3.131 \(\int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx\)

Optimal. Leaf size=731 \[ \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}+\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}+\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )+\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right ) \]

[Out]

1/8*a^2*(1-1/a/x)^(7/8)*(1+1/a/x)^(1/8)+1/2*a^2*(1-1/a/x)^(7/8)*(1+1/a/x)^(9/8)-1/32*a^2*arctan((-2*(1-1/a/x)^
(1/8)/(1+1/a/x)^(1/8)+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)+1/32*a^2*arctan((2*(1-1/a/x)^(1/
8)/(1+1/a/x)^(1/8)+(2+2^(1/2))^(1/2))/(2-2^(1/2))^(1/2))*(2-2^(1/2))^(1/2)+1/64*a^2*ln(1+(1-1/a/x)^(1/4)/(1+1/
a/x)^(1/4)-(1-1/a/x)^(1/8)*(2-2^(1/2))^(1/2)/(1+1/a/x)^(1/8))*(2-2^(1/2))^(1/2)-1/64*a^2*ln(1+(1-1/a/x)^(1/4)/
(1+1/a/x)^(1/4)+(1-1/a/x)^(1/8)*(2-2^(1/2))^(1/2)/(1+1/a/x)^(1/8))*(2-2^(1/2))^(1/2)-1/32*a^2*arctan((-2*(1-1/
a/x)^(1/8)/(1+1/a/x)^(1/8)+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)+1/32*a^2*arctan((2*(1-1/a/x
)^(1/8)/(1+1/a/x)^(1/8)+(2-2^(1/2))^(1/2))/(2+2^(1/2))^(1/2))*(2+2^(1/2))^(1/2)+1/64*a^2*ln(1+(1-1/a/x)^(1/4)/
(1+1/a/x)^(1/4)-(1-1/a/x)^(1/8)*(2+2^(1/2))^(1/2)/(1+1/a/x)^(1/8))*(2+2^(1/2))^(1/2)-1/64*a^2*ln(1+(1-1/a/x)^(
1/4)/(1+1/a/x)^(1/4)+(1-1/a/x)^(1/8)*(2+2^(1/2))^(1/2)/(1+1/a/x)^(1/8))*(2+2^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.66, antiderivative size = 731, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6171, 80, 50, 63, 331, 299, 1122, 1169, 634, 618, 204, 628} \[ \frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (\frac {1}{a x}+1\right )^{9/8}+\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{\frac {1}{a x}+1}+\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )+\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+1\right )-\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{\frac {1}{a x}+1}}+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^(ArcCoth[a*x]/4)/x^3,x]

[Out]

(a^2*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8))/8 + (a^2*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(9/8))/2 - (Sqrt[2 +
Sqrt[2]]*a^2*ArcTan[(Sqrt[2 - Sqrt[2]] - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 + Sqrt[2]]])/32 -
(Sqrt[2 - Sqrt[2]]*a^2*ArcTan[(Sqrt[2 + Sqrt[2]] - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sqrt[2 - Sqrt
[2]]])/32 + (Sqrt[2 + Sqrt[2]]*a^2*ArcTan[(Sqrt[2 - Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/Sq
rt[2 + Sqrt[2]]])/32 + (Sqrt[2 - Sqrt[2]]*a^2*ArcTan[(Sqrt[2 + Sqrt[2]] + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x)
)^(1/8))/Sqrt[2 - Sqrt[2]]])/32 + (Sqrt[2 - Sqrt[2]]*a^2*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sq
rt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)])/64 - (Sqrt[2 - Sqrt[2]]*a^2*Log[1 + (1 - 1/(a*x))^(
1/4)/(1 + 1/(a*x))^(1/4) + (Sqrt[2 - Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)])/64 + (Sqrt[2 + Sqrt[2
]]*a^2*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (Sqrt[2 + Sqrt[2]]*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))
^(1/8)])/64 - (Sqrt[2 + Sqrt[2]]*a^2*Log[1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (Sqrt[2 + Sqrt[2]]*(1 -
1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)])/64

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 299

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[a/b, 4]], s = Denominator[Rt[a/b,
4]]}, Dist[s^3/(2*Sqrt[2]*b*r), Int[x^(m - n/4)/(r^2 - Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x] - Dist[s^3/
(2*Sqrt[2]*b*r), Int[x^(m - n/4)/(r^2 + Sqrt[2]*r*s*x^(n/4) + s^2*x^(n/2)), x], x]] /; FreeQ[{a, b}, x] && IGt
Q[n/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && GtQ[a/b, 0]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d^3*(d*x)^(m - 3)*(a + b*
x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 6171

