3.65 \(\int \frac {e^{\frac {1}{2} \coth ^{-1}(a x)}}{x^2} \, dx\)

Optimal. Leaf size=267 \[ a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}+\frac {a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}-\frac {a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{\sqrt {2}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{\sqrt {2}} \]

[Out]

a*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)+1/2*a*arctan(-1+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4))*2^(1/2)+1/2*a*arcta
n(1+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4))*2^(1/2)+1/4*a*ln(1-(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4)+(1-1/a
/x)^(1/2)/(1+1/a/x)^(1/2))*2^(1/2)-1/4*a*ln(1+(1-1/a/x)^(1/4)*2^(1/2)/(1+1/a/x)^(1/4)+(1-1/a/x)^(1/2)/(1+1/a/x
)^(1/2))*2^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6171, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}+\frac {a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}-\frac {a \log \left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {\frac {1}{a x}+1}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{2 \sqrt {2}}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}\right )}{\sqrt {2}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{\frac {1}{a x}+1}}+1\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(ArcCoth[a*x]/2)/x^2,x]

[Out]

a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4) - (a*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/
Sqrt[2] + (a*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] + (a*Log[1 + Sqrt[1 - 1/(a
*x)]/Sqrt[1 + 1/(a*x)] - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2]) - (a*Log[1 + Sqrt[1 -
1/(a*x)]/Sqrt[1 + 1/(a*x)] + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/(2*Sqrt[2])

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 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 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

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

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}{2} \coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [4]{1+\frac {x}{a}}}{\sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+(2 a) \operatorname {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-\frac {1}{a x}}\right )\\ &=a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+(2 a) \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )\\ &=a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-a \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )+a \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )\\ &=a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}\\ &=a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}+\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}-\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}\\ &=a \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}+\frac {a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{\sqrt {2}}+\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}-\frac {a \log \left (1+\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {2} \sqrt [4]{1-\frac {1}{a x}}}{\sqrt [4]{1+\frac {1}{a x}}}\right )}{2 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 148, normalized size = 0.55 \[ a \left (\frac {2 e^{\frac {1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}+1}+\frac {\log \left (-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+e^{\coth ^{-1}(a x)}+1\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1-\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (\sqrt {2} e^{\frac {1}{2} \coth ^{-1}(a x)}+1\right )}{\sqrt {2}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(ArcCoth[a*x]/2)/x^2,x]

[Out]

a*((2*E^(ArcCoth[a*x]/2))/(1 + E^(2*ArcCoth[a*x])) + ArcTan[1 - Sqrt[2]*E^(ArcCoth[a*x]/2)]/Sqrt[2] - ArcTan[1
+ Sqrt[2]*E^(ArcCoth[a*x]/2)]/Sqrt[2] + Log[1 - Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]]/(2*Sqrt[2]) - Lo
g[1 + Sqrt[2]*E^(ArcCoth[a*x]/2) + E^ArcCoth[a*x]]/(2*Sqrt[2]))

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fricas [A]  time = 0.47, size = 396, normalized size = 1.48 \[ -\frac {4 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \arctan \left (-\frac {a^{4} + \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - \sqrt {2} \sqrt {a^{6} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {a^{4}} a^{4} + \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} {\left (a^{4}\right )}^{\frac {1}{4}}}{a^{4}}\right ) + 4 \, \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \arctan \left (\frac {a^{4} - \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {2} \sqrt {a^{6} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {a^{4}} a^{4} - \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}} {\left (a^{4}\right )}^{\frac {1}{4}}}{a^{4}}\right ) + \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \log \left (a^{6} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {a^{4}} a^{4} + \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - \sqrt {2} {\left (a^{4}\right )}^{\frac {1}{4}} x \log \left (a^{6} \sqrt {\frac {a x - 1}{a x + 1}} + \sqrt {a^{4}} a^{4} - \sqrt {2} {\left (a^{4}\right )}^{\frac {3}{4}} a^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 4 \, {\left (a x + 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{4 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/4)/x^2,x, algorithm="fricas")

[Out]

-1/4*(4*sqrt(2)*(a^4)^(1/4)*x*arctan(-(a^4 + sqrt(2)*(a^4)^(1/4)*a^3*((a*x - 1)/(a*x + 1))^(1/4) - sqrt(2)*sqr
t(a^6*sqrt((a*x - 1)/(a*x + 1)) + sqrt(a^4)*a^4 + sqrt(2)*(a^4)^(3/4)*a^3*((a*x - 1)/(a*x + 1))^(1/4))*(a^4)^(
1/4))/a^4) + 4*sqrt(2)*(a^4)^(1/4)*x*arctan((a^4 - sqrt(2)*(a^4)^(1/4)*a^3*((a*x - 1)/(a*x + 1))^(1/4) + sqrt(
2)*sqrt(a^6*sqrt((a*x - 1)/(a*x + 1)) + sqrt(a^4)*a^4 - sqrt(2)*(a^4)^(3/4)*a^3*((a*x - 1)/(a*x + 1))^(1/4))*(
a^4)^(1/4))/a^4) + sqrt(2)*(a^4)^(1/4)*x*log(a^6*sqrt((a*x - 1)/(a*x + 1)) + sqrt(a^4)*a^4 + sqrt(2)*(a^4)^(3/
4)*a^3*((a*x - 1)/(a*x + 1))^(1/4)) - sqrt(2)*(a^4)^(1/4)*x*log(a^6*sqrt((a*x - 1)/(a*x + 1)) + sqrt(a^4)*a^4
- sqrt(2)*(a^4)^(3/4)*a^3*((a*x - 1)/(a*x + 1))^(1/4)) - 4*(a*x + 1)*((a*x - 1)/(a*x + 1))^(3/4))/x

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giac [A]  time = 0.19, size = 186, normalized size = 0.70 \[ \frac {1}{4} \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {8 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/4)/x^2,x, algorithm="giac")

[Out]

1/4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4))) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(
sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4))) - sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(
a*x + 1)) + 1) + sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) + 8*((a*x -
1)/(a*x + 1))^(3/4)/((a*x - 1)/(a*x + 1) + 1))*a

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maple [F]  time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}} x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(1/4)/x^2,x)

[Out]

int(1/((a*x-1)/(a*x+1))^(1/4)/x^2,x)

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maxima [A]  time = 0.41, size = 186, normalized size = 0.70 \[ \frac {1}{4} \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + \sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \frac {8 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{\frac {a x - 1}{a x + 1} + 1}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(1/4)/x^2,x, algorithm="maxima")

[Out]

1/4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4))) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(
sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4))) - sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(
a*x + 1)) + 1) + sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) + 8*((a*x -
1)/(a*x + 1))^(3/4)/((a*x - 1)/(a*x + 1) + 1))*a

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mupad [B]  time = 1.19, size = 87, normalized size = 0.33 \[ {\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )-{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )+\frac {2\,a\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4}}{\frac {a\,x-1}{a\,x+1}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*((a*x - 1)/(a*x + 1))^(1/4)),x)

[Out]

(-1)^(1/4)*a*atan((-1)^(1/4)*((a*x - 1)/(a*x + 1))^(1/4)) - (-1)^(1/4)*a*atanh((-1)^(1/4)*((a*x - 1)/(a*x + 1)
)^(1/4)) + (2*a*((a*x - 1)/(a*x + 1))^(3/4))/((a*x - 1)/(a*x + 1) + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt [4]{\frac {a x - 1}{a x + 1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/4)/x**2,x)

[Out]

Integral(1/(x**2*((a*x - 1)/(a*x + 1))**(1/4)), x)

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