3.84 \(\int \frac{\cot ^{-1}(a x^2)}{x^4} \, dx\)

Optimal. Leaf size=150 \[ \frac{a^{3/2} \log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{6 \sqrt{2}}-\frac{a^{3/2} \log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{6 \sqrt{2}}-\frac{a^{3/2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2}}+\frac{a^{3/2} \tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{3 \sqrt{2}}-\frac{\cot ^{-1}\left (a x^2\right )}{3 x^3}+\frac{2 a}{3 x} \]

[Out]

(2*a)/(3*x) - ArcCot[a*x^2]/(3*x^3) - (a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[a]*x])/(3*Sqrt[2]) + (a^(3/2)*ArcTan[1
+ Sqrt[2]*Sqrt[a]*x])/(3*Sqrt[2]) + (a^(3/2)*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2])/(6*Sqrt[2]) - (a^(3/2)*Log[1
+ Sqrt[2]*Sqrt[a]*x + a*x^2])/(6*Sqrt[2])

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Rubi [A]  time = 0.0916708, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5034, 325, 297, 1162, 617, 204, 1165, 628} \[ \frac{a^{3/2} \log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{6 \sqrt{2}}-\frac{a^{3/2} \log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{6 \sqrt{2}}-\frac{a^{3/2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2}}+\frac{a^{3/2} \tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{3 \sqrt{2}}-\frac{\cot ^{-1}\left (a x^2\right )}{3 x^3}+\frac{2 a}{3 x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x^2]/x^4,x]

[Out]

(2*a)/(3*x) - ArcCot[a*x^2]/(3*x^3) - (a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[a]*x])/(3*Sqrt[2]) + (a^(3/2)*ArcTan[1
+ Sqrt[2]*Sqrt[a]*x])/(3*Sqrt[2]) + (a^(3/2)*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2])/(6*Sqrt[2]) - (a^(3/2)*Log[1
+ Sqrt[2]*Sqrt[a]*x + a*x^2])/(6*Sqrt[2])

Rule 5034

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCot
[c*x^n]))/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x]
/; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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]

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}\left (a x^2\right )}{x^4} \, dx &=-\frac{\cot ^{-1}\left (a x^2\right )}{3 x^3}-\frac{1}{3} (2 a) \int \frac{1}{x^2 \left (1+a^2 x^4\right )} \, dx\\ &=\frac{2 a}{3 x}-\frac{\cot ^{-1}\left (a x^2\right )}{3 x^3}+\frac{1}{3} \left (2 a^3\right ) \int \frac{x^2}{1+a^2 x^4} \, dx\\ &=\frac{2 a}{3 x}-\frac{\cot ^{-1}\left (a x^2\right )}{3 x^3}-\frac{1}{3} a^2 \int \frac{1-a x^2}{1+a^2 x^4} \, dx+\frac{1}{3} a^2 \int \frac{1+a x^2}{1+a^2 x^4} \, dx\\ &=\frac{2 a}{3 x}-\frac{\cot ^{-1}\left (a x^2\right )}{3 x^3}+\frac{1}{6} a \int \frac{1}{\frac{1}{a}-\frac{\sqrt{2} x}{\sqrt{a}}+x^2} \, dx+\frac{1}{6} a \int \frac{1}{\frac{1}{a}+\frac{\sqrt{2} x}{\sqrt{a}}+x^2} \, dx+\frac{a^{3/2} \int \frac{\frac{\sqrt{2}}{\sqrt{a}}+2 x}{-\frac{1}{a}-\frac{\sqrt{2} x}{\sqrt{a}}-x^2} \, dx}{6 \sqrt{2}}+\frac{a^{3/2} \int \frac{\frac{\sqrt{2}}{\sqrt{a}}-2 x}{-\frac{1}{a}+\frac{\sqrt{2} x}{\sqrt{a}}-x^2} \, dx}{6 \sqrt{2}}\\ &=\frac{2 a}{3 x}-\frac{\cot ^{-1}\left (a x^2\right )}{3 x^3}+\frac{a^{3/2} \log \left (1-\sqrt{2} \sqrt{a} x+a x^2\right )}{6 \sqrt{2}}-\frac{a^{3/2} \log \left (1+\sqrt{2} \sqrt{a} x+a x^2\right )}{6 \sqrt{2}}+\frac{a^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2}}-\frac{a^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2}}\\ &=\frac{2 a}{3 x}-\frac{\cot ^{-1}\left (a x^2\right )}{3 x^3}-\frac{a^{3/2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2}}+\frac{a^{3/2} \tan ^{-1}\left (1+\sqrt{2} \sqrt{a} x\right )}{3 \sqrt{2}}+\frac{a^{3/2} \log \left (1-\sqrt{2} \sqrt{a} x+a x^2\right )}{6 \sqrt{2}}-\frac{a^{3/2} \log \left (1+\sqrt{2} \sqrt{a} x+a x^2\right )}{6 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0514186, size = 146, normalized size = 0.97 \[ \frac{a x^2 \left (\sqrt{2} \sqrt{a} x \log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )-\sqrt{2} \sqrt{a} x \log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )-2 \sqrt{2} \sqrt{a} x \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )+2 \sqrt{2} \sqrt{a} x \tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )+8\right )-4 \cot ^{-1}\left (a x^2\right )}{12 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x^2]/x^4,x]