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2
)), x], x, 1/x] /; FreeQ[{a, n}, x] &&  !IntegerQ[n] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {e^{\frac {1}{4} \coth ^{-1}(a x)}}{x^3} \, dx &=-\operatorname {Subst}\left (\int \frac {x \sqrt [8]{1+\frac {x}{a}}}{\sqrt [8]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{8} a \operatorname {Subst}\left (\int \frac {\sqrt [8]{1+\frac {x}{a}}}{\sqrt [8]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{32} a \operatorname {Subst}\left (\int \frac {1}{\sqrt [8]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/8}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-\frac {1}{a x}}\right )\\ &=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {x^6}{1+x^8} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )\\ &=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {a^2 \operatorname {Subst}\left (\int \frac {x^4}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}-\frac {a^2 \operatorname {Subst}\left (\int \frac {x^4}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}\\ &=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{8 \sqrt {2}}\\ &=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )}{16 \sqrt {2 \left (2+\sqrt {2}\right )}}\\ &=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {1}{32} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{32} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{64} \left (\sqrt {2-\sqrt {2}} a^2\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{64} \left (\sqrt {2-\sqrt {2}} a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{64} \left (\sqrt {2+\sqrt {2}} a^2\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{64} \left (\sqrt {2+\sqrt {2}} a^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{32} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{32} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )\\ &=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}+\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{16} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{16} \left (\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{16} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{16} \left (\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )\\ &=\frac {1}{8} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \sqrt [8]{1+\frac {1}{a x}}+\frac {1}{2} a^2 \left (1-\frac {1}{a x}\right )^{7/8} \left (1+\frac {1}{a x}\right )^{9/8}-\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )-\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2+\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2+\sqrt {2}}}\right )+\frac {1}{32} \sqrt {2-\sqrt {2}} a^2 \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}}{\sqrt {2-\sqrt {2}}}\right )+\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{64} \sqrt {2-\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )+\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )-\frac {1}{64} \sqrt {2+\sqrt {2}} a^2 \log \left (1+\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-\frac {1}{a x}}}{\sqrt [8]{1+\frac {1}{a x}}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 72, normalized size = 0.10 \[ -\frac {a^2 e^{\frac {1}{4} \coth ^{-1}(a x)} \left (\left (e^{2 \coth ^{-1}(a x)}+1\right )^2 \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-e^{2 \coth ^{-1}(a x)}\right )-9 e^{2 \coth ^{-1}(a x)}-1\right )}{4 \left (e^{2 \coth ^{-1}(a x)}+1\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(ArcCoth[a*x]/4)/x^3,x]

[Out]

-1/4*(a^2*E^(ArcCoth[a*x]/4)*(-1 - 9*E^(2*ArcCoth[a*x]) + (1 + E^(2*ArcCoth[a*x]))^2*Hypergeometric2F1[1/8, 1,
9/8, -E^(2*ArcCoth[a*x])]))/(1 + E^(2*ArcCoth[a*x]))^2

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fricas [B]  time = 0.94, size = 2931, normalized size = 4.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x, algorithm="fricas")

[Out]

