac {32 x^2 \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {40 x^4 \sqrt {1+a^2 x^2}}{3 a \sqrt {\text {arcsinh}(a x)}}-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{30 a^5}+\frac {9 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{20 a^5}-\frac {5 \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{12 a^5}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{30 a^5}-\frac {9 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{20 a^5}+\frac {5 \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )}{12 a^5} \]
-16/15*x^3/a^2/arcsinh(a*x)^(3/2)-4/3*x^5/arcsinh(a*x)^(3/2)-1/30*erf(arcs inh(a*x)^(1/2))*Pi^(1/2)/a^5+1/30*erfi(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^5+9/ 20*erf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5-9/20*erfi(3^(1/2)* arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^5-5/12*erf(5^(1/2)*arcsinh(a*x)^(1/ 2))*5^(1/2)*Pi^(1/2)/a^5+5/12*erfi(5^(1/2)*arcsinh(a*x)^(1/2))*5^(1/2)*Pi^ (1/2)/a^5-2/5*x^4*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(5/2)-32/5*x^2*(a^2*x^2 +1)^(1/2)/a^3/arcsinh(a*x)^(1/2)-40/3*x^4*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x) ^(1/2)
Time = 0.57 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.17 \[ \int \frac {x^4}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {-2 e^{\text {arcsinh}(a x)} \left (3+2 \text {arcsinh}(a x)+4 \text {arcsinh}(a x)^2\right )+9 e^{3 \text {arcsinh}(a x)} \left (1+2 \text {arcsinh}(a x)+12 \text {arcsinh}(a x)^2\right )-e^{5 \text {arcsinh}(a x)} \left (3+10 \text {arcsinh}(a x)+100 \text {arcsinh}(a x)^2\right )+100 \sqrt {5} (-\text {arcsinh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-5 \text {arcsinh}(a x)\right )-108 \sqrt {3} (-\text {arcsinh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )+8 (-\text {arcsinh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )+e^{-\text {arcsinh}(a x)} \left (-6+4 \text {arcsinh}(a x)-8 \text {arcsinh}(a x)^2+8 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{5/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )+9 e^{-3 \text {arcsinh}(a x)} \left (1-2 \text {arcsinh}(a x)+12 \text {arcsinh}(a x)^2-12 \sqrt {3} e^{3 \text {arcsinh}(a x)} \text {arcsinh}(a x)^{5/2} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )\right )+e^{-5 \text {arcsinh}(a x)} \left (-3+10 \text {arcsinh}(a x)-100 \text {arcsinh}(a x)^2+100 \sqrt {5} e^{5 \text {arcsinh}(a x)} \text {arcsinh}(a x)^{5/2} \Gamma \left (\frac {1}{2},5 \text {arcsinh}(a x)\right )\right )}{240 a^5 \text {arcsinh}(a x)^{5/2}} \]
(-2*E^ArcSinh[a*x]*(3 + 2*ArcSinh[a*x] + 4*ArcSinh[a*x]^2) + 9*E^(3*ArcSin h[a*x])*(1 + 2*ArcSinh[a*x] + 12*ArcSinh[a*x]^2) - E^(5*ArcSinh[a*x])*(3 + 10*ArcSinh[a*x] + 100*ArcSinh[a*x]^2) + 100*Sqrt[5]*(-ArcSinh[a*x])^(5/2) *Gamma[1/2, -5*ArcSinh[a*x]] - 108*Sqrt[3]*(-ArcSinh[a*x])^(5/2)*Gamma[1/2 , -3*ArcSinh[a*x]] + 8*(-ArcSinh[a*x])^(5/2)*Gamma[1/2, -ArcSinh[a*x]] + ( -6 + 4*ArcSinh[a*x] - 8*ArcSinh[a*x]^2 + 8*E^ArcSinh[a*x]*ArcSinh[a*x]^(5/ 2)*Gamma[1/2, ArcSinh[a*x]])/E^ArcSinh[a*x] + (9*(1 - 2*ArcSinh[a*x] + 12* ArcSinh[a*x]^2 - 12*Sqrt[3]*E^(3*ArcSinh[a*x])*ArcSinh[a*x]^(5/2)*Gamma[1/ 2, 3*ArcSinh[a*x]]))/E^(3*ArcSinh[a*x]) + (-3 + 10*ArcSinh[a*x] - 100*ArcS inh[a*x]^2 + 100*Sqrt[5]*E^(5*ArcSinh[a*x])*ArcSinh[a*x]^(5/2)*Gamma[1/2, 5*ArcSinh[a*x]])/E^(5*ArcSinh[a*x]))/(240*a^5*ArcSinh[a*x]^(5/2))
Time = 0.86 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.38, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6194, 6233, 6193, 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.
| \(\displaystyle \int \frac {x^4}{\text {arcsinh}(a x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle 2 a \int \frac {x^5}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{5/2}}dx+\frac {8 \int \frac {x^3}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{5/2}}dx}{5 a}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle 2 a \left (\frac {10 \int \frac {x^4}{\text {arcsinh}(a x)^{3/2}}dx}{3 a}-\frac {2 x^5}{3 a \text {arcsinh}(a x)^{3/2}}\right )+\frac {8 \left (\frac {2 \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}}dx}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle 2 a \left (\frac {10 \left (\frac {2 \int \left (\frac {a x}{8 \sqrt {\text {arcsinh}(a x)}}-\frac {9 \sinh (3 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}+\frac {5 \sinh (5 \text {arcsinh}(a x))}{16 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^5}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^5}{3 a \text {arcsinh}(a x)^{3/2}}\right )+\frac {8 \left (\frac {2 \left (\frac {2 \int \left (\frac {3 \sinh (3 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}-\frac {a x}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 x^4 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+2 a \left (\frac {10 \left (\frac {2 \left (-\frac {1}{16} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{32} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{32} \sqrt {5 \pi } \text {erf}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{16} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {3}{32} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {5 \pi } \text {erfi}\left (\sqrt {5} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^5}-\frac {2 x^4 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^5}{3 a \text {arcsinh}(a x)^{3/2}}\right )+\frac {8 \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}\) |
(-2*x^4*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) + (8*((-2*x^3)/(3*a*Ar cSinh[a*x]^(3/2)) + (2*((-2*x^2*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) + (2*((Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/8 - (Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ ArcSinh[a*x]]])/8 - (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/8 + (Sqrt[3*Pi]*Er fi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/8))/a^3))/a))/(5*a) + 2*a*((-2*x^5)/(3*a*A rcSinh[a*x]^(3/2)) + (10*((-2*x^4*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]] ) + (2*(-1/16*(Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]]) + (3*Sqrt[3*Pi]*Erf[Sqrt[ 3]*Sqrt[ArcSinh[a*x]]])/32 - (Sqrt[5*Pi]*Erf[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/ 32 + (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/16 - (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*S qrt[ArcSinh[a*x]]])/32 + (Sqrt[5*Pi]*Erfi[Sqrt[5]*Sqrt[ArcSinh[a*x]]])/32) )/a^5))/(3*a))
3.2.11.3.1 Defintions of rubi rules used
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]
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_)*((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]
\[\int \frac {x^{4}}{\operatorname {arcsinh}\left (a x \right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {x^4}{\text {arcsinh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^4}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^{4}}{\operatorname {asinh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^4}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {x^4}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^4}{{\mathrm {asinh}\left (a\,x\right )}^{7/2}} \,d x \]