Integrand size = 12, antiderivative size = 222 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {8 x}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {4 x^3}{5 \text {arcsinh}(a x)^{3/2}}-\frac {16 \sqrt {1+a^2 x^2}}{15 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {24 x^2 \sqrt {1+a^2 x^2}}{5 a \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a^3}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{5 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{5 a^3} \]
-8/15*x/a^2/arcsinh(a*x)^(3/2)-4/5*x^3/arcsinh(a*x)^(3/2)+1/15*erf(arcsinh (a*x)^(1/2))*Pi^(1/2)/a^3-1/15*erfi(arcsinh(a*x)^(1/2))*Pi^(1/2)/a^3-3/5*e rf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3+3/5*erfi(3^(1/2)*arcsi nh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3-2/5*x^2*(a^2*x^2+1)^(1/2)/a/arcsinh(a* x)^(5/2)-16/15*(a^2*x^2+1)^(1/2)/a^3/arcsinh(a*x)^(1/2)-24/5*x^2*(a^2*x^2+ 1)^(1/2)/a/arcsinh(a*x)^(1/2)
Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {e^{\text {arcsinh}(a x)} \left (3+2 \text {arcsinh}(a x)+4 \text {arcsinh}(a x)^2\right )-3 e^{3 \text {arcsinh}(a x)} \left (1+2 \text {arcsinh}(a x)+12 \text {arcsinh}(a x)^2\right )+36 \sqrt {3} (-\text {arcsinh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )-4 (-\text {arcsinh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )+e^{-\text {arcsinh}(a x)} \left (3-2 \text {arcsinh}(a x)+4 \text {arcsinh}(a x)^2-4 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{5/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )+e^{-3 \text {arcsinh}(a x)} \left (-3+6 \text {arcsinh}(a x)-36 \text {arcsinh}(a x)^2+36 \sqrt {3} e^{3 \text {arcsinh}(a x)} \text {arcsinh}(a x)^{5/2} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )\right )}{60 a^3 \text {arcsinh}(a x)^{5/2}} \]
(E^ArcSinh[a*x]*(3 + 2*ArcSinh[a*x] + 4*ArcSinh[a*x]^2) - 3*E^(3*ArcSinh[a *x])*(1 + 2*ArcSinh[a*x] + 12*ArcSinh[a*x]^2) + 36*Sqrt[3]*(-ArcSinh[a*x]) ^(5/2)*Gamma[1/2, -3*ArcSinh[a*x]] - 4*(-ArcSinh[a*x])^(5/2)*Gamma[1/2, -A rcSinh[a*x]] + (3 - 2*ArcSinh[a*x] + 4*ArcSinh[a*x]^2 - 4*E^ArcSinh[a*x]*A rcSinh[a*x]^(5/2)*Gamma[1/2, ArcSinh[a*x]])/E^ArcSinh[a*x] + (-3 + 6*ArcSi nh[a*x] - 36*ArcSinh[a*x]^2 + 36*Sqrt[3]*E^(3*ArcSinh[a*x])*ArcSinh[a*x]^( 5/2)*Gamma[1/2, 3*ArcSinh[a*x]])/E^(3*ArcSinh[a*x]))/(60*a^3*ArcSinh[a*x]^ (5/2))
Result contains complex when optimal does not.
Time = 1.36 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.31, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6194, 6233, 6188, 6193, 2009, 6234, 3042, 26, 3789, 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.
| \(\displaystyle \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {4 \int \frac {x}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{5/2}}dx}{5 a}+\frac {6}{5} a \int \frac {x^3}{\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^{5/2}}dx-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {6}{5} a \left (\frac {2 \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}}dx}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )+\frac {4 \left (\frac {2 \int \frac {1}{\text {arcsinh}(a x)^{3/2}}dx}{3 a}-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6188 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (2 a \int \frac {x}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}+\frac {6}{5} a \left (\frac {2 \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}}dx}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (2 a \int \frac {x}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}+\frac {6}{5} a \left (\frac {2 \left (\frac {2 \int \left (\frac {3 \sinh (3 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}-\frac {a x}{4 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (2 a \int \frac {x}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (\frac {2 \int \frac {a x}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a}-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{3 a}-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}\right )}{5 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}+\frac {2 \int -\frac {i \sin (i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a}\right )}{3 a}\right )}{5 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \int \frac {\sin (i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a}\right )}{3 a}\right )}{5 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \int \frac {e^{\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-\text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{a}\right )}{3 a}\right )}{5 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (i \int e^{\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a}\right )}{3 a}\right )}{5 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-\text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{a}\right )}{3 a}\right )}{5 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 a \text {arcsinh}(a x)^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}-\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )\right )}{a}\right )}{3 a}\right )}{5 a}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {6}{5} a \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} \sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{8} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )\right )}{a^3}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)}}\right )}{a}-\frac {2 x^3}{3 a \text {arcsinh}(a x)^{3/2}}\right )\) |
(-2*x^2*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) + (4*((-2*x)/(3*a*ArcS inh[a*x]^(3/2)) + (2*((-2*Sqrt[1 + a^2*x^2])/(a*Sqrt[ArcSinh[a*x]]) - ((2* I)*((-1/2*I)*Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]] + (I/2)*Sqrt[Pi]*Erfi[Sqrt[A rcSinh[a*x]]]))/a))/(3*a)))/(5*a) + (6*a*((-2*x^3)/(3*a*ArcSinh[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]*Erfi[Sqrt[3]*Sqrt[ ArcSinh[a*x]]])/8))/a^3))/a))/5
3.2.13.3.1 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_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
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_)^(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[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \frac {x^{2}}{\operatorname {arcsinh}\left (a x \right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^{7/2}} \,d x \]