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

Optimal. Leaf size=312 \[ \frac{a^2 \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{32 \sqrt{2}}-\frac{a^2 \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{32 \sqrt{2}}-\frac{1}{16} a^2 \tan ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{16 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{16 \sqrt{2}}-\frac{1}{16} a^2 \tanh ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac{(1-a x)^{7/8} (a x+1)^{9/8}}{2 x^2}-\frac{a (1-a x)^{7/8} \sqrt [8]{a x+1}}{8 x} \]

[Out]

-(a*(1 - a*x)^(7/8)*(1 + a*x)^(1/8))/(8*x) - ((1 - a*x)^(7/8)*(1 + a*x)^(9/8))/(2*x^2) - (a^2*ArcTan[(1 + a*x)
^(1/8)/(1 - a*x)^(1/8)])/16 + (a^2*ArcTan[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)])/(16*Sqrt[2]) - (a^2*
ArcTan[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)])/(16*Sqrt[2]) - (a^2*ArcTanh[(1 + a*x)^(1/8)/(1 - a*x)^(
1/8)])/16 + (a^2*Log[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/(32*Sqr
t[2]) - (a^2*Log[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/(32*Sqrt[2]
)

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Rubi [A]  time = 0.152656, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6126, 96, 94, 93, 214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ \frac{a^2 \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{32 \sqrt{2}}-\frac{a^2 \log \left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}+\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{32 \sqrt{2}}-\frac{1}{16} a^2 \tan ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )+\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )}{16 \sqrt{2}}-\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}+1\right )}{16 \sqrt{2}}-\frac{1}{16} a^2 \tanh ^{-1}\left (\frac{\sqrt [8]{a x+1}}{\sqrt [8]{1-a x}}\right )-\frac{(1-a x)^{7/8} (a x+1)^{9/8}}{2 x^2}-\frac{a (1-a x)^{7/8} \sqrt [8]{a x+1}}{8 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(1 - a*x)^(7/8)*(1 + a*x)^(1/8))/(8*x) - ((1 - a*x)^(7/8)*(1 + a*x)^(9/8))/(2*x^2) - (a^2*ArcTan[(1 + a*x)
^(1/8)/(1 - a*x)^(1/8)])/16 + (a^2*ArcTan[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)])/(16*Sqrt[2]) - (a^2*
ArcTan[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8)])/(16*Sqrt[2]) - (a^2*ArcTanh[(1 + a*x)^(1/8)/(1 - a*x)^(
1/8)])/16 + (a^2*Log[1 - (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/(32*Sqr
t[2]) - (a^2*Log[1 + (Sqrt[2]*(1 + a*x)^(1/8))/(1 - a*x)^(1/8) + (1 + a*x)^(1/4)/(1 - a*x)^(1/4)])/(32*Sqrt[2]
)

Rule 6126

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

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 214

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

Rule 212

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

Rule 206

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

Rule 203

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

Rule 211

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

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 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 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 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])

Rubi steps

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

Mathematica [C]  time = 0.0209067, size = 73, normalized size = 0.23 \[ -\frac{(1-a x)^{7/8} \left (2 a^2 x^2 \text{Hypergeometric2F1}\left (\frac{7}{8},1,\frac{15}{8},\frac{1-a x}{a x+1}\right )+7 \left (5 a^2 x^2+9 a x+4\right )\right )}{56 x^2 (a x+1)^{7/8}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((1 - a*x)^(7/8)*(7*(4 + 9*a*x + 5*a^2*x^2) + 2*a^2*x^2*Hypergeometric2F1[7/8, 1, 15/8, (1 - a*x)/(1 + a*x)])
)/(56*x^2*(1 + a*x)^(7/8))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\sqrt [4]{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)/x^3,x)

[Out]

int(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{1}{4}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)/x^3,x, algorithm="maxima")

[Out]

integrate(((a*x + 1)/sqrt(-a^2*x^2 + 1))^(1/4)/x^3, x)

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Fricas [B]  time = 2.37574, size = 1423, normalized size = 4.56 \begin{align*} -\frac{4 \, a^{2} x^{2} \arctan \left (\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}}\right ) + 2 \, a^{2} x^{2} \log \left (\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + 1\right ) - 2 \, a^{2} x^{2} \log \left (\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} - 1\right ) - 4 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \arctan \left (-\frac{a^{8} + \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} - \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} \sqrt{a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}}}{a^{8}}\right ) - 4 \, \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \arctan \left (\frac{a^{8} - \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{2}{\left (a^{8}\right )}^{\frac{3}{4}} \sqrt{a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}}}{a^{8}}\right ) + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \log \left (a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}\right ) - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} x^{2} \log \left (a^{4} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - \sqrt{2}{\left (a^{8}\right )}^{\frac{1}{4}} a^{2} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}} + \sqrt{a^{8}}\right ) - 8 \,{\left (5 \, a^{2} x^{2} - a x - 4\right )} \left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}\right )^{\frac{1}{4}}}{64 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)/x^3,x, algorithm="fricas")

[Out]

-1/64*(4*a^2*x^2*arctan((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4)) + 2*a^2*x^2*log((-sqrt(-a^2*x^2 + 1)/(a*x - 1))
^(1/4) + 1) - 2*a^2*x^2*log((-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - 1) - 4*sqrt(2)*(a^8)^(1/4)*x^2*arctan(-(a^
8 + sqrt(2)*(a^8)^(3/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) - sqrt(2)*(a^8)^(3/4)*sqrt(a^4*sqrt(-sqrt(-a
^2*x^2 + 1)/(a*x - 1)) + sqrt(2)*(a^8)^(1/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^8)))/a^8) - 4*
sqrt(2)*(a^8)^(1/4)*x^2*arctan((a^8 - sqrt(2)*(a^8)^(3/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(2)*
(a^8)^(3/4)*sqrt(a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(2)*(a^8)^(1/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x -
1))^(1/4) + sqrt(a^8)))/a^8) + sqrt(2)*(a^8)^(1/4)*x^2*log(a^4*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)) + sqrt(2)*
(a^8)^(1/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^8)) - sqrt(2)*(a^8)^(1/4)*x^2*log(a^4*sqrt(-sqr
t(-a^2*x^2 + 1)/(a*x - 1)) - sqrt(2)*(a^8)^(1/4)*a^2*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4) + sqrt(a^8)) - 8*(5
*a^2*x^2 - a*x - 4)*(-sqrt(-a^2*x^2 + 1)/(a*x - 1))^(1/4))/x^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/4)/x**3,x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/4)/x^3,x, algorithm="giac")

[Out]

Timed out