3.1.24 \(\int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx\) [24]

Optimal. Leaf size=93 \[ -\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )+\frac {11}{2} a^3 \csc ^{-1}(a x) \]

[Out]

11/2*a^3*arccsc(a*x)-(a+1/x)^3/(1-1/a^2/x^2)^(1/2)-1/3*a*(3*a+1/x)^2*(1-1/a^2/x^2)^(1/2)-1/6*a^2*(28*a+3/x)*(1
-1/a^2/x^2)^(1/2)

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Rubi [A]
time = 0.48, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6304, 1647, 1607, 12, 866, 1649, 1668, 794, 222} \begin {gather*} \frac {11}{2} a^3 \csc ^{-1}(a x)-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + x^(-1))^3/Sqrt[1 - 1/(a^2*x^2)]) - (a*Sqrt[1 - 1/(a^2*x^2)]*(3*a + x^(-1))^2)/3 - (a^2*Sqrt[1 - 1/(a^2*
x^2)]*(28*a + 3/x))/6 + (11*a^3*ArcCsc[a*x])/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 222

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

Rule 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 866

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

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d*e, Int[(d + e*x)^(m - 1)*
PolynomialQuotient[Pq, a*e + c*d*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && EqQ[c*d^2 + a*e^2, 0] && EqQ[PolynomialRemainder[Pq, a*e + c*d*x, x], 0]

Rule 1649

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a*e + c*d*x, x], f = PolynomialRemainder[Pq, a*e + c*d*x, x]}, Simp[(-d)*f*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2
*a*e*(p + 1))), x] + Dist[d/(2*a*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)
*Q + f*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] && ILtQ[p
+ 1/2, 0] && GtQ[m, 0]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 6304

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

Rubi steps

\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{x^4} \, dx &=-\text {Subst}\left (\int \frac {x^2 \left (1+\frac {x}{a}\right )^2}{\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}} \left (-a x^2-x^3\right )}{\left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x) x^2 \sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {\text {Subst}\left (\int \frac {a^2 x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=-\text {Subst}\left (\int \frac {x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \frac {x^2 \left (1+\frac {x}{a}\right )^3}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}+\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^2 \left (3 a^2+a x\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{3} \text {Subst}\left (\int \frac {\left (-5-\frac {3 x}{a}\right ) \left (3 a^2+a x\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )+\frac {1}{2} \left (11 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\left (a+\frac {1}{x}\right )^3}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {1}{3} a \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a+\frac {1}{x}\right )^2-\frac {1}{6} a^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (28 a+\frac {3}{x}\right )+\frac {11}{2} a^3 \csc ^{-1}(a x)\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 66, normalized size = 0.71 \begin {gather*} \frac {1}{6} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (2+7 a x+19 a^2 x^2-52 a^3 x^3\right )}{x^2 (-1+a x)}+33 a^2 \text {ArcSin}\left (\frac {1}{a x}\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*((Sqrt[1 - 1/(a^2*x^2)]*(2 + 7*a*x + 19*a^2*x^2 - 52*a^3*x^3))/(x^2*(-1 + a*x)) + 33*a^2*ArcSin[1/(a*x)]))/
6

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(665\) vs. \(2(81)=162\).
time = 0.10, size = 666, normalized size = 7.16

