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

Optimal. Leaf size=320 \[ -\frac{8 \sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}}+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right ) \]

[Out]

(-8*(1 + 1/(a*x))^(1/4))/(1 - 1/(a*x))^(1/4) + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^
(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] + 2*ArcTan[(1 + 1/(a*x))^(1/4)/
(1 - 1/(a*x))^(1/4)] + 2*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] - 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)]/Sqrt[2] + 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)]/Sqrt[2]

________________________________________________________________________________________

Rubi [A]  time = 0.297283, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 16, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.143, Rules used = {6171, 98, 21, 105, 63, 331, 297, 1162, 617, 204, 1165, 628, 93, 212, 206, 203} \[ -\frac{8 \sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}-\frac{\log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{\frac{1}{a x}+1}}+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )}{\sqrt{2}}+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{\frac{1}{a x}+1}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(1 + 1/(a*x))^(1/4))/(1 - 1/(a*x))^(1/4) + Sqrt[2]*ArcTan[1 - (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^
(1/4)] - Sqrt[2]*ArcTan[1 + (Sqrt[2]*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)] + 2*ArcTan[(1 + 1/(a*x))^(1/4)/
(1 - 1/(a*x))^(1/4)] + 2*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)] - 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)]/Sqrt[2] + 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)]/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 98

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 105

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

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 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 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 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])

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 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 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])

Rubi steps

\begin{align*} \int \frac{e^{\frac{5}{2} \coth ^{-1}(a x)}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/4}}{x \left (1-\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{8 \sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}+(4 a) \operatorname{Subst}\left (\int \frac{-\frac{1}{4 a}+\frac{x}{4 a^2}}{x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{8 \sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/4}}{x \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{8 \sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}+\frac{\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}-\operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{8 \sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}-4 \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-\frac{1}{a x}}\right )-4 \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{8 \sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}+2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )+2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )-4 \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 )\\ &=-\frac{8 \sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}+2 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )+2 \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 )-2 \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{8 \sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}+2 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )-\frac{\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 )}{\sqrt{2}}-\frac{\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 )}{\sqrt{2}}-\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 )-\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{8 \sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}+2 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )-\frac{\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 )}{\sqrt{2}}+\frac{\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 )}{\sqrt{2}}-\sqrt{2} \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} \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 )\\ &=-\frac{8 \sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )-\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-\frac{1}{a x}}}{\sqrt [4]{1+\frac{1}{a x}}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )+2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )-\frac{\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 )}{\sqrt{2}}+\frac{\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 )}{\sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0710473, size = 30, normalized size = 0.09 \[ 8 e^{\frac{1}{2} \coth ^{-1}(a x)} \left (\text{Hypergeometric2F1}\left (\frac{1}{8},1,\frac{9}{8},e^{4 \coth ^{-1}(a x)}\right )-1\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.135, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \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(1/((a*x-1)/(a*x+1))^(5/4)/x,x)

[Out]

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

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Maxima [A]  time = 1.53527, size = 329, normalized size = 1.03 \begin{align*} -\frac{1}{2} \, a{\left (\frac{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 )}{a} + \frac{4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a} - \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a} + \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a} + \frac{16}{a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*((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))/a + 4*
arctan(((a*x - 1)/(a*x + 1))^(1/4))/a - 2*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a + 2*log(((a*x - 1)/(a*x + 1))
^(1/4) - 1)/a + 16/(a*((a*x - 1)/(a*x + 1))^(1/4)))

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Fricas [A]  time = 1.67634, size = 975, normalized size = 3.05 \begin{align*} \frac{4 \, \sqrt{2}{\left (a x - 1\right )} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a x - 1}{a x + 1}} + 1} - \sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right ) + 4 \, \sqrt{2}{\left (a x - 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 4 \, \sqrt{\frac{a x - 1}{a x + 1}} + 4} - \sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + \sqrt{2}{\left (a x - 1\right )} \log \left (4 \, \sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 4 \, \sqrt{\frac{a x - 1}{a x + 1}} + 4\right ) - \sqrt{2}{\left (a x - 1\right )} \log \left (-4 \, \sqrt{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 4 \, \sqrt{\frac{a x - 1}{a x + 1}} + 4\right ) - 4 \,{\left (a x - 1\right )} \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) + 2 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) - 2 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right ) - 16 \,{\left (a x + 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{2 \,{\left (a x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*sqrt(2)*(a*x - 1)*arctan(sqrt(2)*sqrt(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) +
1) - sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) - 1) + 4*sqrt(2)*(a*x - 1)*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*((a*x
- 1)/(a*x + 1))^(1/4) + 4*sqrt((a*x - 1)/(a*x + 1)) + 4) - sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + 1) + sqrt(2)*
(a*x - 1)*log(4*sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + 4*sqrt((a*x - 1)/(a*x + 1)) + 4) - sqrt(2)*(a*x - 1)*log
(-4*sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + 4*sqrt((a*x - 1)/(a*x + 1)) + 4) - 4*(a*x - 1)*arctan(((a*x - 1)/(a*
x + 1))^(1/4)) + 2*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) - 2*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4
) - 1) - 16*(a*x + 1)*((a*x - 1)/(a*x + 1))^(3/4))/(a*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(5/4)/x,x)

[Out]

Integral(1/(x*((a*x - 1)/(a*x + 1))**(5/4)), x)

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Giac [A]  time = 1.1944, size = 340, normalized size = 1.06 \begin{align*} -\frac{1}{2} \, a{\left (\frac{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 )}{a} + \frac{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 )}{a} - \frac{\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 )}{a} + \frac{\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 )}{a} + \frac{4 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a} - \frac{2 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a} + \frac{2 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a} + \frac{16}{a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*((a*x - 1)/(a*x + 1))^(1/4)))/a + 2*sqrt(2)*arctan(-1/2*sqrt
(2)*(sqrt(2) - 2*((a*x - 1)/(a*x + 1))^(1/4)))/a - sqrt(2)*log(sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x
- 1)/(a*x + 1)) + 1)/a + sqrt(2)*log(-sqrt(2)*((a*x - 1)/(a*x + 1))^(1/4) + sqrt((a*x - 1)/(a*x + 1)) + 1)/a
+ 4*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a - 2*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a + 2*log(abs(((a*x - 1)/(a
*x + 1))^(1/4) - 1))/a + 16/(a*((a*x - 1)/(a*x + 1))^(1/4)))