### 3.110 $$\int \frac{e^{-\frac{5}{2} \coth ^{-1}(a x)}}{x^2} \, dx$$

Optimal. Leaf size=299 $\frac{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{\frac{1}{a x}+1}}+5 a \left (\frac{1}{a x}+1\right )^{3/4} \sqrt [4]{1-\frac{1}{a x}}+\frac{5 a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}-\frac{5 a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}+\frac{5 a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{\sqrt{2}}-\frac{5 a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}}$

[Out]

(4*a*(1 - 1/(a*x))^(5/4))/(1 + 1/(a*x))^(1/4) + 5*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4) + (5*a*ArcTan[1 -
(Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] - (5*a*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1
+ 1/(a*x))^(1/4)])/Sqrt[2] + (5*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]) - (5*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])

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Rubi [A]  time = 0.244575, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.786, Rules used = {6171, 47, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} $\frac{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{\frac{1}{a x}+1}}+5 a \left (\frac{1}{a x}+1\right )^{3/4} \sqrt [4]{1-\frac{1}{a x}}+\frac{5 a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}-\frac{5 a \log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{2 \sqrt{2}}+\frac{5 a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )}{\sqrt{2}}-\frac{5 a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(E^((5*ArcCoth[a*x])/2)*x^2),x]

[Out]

(4*a*(1 - 1/(a*x))^(5/4))/(1 + 1/(a*x))^(1/4) + 5*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4) + (5*a*ArcTan[1 -
(Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)])/Sqrt[2] - (5*a*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1
+ 1/(a*x))^(1/4)])/Sqrt[2] + (5*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]) - (5*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 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]

Rule 47

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

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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

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{5}{2} \coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{5/4}}{\left (1+\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac{1}{a x}}}+5 \operatorname{Subst}\left (\int \frac{\sqrt [4]{1-\frac{x}{a}}}{\sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac{1}{a x}}}+5 a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{5}{2} \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right )^{3/4} \sqrt [4]{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac{1}{a x}}}+5 a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}-(10 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-\frac{1}{a x}}\right )\\ &=\frac{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac{1}{a x}}}+5 a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}-(10 a) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )\\ &=\frac{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac{1}{a x}}}+5 a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}-(5 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 )-(5 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 )\\ &=\frac{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac{1}{a x}}}+5 a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}-\frac{1}{2} (5 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} (5 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{(5 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{(5 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{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac{1}{a x}}}+5 a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{5 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{5 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{(5 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{(5 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{4 a \left (1-\frac{1}{a x}\right )^{5/4}}{\sqrt [4]{1+\frac{1}{a x}}}+5 a \sqrt [4]{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/4}+\frac{5 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{5 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{5 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{5 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*}

Mathematica [C]  time = 0.0820784, size = 31, normalized size = 0.1 $8 a e^{-\frac{1}{2} \coth ^{-1}(a x)} \text{Hypergeometric2F1}\left (-\frac{1}{4},2,\frac{3}{4},-e^{2 \coth ^{-1}(a x)}\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^((5*ArcCoth[a*x])/2)*x^2),x]

[Out]

(8*a*Hypergeometric2F1[-1/4, 2, 3/4, -E^(2*ArcCoth[a*x])])/E^(ArcCoth[a*x]/2)

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Maple [F]  time = 0.353, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{5}{4}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x-1)/(a*x+1))^(5/4)/x^2,x)

[Out]

int(((a*x-1)/(a*x+1))^(5/4)/x^2,x)

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Maxima [A]  time = 1.5397, size = 275, normalized size = 0.92 \begin{align*} -\frac{1}{4} \,{\left (10 \, \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 ) + 10 \, \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 ) + 5 \, \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 ) - 5 \, \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 ) - 32 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \frac{8 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\frac{a x - 1}{a x + 1} + 1}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(5/4)/x^2,x, algorithm="maxima")

[Out]