[Out]

(-4*ArcCot[a*x^2] + a*x^2*(8 - 2*Sqrt[2]*Sqrt[a]*x*ArcTan[1 - Sqrt[2]*Sqrt[a]*x] + 2*Sqrt[2]*Sqrt[a]*x*ArcTan[
1 + Sqrt[2]*Sqrt[a]*x] + Sqrt[2]*Sqrt[a]*x*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] - Sqrt[2]*Sqrt[a]*x*Log[1 + Sqrt
[2]*Sqrt[a]*x + a*x^2]))/(12*x^3)

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Maple [A]  time = 0.043, size = 121, normalized size = 0.8 \begin{align*} -{\frac{{\rm arccot} \left (a{x}^{2}\right )}{3\,{x}^{3}}}+{\frac{a\sqrt{2}}{12}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{a}^{-2}}x\sqrt{2}+\sqrt{{a}^{-2}} \right ) \left ({x}^{2}+\sqrt [4]{{a}^{-2}}x\sqrt{2}+\sqrt{{a}^{-2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{a}^{-2}}}}}+{\frac{a\sqrt{2}}{6}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{-2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{a}^{-2}}}}}+{\frac{a\sqrt{2}}{6}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{-2}}}}}-1 \right ){\frac{1}{\sqrt [4]{{a}^{-2}}}}}+{\frac{2\,a}{3\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x^2)/x^4,x)

[Out]

-1/3*arccot(a*x^2)/x^3+1/12*a/(1/a^2)^(1/4)*2^(1/2)*ln((x^2-(1/a^2)^(1/4)*x*2^(1/2)+(1/a^2)^(1/2))/(x^2+(1/a^2
)^(1/4)*x*2^(1/2)+(1/a^2)^(1/2)))+1/6*a/(1/a^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/a^2)^(1/4)*x+1)+1/6*a/(1/a^2)^
(1/4)*2^(1/2)*arctan(2^(1/2)/(1/a^2)^(1/4)*x-1)+2/3*a/x

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Maxima [B]  time = 1.47613, size = 356, normalized size = 2.37 \begin{align*} -\frac{1}{12} \,{\left (a^{2}{\left (\frac{\sqrt{2} \log \left (\sqrt{a^{2}} x^{2} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (a^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\sqrt{a^{2}} x^{2} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (a^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{a^{2}} x - \sqrt{2} \sqrt{-\sqrt{a^{2}}} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{a^{2}} x + \sqrt{2} \sqrt{-\sqrt{a^{2}}} + \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{a^{2}} \sqrt{-\sqrt{a^{2}}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{a^{2}} x - \sqrt{2} \sqrt{-\sqrt{a^{2}}} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{a^{2}} x + \sqrt{2} \sqrt{-\sqrt{a^{2}}} - \sqrt{2}{\left (a^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{a^{2}} \sqrt{-\sqrt{a^{2}}}}\right )} - \frac{8}{x}\right )} a - \frac{\operatorname{arccot}\left (a x^{2}\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^2)/x^4,x, algorithm="maxima")

[Out]

-1/12*(a^2*(sqrt(2)*log(sqrt(a^2)*x^2 + sqrt(2)*(a^2)^(1/4)*x + 1)/(a^2)^(3/4) - sqrt(2)*log(sqrt(a^2)*x^2 - s
qrt(2)*(a^2)^(1/4)*x + 1)/(a^2)^(3/4) - sqrt(2)*log((2*sqrt(a^2)*x - sqrt(2)*sqrt(-sqrt(a^2)) + sqrt(2)*(a^2)^
(1/4))/(2*sqrt(a^2)*x + sqrt(2)*sqrt(-sqrt(a^2)) + sqrt(2)*(a^2)^(1/4)))/(sqrt(a^2)*sqrt(-sqrt(a^2))) - sqrt(2
)*log((2*sqrt(a^2)*x - sqrt(2)*sqrt(-sqrt(a^2)) - sqrt(2)*(a^2)^(1/4))/(2*sqrt(a^2)*x + sqrt(2)*sqrt(-sqrt(a^2
)) - sqrt(2)*(a^2)^(1/4)))/(sqrt(a^2)*sqrt(-sqrt(a^2)))) - 8/x)*a - 1/3*arccot(a*x^2)/x^3

