\(\int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx\) [113]

Optimal result

Integrand size = 12, antiderivative size = 222 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 x^2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {8 x}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {4 x^3}{5 \text {arcsinh}(a x)^{3/2}}-\frac {16 \sqrt {1+a^2 x^2}}{15 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {24 x^2 \sqrt {1+a^2 x^2}}{5 a \sqrt {\text {arcsinh}(a x)}}+\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a^3}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{5 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{5 a^3} \]

[Out]

-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*erf(3^(1/2)*arcsinh(a*x)^(1/2))*3^(1/2)*Pi^(1/2)/a^3+3/5*erfi(3^(1/2)*arc
sinh(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)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5779, 5818, 5778, 3389, 2211, 2235, 2236, 5773, 5819} \[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a^3}-\frac {3 \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{5 a^3}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\text {arcsinh}(a x)}\right )}{15 a^3}+\frac {3 \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {\text {arcsinh}(a x)}\right )}{5 a^3}-\frac {24 x^2 \sqrt {a^2 x^2+1}}{5 a \sqrt {\text {arcsinh}(a x)}}-\frac {2 x^2 \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {8 x}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {16 \sqrt {a^2 x^2+1}}{15 a^3 \sqrt {\text {arcsinh}(a x)}}-\frac {4 x^3}{5 \text {arcsinh}(a x)^{3/2}} \]

[In]

Int[x^2/ArcSinh[a*x]^(7/2),x]

[Out]

(-2*x^2*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - (8*x)/(15*a^2*ArcSinh[a*x]^(3/2)) - (4*x^3)/(5*ArcSinh[a
*x]^(3/2)) - (16*Sqrt[1 + a^2*x^2])/(15*a^3*Sqrt[ArcSinh[a*x]]) - (24*x^2*Sqrt[1 + a^2*x^2])/(5*a*Sqrt[ArcSinh
[a*x]]) + (Sqrt[Pi]*Erf[Sqrt[ArcSinh[a*x]]])/(15*a^3) - (3*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(5*a^3)
- (Sqrt[Pi]*Erfi[Sqrt[ArcSinh[a*x]]])/(15*a^3) + (3*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[ArcSinh[a*x]]])/(5*a^3)

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[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]

Rule 2235

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]

Rule 2236

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] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3389

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 5773

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] - Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; F
reeQ[{a, b, c}, x] && LtQ[n, -1]

Rule 5778

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSi
nh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Si
nh[-a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}
, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5779

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSi
nh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Dist[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] - Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^
2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5818

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] - Dist[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]

Rule 5819

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {4 \int \frac {x}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (6 a) \int \frac {x^3}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{5/2}} \, 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 {12}{5} \int \frac {x^2}{\text {arcsinh}(a x)^{3/2}} \, dx+\frac {8 \int \frac {1}{\text {arcsinh}(a x)^{3/2}} \, dx}{15 a^2} \\ & = -\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 {24 \text {Subst}\left (\int \left (-\frac {\sinh (x)}{4 \sqrt {x}}+\frac {3 \sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\text {arcsinh}(a x)\right )}{5 a^3}+\frac {16 \int \frac {x}{\sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \, dx}{15 a} \\ & = -\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 {16 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^3}-\frac {6 \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^3}+\frac {18 \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^3} \\ & = -\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 {8 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^3}+\frac {8 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^3}+\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^3}-\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^3}-\frac {9 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^3}+\frac {9 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{5 a^3} \\ & = -\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 {16 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^3}+\frac {16 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^3}+\frac {6 \text {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{5 a^3}-\frac {6 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{5 a^3}-\frac {18 \text {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{5 a^3}+\frac {18 \text {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{5 a^3} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {e^{\text {arcsinh}(a x)} \left (3+2 \text {arcsinh}(a x)+4 \text {arcsinh}(a x)^2\right )-3 e^{3 \text {arcsinh}(a x)} \left (1+2 \text {arcsinh}(a x)+12 \text {arcsinh}(a x)^2\right )+36 \sqrt {3} (-\text {arcsinh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-3 \text {arcsinh}(a x)\right )-4 (-\text {arcsinh}(a x))^{5/2} \Gamma \left (\frac {1}{2},-\text {arcsinh}(a x)\right )+e^{-\text {arcsinh}(a x)} \left (3-2 \text {arcsinh}(a x)+4 \text {arcsinh}(a x)^2-4 e^{\text {arcsinh}(a x)} \text {arcsinh}(a x)^{5/2} \Gamma \left (\frac {1}{2},\text {arcsinh}(a x)\right )\right )+e^{-3 \text {arcsinh}(a x)} \left (-3+6 \text {arcsinh}(a x)-36 \text {arcsinh}(a x)^2+36 \sqrt {3} e^{3 \text {arcsinh}(a x)} \text {arcsinh}(a x)^{5/2} \Gamma \left (\frac {1}{2},3 \text {arcsinh}(a x)\right )\right )}{60 a^3 \text {arcsinh}(a x)^{5/2}} \]

[In]

Integrate[x^2/ArcSinh[a*x]^(7/2),x]

[Out]

(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*ArcSin
h[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,
-ArcSinh[a*x]] + (3 - 2*ArcSinh[a*x] + 4*ArcSinh[a*x]^2 - 4*E^ArcSinh[a*x]*ArcSinh[a*x]^(5/2)*Gamma[1/2, ArcSi
nh[a*x]])/E^ArcSinh[a*x] + (-3 + 6*ArcSinh[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))

Maple [F]

\[\int \frac {x^{2}}{\operatorname {arcsinh}\left (a x \right )^{\frac {7}{2}}}d x\]

[In]

int(x^2/arcsinh(a*x)^(7/2),x)

[Out]

int(x^2/arcsinh(a*x)^(7/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^2/arcsinh(a*x)^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^{2}}{\operatorname {asinh}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]

[In]

integrate(x**2/asinh(a*x)**(7/2),x)

[Out]

Integral(x**2/asinh(a*x)**(7/2), x)

Maxima [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate(x^2/arcsinh(a*x)^(7/2),x, algorithm="maxima")

[Out]

integrate(x^2/arcsinh(a*x)^(7/2), x)

Giac [F]

\[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\int { \frac {x^{2}}{\operatorname {arsinh}\left (a x\right )^{\frac {7}{2}}} \,d x } \]

[In]

integrate(x^2/arcsinh(a*x)^(7/2),x, algorithm="giac")

[Out]

integrate(x^2/arcsinh(a*x)^(7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\text {arcsinh}(a x)^{7/2}} \, dx=\int \frac {x^2}{{\mathrm {asinh}\left (a\,x\right )}^{7/2}} \,d x \]

[In]

int(x^2/asinh(a*x)^(7/2),x)

[Out]

int(x^2/asinh(a*x)^(7/2), x)