-1/4*(10*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4))) + 10*sqrt(2)*arctan(-1/2*sqrt(2
)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4))) + 5*sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x -
1)/(a*x + 1)) + 1) - 5*sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) - 32
*((a*x - 1)/(a*x + 1))^(1/4) - 8*((a*x - 1)/(a*x + 1))^(1/4)/((a*x - 1)/(a*x + 1) + 1))*a

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Fricas [A]  time = 1.71596, size = 1023, normalized size = 3.42 \begin{align*} \frac{20 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \arctan \left (-\frac{a^{4} + \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} \sqrt{a^{2} \sqrt{\frac{a x - 1}{a x + 1}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}}}{a^{4}}\right ) + 20 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \arctan \left (\frac{a^{4} - \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{2}{\left (a^{4}\right )}^{\frac{3}{4}} \sqrt{a^{2} \sqrt{\frac{a x - 1}{a x + 1}} - \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{a^{4}}}}{a^{4}}\right ) - 5 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \log \left (25 \, a^{2} \sqrt{\frac{a x - 1}{a x + 1}} + 25 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 25 \, \sqrt{a^{4}}\right ) + 5 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} x \log \left (25 \, a^{2} \sqrt{\frac{a x - 1}{a x + 1}} - 25 \, \sqrt{2}{\left (a^{4}\right )}^{\frac{1}{4}} a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 25 \, \sqrt{a^{4}}\right ) + 4 \,{\left (9 \, a x + 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{4 \, x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(5/4)/x^2,x, algorithm="fricas")

[Out]

1/4*(20*sqrt(2)*(a^4)^(1/4)*x*arctan(-(a^4 + sqrt(2)*(a^4)^(3/4)*a*((a*x - 1)/(a*x + 1))^(1/4) - sqrt(2)*(a^4)
^(3/4)*sqrt(a^2*sqrt((a*x - 1)/(a*x + 1)) + sqrt(2)*(a^4)^(1/4)*a*((a*x - 1)/(a*x + 1))^(1/4) + sqrt(a^4)))/a^
4) + 20*sqrt(2)*(a^4)^(1/4)*x*arctan((a^4 - sqrt(2)*(a^4)^(3/4)*a*((a*x - 1)/(a*x + 1))^(1/4) + sqrt(2)*(a^4)^
(3/4)*sqrt(a^2*sqrt((a*x - 1)/(a*x + 1)) - sqrt(2)*(a^4)^(1/4)*a*((a*x - 1)/(a*x + 1))^(1/4) + sqrt(a^4)))/a^4
) - 5*sqrt(2)*(a^4)^(1/4)*x*log(25*a^2*sqrt((a*x - 1)/(a*x + 1)) + 25*sqrt(2)*(a^4)^(1/4)*a*((a*x - 1)/(a*x +
1))^(1/4) + 25*sqrt(a^4)) + 5*sqrt(2)*(a^4)^(1/4)*x*log(25*a^2*sqrt((a*x - 1)/(a*x + 1)) - 25*sqrt(2)*(a^4)^(1
/4)*a*((a*x - 1)/(a*x + 1))^(1/4) + 25*sqrt(a^4)) + 4*(9*a*x + 1)*((a*x - 1)/(a*x + 1))^(1/4))/x

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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*x+1))**(5/4)/x**2,x)

[Out]

Timed out

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Giac [A]  time = 1.19687, size = 275, normalized size = 0.92 \begin{align*} -\frac{1}{4} \,{\left (10 \, \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 ) + 10 \, \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 ) + 5 \, \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 ) - 5 \, \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 ) - 32 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - \frac{8 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}{\frac{a x - 1}{a x + 1} + 1}\right )} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x-1)/(a*x+1))^(5/4)/x^2,x, algorithm="giac")

[Out]

-1/4*(10*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4))) + 10*sqrt(2)*arctan(-1/2*sqrt(2
)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4))) + 5*sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x -
1)/(a*x + 1)) + 1) - 5*sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1) - 32
*((a*x - 1)/(a*x + 1))^(1/4) - 8*((a*x - 1)/(a*x + 1))^(1/4)/((a*x - 1)/(a*x + 1) + 1))*a