-1/256*(8*(a^16)^(1/8)*x^2*sqrt(-sqrt(2) + 2)*arctan(-(2*(a^16)^(1/8)*a^14*((a*x - 1)/(a*x + 1))^(1/8) + (a^16
*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) + 2) - 2*sqrt(a^28*((a*x - 1)/(a*x + 1))^(1/4) + (a^16)^(3/4)*a^16 + (a^1
6)^(7/8)*(a^14*(sqrt(2) + 2) - a^14)*sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8))*(a^16)^(1/8))/(a^16*(sqrt
(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2) + 2))) + 8*(a^16)^(1/8)*x^2*sqrt(-sqrt(2) + 2)*arctan(-(2*(a^16)^(1/8)*a^
14*((a*x - 1)/(a*x + 1))^(1/8) - (a^16*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) + 2) - 2*sqrt(a^28*((a*x - 1)/(a*x
+ 1))^(1/4) + (a^16)^(3/4)*a^16 - (a^16)^(7/8)*(a^14*(sqrt(2) + 2) - a^14)*sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x
+ 1))^(1/8))*(a^16)^(1/8))/(a^16*(sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2) + 2))) + 8*(a^16)^(1/8)*x^2*sqrt(sq
rt(2) + 2)*arctan(-(a^16*(sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2) + 2) + 2*(a^16)^(1/8)*a^14*((a*x - 1)/(a*x
+ 1))^(1/8) - 2*sqrt(a^28*((a*x - 1)/(a*x + 1))^(1/4) + (a^16)^(3/4)*a^16 + (a^16)^(7/8)*(a^14*(sqrt(2) + 2)^(
3/2) - 3*a^14*sqrt(sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8))*(a^16)^(1/8))/((a^16*(sqrt(2) + 2) - a^16)*sqrt(
-sqrt(2) + 2))) + 8*(a^16)^(1/8)*x^2*sqrt(sqrt(2) + 2)*arctan((a^16*(sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2)
+ 2) - 2*(a^16)^(1/8)*a^14*((a*x - 1)/(a*x + 1))^(1/8) + 2*sqrt(a^28*((a*x - 1)/(a*x + 1))^(1/4) + (a^16)^(3/4
)*a^16 - (a^16)^(7/8)*(a^14*(sqrt(2) + 2)^(3/2) - 3*a^14*sqrt(sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8))*(a^16
)^(1/8))/((a^16*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) + 2))) + 2*(a^16)^(1/8)*x^2*sqrt(sqrt(2) + 2)*log(a^28*((a
*x - 1)/(a*x + 1))^(1/4) + (a^16)^(3/4)*a^16 + (a^16)^(7/8)*(a^14*(sqrt(2) + 2) - a^14)*sqrt(-sqrt(2) + 2)*((a
*x - 1)/(a*x + 1))^(1/8)) - 2*(a^16)^(1/8)*x^2*sqrt(sqrt(2) + 2)*log(a^28*((a*x - 1)/(a*x + 1))^(1/4) + (a^16)
^(3/4)*a^16 - (a^16)^(7/8)*(a^14*(sqrt(2) + 2) - a^14)*sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8)) + 2*(a^
16)^(1/8)*x^2*sqrt(-sqrt(2) + 2)*log(a^28*((a*x - 1)/(a*x + 1))^(1/4) + (a^16)^(3/4)*a^16 + (a^16)^(7/8)*(a^14
*(sqrt(2) + 2)^(3/2) - 3*a^14*sqrt(sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8)) - 2*(a^16)^(1/8)*x^2*sqrt(-sqrt(
2) + 2)*log(a^28*((a*x - 1)/(a*x + 1))^(1/4) + (a^16)^(3/4)*a^16 - (a^16)^(7/8)*(a^14*(sqrt(2) + 2)^(3/2) - 3*
a^14*sqrt(sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8)) + 4*(a^16)^(1/8)*(sqrt(2)*x^2*sqrt(sqrt(2) + 2) + sqrt(2)
*x^2*sqrt(-sqrt(2) + 2))*arctan(-(a^16*(sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2) + 2) + 2*sqrt(2)*(a^16)^(1/8)
*a^14*((a*x - 1)/(a*x + 1))^(1/8) - (a^16*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) + 2) - sqrt(2)*sqrt(4*a^28*((a*x
- 1)/(a*x + 1))^(1/4) + 4*(a^16)^(3/4)*a^16 + 2*(a^16)^(7/8)*(sqrt(2)*a^14*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^
14*sqrt(sqrt(2) + 2) - (sqrt(2)*a^14*(sqrt(2) + 2) - sqrt(2)*a^14)*sqrt(-sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(
1/8))*(a^16)^(1/8))/(a^16*(sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2) + 2) + (a^16*(sqrt(2) + 2) - a^16)*sqrt(-s
qrt(2) + 2))) + 4*(a^16)^(1/8)*(sqrt(2)*x^2*sqrt(sqrt(2) + 2) + sqrt(2)*x^2*sqrt(-sqrt(2) + 2))*arctan((a^16*(
sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2) + 2) - 2*sqrt(2)*(a^16)^(1/8)*a^14*((a*x - 1)/(a*x + 1))^(1/8) - (a^1
