Integrand size = 12, antiderivative size = 229 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 x^3 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {16 x^4}{15 \text {arcsinh}(a x)^{3/2}}-\frac {16 x \sqrt {1+a^2 x^2}}{5 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {128 x^3 \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {16 \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4}-\frac {4 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^4} \]
-4/5*x^2/a^2/arcsinh(a*x)^(3/2)-16/15*x^4/arcsinh(a*x)^(3/2)+16/15*erf(2*a rcsinh(a*x)^(1/2))*Pi^(1/2)/a^4+16/15*erfi(2*arcsinh(a*x)^(1/2))*Pi^(1/2)/ a^4-4/15*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-4/15*erfi(2^ (1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^4-2/5*x^3*(a^2*x^2+1)^(1/2)/a /arcsinh(a*x)^(5/2)-16/5*x*(a^2*x^2+1)^(1/2)/a^3/arcsinh(a*x)^(1/2)-128/15 *x^3*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(1/2)
Time = 0.54 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.92 \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {4 \text {arcsinh}(a x) \left (e^{-2 \text {arcsinh}(a x)} (1-4 \text {arcsinh}(a x))+e^{2 \text {arcsinh}(a x)} (1+4 \text {arcsinh}(a x))+4 \sqrt {2} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )+4 \sqrt {2} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )\right )-4 \text {arcsinh}(a x) \left (e^{-4 \text {arcsinh}(a x)} (1-8 \text {arcsinh}(a x))+e^{4 \text {arcsinh}(a x)} (1+8 \text {arcsinh}(a x))+16 (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )+16 \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )\right )+6 \sinh (2 \text {arcsinh}(a x))-3 \sinh (4 \text {arcsinh}(a x))}{60 a^4 \text {arcsinh}(a x)^{5/2}} \]
(4*ArcSinh[a*x]*((1 - 4*ArcSinh[a*x])/E^(2*ArcSinh[a*x]) + E^(2*ArcSinh[a* x])*(1 + 4*ArcSinh[a*x]) + 4*Sqrt[2]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -2*A rcSinh[a*x]] + 4*Sqrt[2]*ArcSinh[a*x]^(3/2)*Gamma[1/2, 2*ArcSinh[a*x]]) - 4*ArcSinh[a*x]*((1 - 8*ArcSinh[a*x])/E^(4*ArcSinh[a*x]) + E^(4*ArcSinh[a*x ])*(1 + 8*ArcSinh[a*x]) + 16*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -4*ArcSinh[a *x]] + 16*ArcSinh[a*x]^(3/2)*Gamma[1/2, 4*ArcSinh[a*x]]) + 6*Sinh[2*ArcSin h[a*x]] - 3*Sinh[4*ArcSinh[a*x]])/(60*a^4*ArcSinh[a*x]^(5/2))
Time = 1.08 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.39, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6194, 6233, 6193, 2009, 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.
| \(\displaystyle \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {6 \int \frac {x^2}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{5/2}}dx}{5 a}+\frac {8}{5} a \int \frac {x^4}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{5/2}}dx-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {6 \left (\frac {4 \int \frac {x}{\text {arcsinh}(a x)^{3/2}}dx}{3 a}-\frac {2 x^2}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}+\frac {8}{5} a \left (\frac {8 \int \frac {x^3}{\text {arcsinh}(a x)^{3/2}}dx}{3 a}-\frac {2 x^4}{3 a \text {arcsinh}(a x)^{3/2}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 \int \frac {\cosh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}+\frac {8}{5} a \left (\frac {8 \left (\frac {2 \int \left (\frac {\cosh (4 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}-\frac {\cosh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arcsinh}(a x)^{3/2}}\right )-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 \int \frac {\cosh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {8}{5} a \left (\frac {8 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {6 \left (-\frac {2 x^2}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {4 \left (-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \int \frac {\sin \left (2 i \text {arcsinh}(a x)+\frac {\pi }{2}\right )}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a^2}\right )}{3 a}\right )}{5 a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {8}{5} a \left (\frac {8 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {6 \left (-\frac {2 x^2}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {4 \left (-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \left (\frac {1}{2} i \int -\frac {i e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {i e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^2}\right )}{3 a}\right )}{5 a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {8}{5} a \left (\frac {8 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 \left (\frac {1}{2} \int \frac {e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)+\frac {1}{2} \int \frac {e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {8}{5} a \left (\frac {8 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 \left (\int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\int e^{2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {8}{5} a \left (\frac {8 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 \left (\int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}+\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {8}{5} a \left (\frac {8 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^2}-\frac {2 x \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^2}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {8}{5} a \left (\frac {8 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^4}-\frac {2 x^3 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x^4}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
(-2*x^3*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) + (8*a*((-2*x^4)/(3*a* ArcSinh[a*x]^(3/2)) + (8*((-2*x^3*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]] ) + (2*((Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[a*x]]])/8 - (Sqrt[Pi/2]*Erf[Sqrt[2]*S qrt[ArcSinh[a*x]]])/4 + (Sqrt[Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/8 - (Sqrt[Pi /2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/4))/a^4))/(3*a)))/5 + (6*((-2*x^2)/( 3*a*ArcSinh[a*x]^(3/2)) + (4*((-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]*Erf i[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/2))/a^2))/(3*a)))/(5*a)
3.2.12.3.1 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]
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]
\[\int \frac {x^{3}}{\operatorname {arcsinh}\left (a x \right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^{3}}{\operatorname {asinh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x^{3}}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
Exception generated. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^3}{{\mathrm {asinh}\left (a\,x\right )}^{7/2}} \,d x \]