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Fricas [B]  time = 2.34024, size = 706, normalized size = 4.71 \begin{align*} -\frac{4 \, \sqrt{2}{\left (a^{6}\right )}^{\frac{1}{4}} x^{3} \arctan \left (-\frac{\sqrt{2}{\left (a^{6}\right )}^{\frac{1}{4}} a^{5} x + a^{6} - \sqrt{2} \sqrt{a^{10} x^{2} + \sqrt{2}{\left (a^{6}\right )}^{\frac{3}{4}} a^{5} x + \sqrt{a^{6}} a^{6}}{\left (a^{6}\right )}^{\frac{1}{4}}}{a^{6}}\right ) + 4 \, \sqrt{2}{\left (a^{6}\right )}^{\frac{1}{4}} x^{3} \arctan \left (-\frac{\sqrt{2}{\left (a^{6}\right )}^{\frac{1}{4}} a^{5} x - a^{6} - \sqrt{2} \sqrt{a^{10} x^{2} - \sqrt{2}{\left (a^{6}\right )}^{\frac{3}{4}} a^{5} x + \sqrt{a^{6}} a^{6}}{\left (a^{6}\right )}^{\frac{1}{4}}}{a^{6}}\right ) + \sqrt{2}{\left (a^{6}\right )}^{\frac{1}{4}} x^{3} \log \left (a^{10} x^{2} + \sqrt{2}{\left (a^{6}\right )}^{\frac{3}{4}} a^{5} x + \sqrt{a^{6}} a^{6}\right ) - \sqrt{2}{\left (a^{6}\right )}^{\frac{1}{4}} x^{3} \log \left (a^{10} x^{2} - \sqrt{2}{\left (a^{6}\right )}^{\frac{3}{4}} a^{5} x + \sqrt{a^{6}} a^{6}\right ) - 8 \, a x^{2} + 4 \, \arctan \left (\frac{1}{a x^{2}}\right )}{12 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^2)/x^4,x, algorithm="fricas")

[Out]

-1/12*(4*sqrt(2)*(a^6)^(1/4)*x^3*arctan(-(sqrt(2)*(a^6)^(1/4)*a^5*x + a^6 - sqrt(2)*sqrt(a^10*x^2 + sqrt(2)*(a
^6)^(3/4)*a^5*x + sqrt(a^6)*a^6)*(a^6)^(1/4))/a^6) + 4*sqrt(2)*(a^6)^(1/4)*x^3*arctan(-(sqrt(2)*(a^6)^(1/4)*a^
5*x - a^6 - sqrt(2)*sqrt(a^10*x^2 - sqrt(2)*(a^6)^(3/4)*a^5*x + sqrt(a^6)*a^6)*(a^6)^(1/4))/a^6) + sqrt(2)*(a^
6)^(1/4)*x^3*log(a^10*x^2 + sqrt(2)*(a^6)^(3/4)*a^5*x + sqrt(a^6)*a^6) - sqrt(2)*(a^6)^(1/4)*x^3*log(a^10*x^2
- sqrt(2)*(a^6)^(3/4)*a^5*x + sqrt(a^6)*a^6) - 8*a*x^2 + 4*arctan(1/(a*x^2)))/x^3

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Sympy [A]  time = 55.3665, size = 1074, normalized size = 7.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x**2)/x**4,x)

[Out]