6*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) + 2) + sqrt(2)*sqrt(4*a^28*((a*x - 1)/(a*x + 1))^(1/4) + 4*(a^16)^(3/4)*
a^16 - 2*(a^16)^(7/8)*(sqrt(2)*a^14*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^14*sqrt(sqrt(2) + 2) - (sqrt(2)*a^14*(sq
rt(2) + 2) - sqrt(2)*a^14)*sqrt(-sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8))*(a^16)^(1/8))/(a^16*(sqrt(2) + 2)^
(3/2) - 3*a^16*sqrt(sqrt(2) + 2) + (a^16*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) + 2))) + 4*(a^16)^(1/8)*(sqrt(2)*
x^2*sqrt(sqrt(2) + 2) - sqrt(2)*x^2*sqrt(-sqrt(2) + 2))*arctan((a^16*(sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2)
+ 2) + 2*sqrt(2)*(a^16)^(1/8)*a^14*((a*x - 1)/(a*x + 1))^(1/8) + (a^16*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) +
2) - sqrt(2)*sqrt(4*a^28*((a*x - 1)/(a*x + 1))^(1/4) + 4*(a^16)^(3/4)*a^16 + 2*(a^16)^(7/8)*(sqrt(2)*a^14*(sqr
t(2) + 2)^(3/2) - 3*sqrt(2)*a^14*sqrt(sqrt(2) + 2) + (sqrt(2)*a^14*(sqrt(2) + 2) - sqrt(2)*a^14)*sqrt(-sqrt(2)
+ 2))*((a*x - 1)/(a*x + 1))^(1/8))*(a^16)^(1/8))/(a^16*(sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2) + 2) - (a^16
*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) + 2))) + 4*(a^16)^(1/8)*(sqrt(2)*x^2*sqrt(sqrt(2) + 2) - sqrt(2)*x^2*sqrt
(-sqrt(2) + 2))*arctan(-(a^16*(sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2) + 2) - 2*sqrt(2)*(a^16)^(1/8)*a^14*((a
*x - 1)/(a*x + 1))^(1/8) + (a^16*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) + 2) + sqrt(2)*sqrt(4*a^28*((a*x - 1)/(a*
x + 1))^(1/4) + 4*(a^16)^(3/4)*a^16 - 2*(a^16)^(7/8)*(sqrt(2)*a^14*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^14*sqrt(s
qrt(2) + 2) + (sqrt(2)*a^14*(sqrt(2) + 2) - sqrt(2)*a^14)*sqrt(-sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8))*(a^
16)^(1/8))/(a^16*(sqrt(2) + 2)^(3/2) - 3*a^16*sqrt(sqrt(2) + 2) - (a^16*(sqrt(2) + 2) - a^16)*sqrt(-sqrt(2) +
2))) + (a^16)^(1/8)*(sqrt(2)*x^2*sqrt(sqrt(2) + 2) + sqrt(2)*x^2*sqrt(-sqrt(2) + 2))*log(4*a^28*((a*x - 1)/(a*
x + 1))^(1/4) + 4*(a^16)^(3/4)*a^16 + 2*(a^16)^(7/8)*(sqrt(2)*a^14*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^14*sqrt(s
qrt(2) + 2) + (sqrt(2)*a^14*(sqrt(2) + 2) - sqrt(2)*a^14)*sqrt(-sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8)) - (
a^16)^(1/8)*(sqrt(2)*x^2*sqrt(sqrt(2) + 2) + sqrt(2)*x^2*sqrt(-sqrt(2) + 2))*log(4*a^28*((a*x - 1)/(a*x + 1))^
(1/4) + 4*(a^16)^(3/4)*a^16 - 2*(a^16)^(7/8)*(sqrt(2)*a^14*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^14*sqrt(sqrt(2) +
2) + (sqrt(2)*a^14*(sqrt(2) + 2) - sqrt(2)*a^14)*sqrt(-sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8)) - (a^16)^(1
/8)*(sqrt(2)*x^2*sqrt(sqrt(2) + 2) - sqrt(2)*x^2*sqrt(-sqrt(2) + 2))*log(4*a^28*((a*x - 1)/(a*x + 1))^(1/4) +
4*(a^16)^(3/4)*a^16 + 2*(a^16)^(7/8)*(sqrt(2)*a^14*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^14*sqrt(sqrt(2) + 2) - (s
qrt(2)*a^14*(sqrt(2) + 2) - sqrt(2)*a^14)*sqrt(-sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8)) + (a^16)^(1/8)*(sqr
t(2)*x^2*sqrt(sqrt(2) + 2) - sqrt(2)*x^2*sqrt(-sqrt(2) + 2))*log(4*a^28*((a*x - 1)/(a*x + 1))^(1/4) + 4*(a^16)
^(3/4)*a^16 - 2*(a^16)^(7/8)*(sqrt(2)*a^14*(sqrt(2) + 2)^(3/2) - 3*sqrt(2)*a^14*sqrt(sqrt(2) + 2) - (sqrt(2)*a
^14*(sqrt(2) + 2) - sqrt(2)*a^14)*sqrt(-sqrt(2) + 2))*((a*x - 1)/(a*x + 1))^(1/8)) - 32*(5*a^2*x^2 + 9*a*x + 4
)*((a*x - 1)/(a*x + 1))^(7/8))/x^2

