\(\int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx\) [114]

Optimal result

Integrand size = 10, antiderivative size = 147 \[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2} \]

[Out]

-4/15/a^2/arcsinh(a*x)^(3/2)-8/15*x^2/arcsinh(a*x)^(3/2)+8/15*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)
/a^2+8/15*erfi(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a^2-2/5*x*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(5/2)-3
2/15*x*(a^2*x^2+1)^(1/2)/a/arcsinh(a*x)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5779, 5818, 5778, 3388, 2211, 2235, 2236, 5783} \[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=\frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}-\frac {32 x \sqrt {a^2 x^2+1}}{15 a \sqrt {\text {arcsinh}(a x)}}-\frac {2 x \sqrt {a^2 x^2+1}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}} \]

[In]

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

[Out]

(-2*x*Sqrt[1 + a^2*x^2])/(5*a*ArcSinh[a*x]^(5/2)) - 4/(15*a^2*ArcSinh[a*x]^(3/2)) - (8*x^2)/(15*ArcSinh[a*x]^(
3/2)) - (32*x*Sqrt[1 + a^2*x^2])/(15*a*Sqrt[ArcSinh[a*x]]) + (8*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(1
5*a^2) + (8*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/(15*a^2)

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 3388

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

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 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}+\frac {2 \int \frac {1}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (4 a) \int \frac {x^2}{\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^{5/2}} \, dx \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}+\frac {16}{15} \int \frac {x}{\text {arcsinh}(a x)^{3/2}} \, dx \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {32 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {16 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^2}+\frac {16 \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\text {arcsinh}(a x)\right )}{15 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {32 \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}+\frac {32 \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\text {arcsinh}(a x)}\right )}{15 a^2} \\ & = -\frac {2 x \sqrt {1+a^2 x^2}}{5 a \text {arcsinh}(a x)^{5/2}}-\frac {4}{15 a^2 \text {arcsinh}(a x)^{3/2}}-\frac {8 x^2}{15 \text {arcsinh}(a x)^{3/2}}-\frac {32 x \sqrt {1+a^2 x^2}}{15 a \sqrt {\text {arcsinh}(a x)}}+\frac {8 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2}+\frac {8 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{15 a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\text {arcsinh}(a x)^{7/2}} \, dx=-\frac {2 \text {arcsinh}(a x) \left (e^{-2 \text {arcsinh}(a x)} (1-4 \text {arcsinh}(a x))+e^{2 \text {arcsinh}(a x)} (1+4 \text {arcsinh}(a x))+4 \sqrt {2} (-\text {arcsinh}(a x))^{3/2} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )+4 \sqrt {2} \text {arcsinh}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )\right )+3 \sinh (2 \text {arcsinh}(a x))}{15 a^2 \text {arcsinh}(a x)^{5/2}} \]

[In]

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

[Out]

-1/15*(2*ArcSinh[a*x]*((1 - 4*ArcSinh[a*x])/E^(2*ArcSinh[a*x]) + E^(2*ArcSinh[a*x])*(1 + 4*ArcSinh[a*x]) + 4*S
qrt[2]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -2*ArcSinh[a*x]] + 4*Sqrt[2]*ArcSinh[a*x]^(3/2)*Gamma[1/2, 2*ArcSinh[a
*x]]) + 3*Sinh[2*ArcSinh[a*x]])/(a^2*ArcSinh[a*x]^(5/2))

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00

method	result	size
default	\(-\frac {\sqrt {2}\, \left (16 \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}} \sqrt {2}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x +4 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, a^{2} x^{2}+3 \sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\, \sqrt {\pi }\, \sqrt {a^{2} x^{2}+1}\, a x -8 \operatorname {arcsinh}\left (a x \right )^{3} \pi \,\operatorname {erf}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )-8 \operatorname {arcsinh}\left (a x \right )^{3} \pi \,\operatorname {erfi}\left (\sqrt {2}\, \sqrt {\operatorname {arcsinh}\left (a x \right )}\right )+2 \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\right )}{15 \sqrt {\pi }\, a^{2} \operatorname {arcsinh}\left (a x \right )^{3}}\)	\(147\)

[In]

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

[Out]

-1/15*2^(1/2)*(16*arcsinh(a*x)^(5/2)*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*a*x+4*arcsinh(a*x)^(3/2)*2^(1/2)*Pi^(1
/2)*a^2*x^2+3*2^(1/2)*arcsinh(a*x)^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*a*x-8*arcsinh(a*x)^3*Pi*erf(2^(1/2)*arcsin
h(a*x)^(1/2))-8*arcsinh(a*x)^3*Pi*erfi(2^(1/2)*arcsinh(a*x)^(1/2))+2*arcsinh(a*x)^(3/2)*2^(1/2)*Pi^(1/2))/Pi^(
1/2)/a^2/arcsinh(a*x)^3

Fricas [F(-2)]

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

[In]

integrate(x/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]

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

[In]

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

[Out]

Integral(x/asinh(a*x)**(7/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(x/arcsinh(a*x)^(7/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(x/arcsinh(a*x)^(7/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x/asinh(a*x)^(7/2), x)