Piecewise((acot((-sqrt(2)/2 - sqrt(2)*I/2)**(-2))/(3*x**3), Eq(a, -1/(x**2*(-sqrt(2)/2 - sqrt(2)*I/2)**2))), (
acot((-sqrt(2)/2 + sqrt(2)*I/2)**(-2))/(3*x**3), Eq(a, -1/(x**2*(-sqrt(2)/2 + sqrt(2)*I/2)**2))), (acot((sqrt(
2)/2 - sqrt(2)*I/2)**(-2))/(3*x**3), Eq(a, -1/(x**2*(sqrt(2)/2 - sqrt(2)*I/2)**2))), (acot((sqrt(2)/2 + sqrt(2
)*I/2)**(-2))/(3*x**3), Eq(a, -1/(x**2*(sqrt(2)/2 + sqrt(2)*I/2)**2))), (-pi/(6*x**3), Eq(a, 0)), (-4*(-1)**(1
/4)*a**15*x**6*(a**(-2))**(37/4)/(-6*(-1)**(1/4)*a**14*x**7*(a**(-2))**(37/4) - 6*(-1)**(1/4)*a**12*x**3*(a**(
-2))**(37/4)) + 2*(-1)**(1/4)*a**14*x**4*(a**(-2))**(37/4)*acot(a*x**2)/(-6*(-1)**(1/4)*a**14*x**7*(a**(-2))**
(37/4) - 6*(-1)**(1/4)*a**12*x**3*(a**(-2))**(37/4)) - 4*(-1)**(1/4)*a**13*x**2*(a**(-2))**(37/4)/(-6*(-1)**(1
/4)*a**14*x**7*(a**(-2))**(37/4) - 6*(-1)**(1/4)*a**12*x**3*(a**(-2))**(37/4)) + 2*(-1)**(1/4)*a**12*(a**(-2))
**(37/4)*acot(a*x**2)/(-6*(-1)**(1/4)*a**14*x**7*(a**(-2))**(37/4) - 6*(-1)**(1/4)*a**12*x**3*(a**(-2))**(37/4
)) - 2*I*a**8*x**7*(a**(-2))**(11/2)*acot(a*x**2)/(-6*(-1)**(1/4)*a**14*x**7*(a**(-2))**(37/4) - 6*(-1)**(1/4)
*a**12*x**3*(a**(-2))**(37/4)) - 2*I*a**6*x**3*(a**(-2))**(11/2)*acot(a*x**2)/(-6*(-1)**(1/4)*a**14*x**7*(a**(
-2))**(37/4) - 6*(-1)**(1/4)*a**12*x**3*(a**(-2))**(37/4)) - 2*x**7*log(x - (-1)**(1/4)*(a**(-2))**(1/4))/(-6*
(-1)**(1/4)*a**17*x**7*(a**(-2))**(37/4) - 6*(-1)**(1/4)*a**15*x**3*(a**(-2))**(37/4)) + x**7*log(x**2 + I*sqr
t(a**(-2)))/(-6*(-1)**(1/4)*a**17*x**7*(a**(-2))**(37/4) - 6*(-1)**(1/4)*a**15*x**3*(a**(-2))**(37/4)) + 2*x**
7*atan((-1)**(3/4)*x/(a**(-2))**(1/4))/(-6*(-1)**(1/4)*a**17*x**7*(a**(-2))**(37/4) - 6*(-1)**(1/4)*a**15*x**3
*(a**(-2))**(37/4)) - 2*x**3*log(x - (-1)**(1/4)*(a**(-2))**(1/4))/(-6*(-1)**(1/4)*a**19*x**7*(a**(-2))**(37/4
) - 6*(-1)**(1/4)*a**17*x**3*(a**(-2))**(37/4)) + x**3*log(x**2 + I*sqrt(a**(-2)))/(-6*(-1)**(1/4)*a**19*x**7*
(a**(-2))**(37/4) - 6*(-1)**(1/4)*a**17*x**3*(a**(-2))**(37/4)) + 2*x**3*atan((-1)**(3/4)*x/(a**(-2))**(1/4))/
(-6*(-1)**(1/4)*a**19*x**7*(a**(-2))**(37/4) - 6*(-1)**(1/4)*a**17*x**3*(a**(-2))**(37/4)), True))

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Giac [A]  time = 1.10934, size = 189, normalized size = 1.26 \begin{align*} \frac{1}{12} \,{\left (2 \, \sqrt{2} \sqrt{{\left | a \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \frac{\sqrt{2}}{\sqrt{{\left | a \right |}}}\right )} \sqrt{{\left | a \right |}}\right ) + 2 \, \sqrt{2} \sqrt{{\left | a \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \frac{\sqrt{2}}{\sqrt{{\left | a \right |}}}\right )} \sqrt{{\left | a \right |}}\right ) - \sqrt{2} \sqrt{{\left | a \right |}} \log \left (x^{2} + \frac{\sqrt{2} x}{\sqrt{{\left | a \right |}}} + \frac{1}{{\left | a \right |}}\right ) + \sqrt{2} \sqrt{{\left | a \right |}} \log \left (x^{2} - \frac{\sqrt{2} x}{\sqrt{{\left | a \right |}}} + \frac{1}{{\left | a \right |}}\right ) + \frac{8}{x}\right )} a - \frac{\arctan \left (\frac{1}{a x^{2}}\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^2)/x^4,x, algorithm="giac")

[Out]

1/12*(2*sqrt(2)*sqrt(abs(a))*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(a)))*sqrt(abs(a))) + 2*sqrt(2)*sqrt(ab
s(a))*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(a)))*sqrt(abs(a))) - sqrt(2)*sqrt(abs(a))*log(x^2 + sqrt(2)*x
/sqrt(abs(a)) + 1/abs(a)) + sqrt(2)*sqrt(abs(a))*log(x^2 - sqrt(2)*x/sqrt(abs(a)) + 1/abs(a)) + 8/x)*a - 1/3*a
rctan(1/(a*x^2))/x^3