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giac [A]  time = 0.47, size = 473, normalized size = 0.65 \[ \frac {1}{64} \, {\left (2 \, a \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, a \sqrt {-\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2}}\right ) + 2 \, a \sqrt {\sqrt {2} + 2} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) + 2 \, a \sqrt {\sqrt {2} + 2} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2}}\right ) - a \sqrt {\sqrt {2} + 2} \log \left (\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + a \sqrt {\sqrt {2} + 2} \log \left (-\sqrt {\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - a \sqrt {-\sqrt {2} + 2} \log \left (\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + a \sqrt {-\sqrt {2} + 2} \log \left (-\sqrt {-\sqrt {2} + 2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}} + \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + \frac {16 \, {\left (\frac {{\left (a x - 1\right )} a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{a x + 1} + 9 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}\right )}}{{\left (\frac {a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x, algorithm="giac")

[Out]

1/64*(2*a*sqrt(-sqrt(2) + 2)*arctan((sqrt(sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(-sqrt(2) + 2)) +
2*a*sqrt(-sqrt(2) + 2)*arctan(-(sqrt(sqrt(2) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(-sqrt(2) + 2)) + 2*a*s
qrt(sqrt(2) + 2)*arctan((sqrt(-sqrt(2) + 2) + 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(sqrt(2) + 2)) + 2*a*sqrt(sqr
t(2) + 2)*arctan(-(sqrt(-sqrt(2) + 2) - 2*((a*x - 1)/(a*x + 1))^(1/8))/sqrt(sqrt(2) + 2)) - a*sqrt(sqrt(2) + 2
)*log(sqrt(sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) + a*sqrt(sqrt(2) + 2)*l
og(-sqrt(sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) - a*sqrt(-sqrt(2) + 2)*lo
g(sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) + a*sqrt(-sqrt(2) + 2)*log
(-sqrt(-sqrt(2) + 2)*((a*x - 1)/(a*x + 1))^(1/8) + ((a*x - 1)/(a*x + 1))^(1/4) + 1) + 16*((a*x - 1)*a*((a*x -
1)/(a*x + 1))^(7/8)/(a*x + 1) + 9*a*((a*x - 1)/(a*x + 1))^(7/8))/((a*x - 1)/(a*x + 1) + 1)^2)*a

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}} x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x)

[Out]

int(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/8)/x^3,x, algorithm="maxima")

[Out]

integrate(1/(x^3*((a*x - 1)/(a*x + 1))^(1/8)), x)

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mupad [B]  time = 1.26, size = 210, normalized size = 0.29 \[ \frac {\frac {9\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/8}}{4}+\frac {a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/8}}{4}}{\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2\,\left (a\,x-1\right )}{a\,x+1}+1}+\frac {{\left (-1\right )}^{1/8}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\right )}{16}+\frac {{\left (-1\right )}^{1/8}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16}+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{32}-\frac {1}{32}{}\mathrm {i}\right )+{\left (-1\right )}^{1/8}\,\sqrt {2}\,a^2\,\mathrm {atan}\left ({\left (-1\right )}^{1/8}\,\sqrt {2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{32}+\frac {1}{32}{}\mathrm {i}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*((a*x - 1)/(a*x + 1))^(1/8)),x)

[Out]

((9*a^2*((a*x - 1)/(a*x + 1))^(7/8))/4 + (a^2*((a*x - 1)/(a*x + 1))^(15/8))/4)/((a*x - 1)^2/(a*x + 1)^2 + (2*(
a*x - 1))/(a*x + 1) + 1) + ((-1)^(1/8)*a^2*atan((-1)^(1/8)*((a*x - 1)/(a*x + 1))^(1/8)))/16 + ((-1)^(1/8)*a^2*
atan((-1)^(1/8)*((a*x - 1)/(a*x + 1))^(1/8)*1i)*1i)/16 + (-1)^(1/8)*2^(1/2)*a^2*atan((-1)^(1/8)*2^(1/2)*((a*x
- 1)/(a*x + 1))^(1/8)*(1/2 - 1i/2))*(1/32 - 1i/32) + (-1)^(1/8)*2^(1/2)*a^2*atan((-1)^(1/8)*2^(1/2)*((a*x - 1)
/(a*x + 1))^(1/8)*(1/2 + 1i/2))*(1/32 + 1i/32)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \sqrt [8]{\frac {a x - 1}{a x + 1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/8)/x**3,x)

[Out]

Integral(1/(x**3*((a*x - 1)/(a*x + 1))**(1/8)), x)

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