method	result	size
risch	\(-\frac {\left (a x -1\right ) \left (28 a^{2} x^{2}+9 a x +2\right )}{6 x^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (-\frac {4 a^{2} \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}+\frac {11 a^{3} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\)	\(135\)
default	\(-\frac {30 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a^{6} x^{6}+30 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}+30 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}-30 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a^{4} x^{4}-93 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}-33 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{5} x^{5}-30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+12 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-60 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}-60 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+51 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+96 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+66 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+60 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+30 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}+30 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-14 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}-33 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}-33 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-30 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-5 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}}{6 x^{3} \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\)	\(666\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(30*(a^2)^(1/2)*(a^2*x^2-1)^(1/2)*a^6*x^6+30*((a*x+1)*(a*x-1))^(1/2)*(a^2)^(1/2)*a^5*x^5+30*ln((a^2*x+(a^
2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^6*x^5-30*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a^4*x^4-93*(a^2*x^2-1)
^(1/2)*(a^2)^(1/2)*a^5*x^5-33*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))*a^5*x^5-30*ln((a^2*x+(a^2*x^2-1)^(1/2)*(
a^2)^(1/2))/(a^2)^(1/2))*a^6*x^5+12*((a*x+1)*(a*x-1))^(3/2)*(a^2)^(1/2)*a^3*x^3-60*(a^2)^(1/2)*((a*x+1)*(a*x-1
))^(1/2)*a^4*x^4-60*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^5*x^4+51*(a^2*x^2-1)^(3/2)*(
a^2)^(1/2)*a^3*x^3+96*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^4*x^4+66*a^4*x^4*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))
+60*ln((a^2*x+(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^5*x^4+30*(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2)*a^3*x
^3+30*ln((a^2*x+(a^2)^(1/2)*((a*x+1)*(a*x-1))^(1/2))/(a^2)^(1/2))*a^4*x^3-14*(a^2*x^2-1)^(3/2)*(a^2)^(1/2)*a^2
*x^2-33*(a^2*x^2-1)^(1/2)*(a^2)^(1/2)*a^3*x^3-33*a^3*x^3*(a^2)^(1/2)*arctan(1/(a^2*x^2-1)^(1/2))-30*ln((a^2*x+
(a^2*x^2-1)^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^4*x^3-5*(a^2)^(1/2)*(a^2*x^2-1)^(3/2)*a*x-2*(a^2*x^2-1)^(3/2)*(a
^2)^(1/2))/x^3/(a^2)^(1/2)/((a*x+1)*(a*x-1))^(1/2)/(a*x+1)/((a*x-1)/(a*x+1))^(3/2)

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Maxima [A]
time = 0.46, size = 154, normalized size = 1.66 \begin {gather*} -\frac {1}{3} \, {\left (33 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {\frac {75 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {88 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {33 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} + 12 \, a^{2}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + \sqrt {\frac {a x - 1}{a x + 1}}}\right )} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(33*a^2*arctan(sqrt((a*x - 1)/(a*x + 1))) + (75*(a*x - 1)*a^2/(a*x + 1) + 88*(a*x - 1)^2*a^2/(a*x + 1)^2
+ 33*(a*x - 1)^3*a^2/(a*x + 1)^3 + 12*a^2)/(((a*x - 1)/(a*x + 1))^(7/2) + 3*((a*x - 1)/(a*x + 1))^(5/2) + 3*((
a*x - 1)/(a*x + 1))^(3/2) + sqrt((a*x - 1)/(a*x + 1))))*a

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Fricas [A]
time = 0.35, size = 96, normalized size = 1.03 \begin {gather*} -\frac {66 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (52 \, a^{4} x^{4} + 33 \, a^{3} x^{3} - 26 \, a^{2} x^{2} - 9 \, a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, {\left (a x^{4} - x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(66*(a^4*x^4 - a^3*x^3)*arctan(sqrt((a*x - 1)/(a*x + 1))) + (52*a^4*x^4 + 33*a^3*x^3 - 26*a^2*x^2 - 9*a*x
- 2)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^4 - x^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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Mupad [B]
time = 1.24, size = 152, normalized size = 1.63 \begin {gather*} -\frac {4\,a^3+\frac {88\,a^3\,{\left (a\,x-1\right )}^2}{3\,{\left (a\,x+1\right )}^2}+\frac {11\,a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {25\,a^3\,\left (a\,x-1\right )}{a\,x+1}}{\sqrt {\frac {a\,x-1}{a\,x+1}}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}+3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}+{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}-11\,a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (4*a^3 + (88*a^3*(a*x - 1)^2)/(3*(a*x + 1)^2) + (11*a^3*(a*x - 1)^3)/(a*x + 1)^3 + (25*a^3*(a*x - 1))/(a*x +
1))/(((a*x - 1)/(a*x + 1))^(1/2) + 3*((a*x - 1)/(a*x + 1))^(3/2) + 3*((a*x - 1)/(a*x + 1))^(5/2) + ((a*x - 1)
/(a*x + 1))^(7/2)) - 11*a^3*atan(((a*x - 1)/(a*x + 1))^(1/2))

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