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

Optimal. Leaf size=152 \[ -\frac{\log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{10 \sqrt{2} a^{5/2}}+\frac{\log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{10 \sqrt{2} a^{5/2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{5 \sqrt{2} a^{5/2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{5 \sqrt{2} a^{5/2}}+\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left (a x^2\right ) \]

[Out]

(2*x^3)/(15*a) + (x^5*ArcCot[a*x^2])/5 + ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(5*Sqrt[2]*a^(5/2)) - ArcTan[1 + Sqrt[2
]*Sqrt[a]*x]/(5*Sqrt[2]*a^(5/2)) - Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(10*Sqrt[2]*a^(5/2)) + Log[1 + Sqrt[2]*S
qrt[a]*x + a*x^2]/(10*Sqrt[2]*a^(5/2))

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Rubi [A]  time = 0.104254, antiderivative size = 152, 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, 321, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )}{10 \sqrt{2} a^{5/2}}+\frac{\log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )}{10 \sqrt{2} a^{5/2}}+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{5 \sqrt{2} a^{5/2}}-\frac{\tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{5 \sqrt{2} a^{5/2}}+\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left (a x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^3)/(15*a) + (x^5*ArcCot[a*x^2])/5 + ArcTan[1 - Sqrt[2]*Sqrt[a]*x]/(5*Sqrt[2]*a^(5/2)) - ArcTan[1 + Sqrt[2
]*Sqrt[a]*x]/(5*Sqrt[2]*a^(5/2)) - Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2]/(10*Sqrt[2]*a^(5/2)) + Log[1 + Sqrt[2]*S
qrt[a]*x + a*x^2]/(10*Sqrt[2]*a^(5/2))

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 x^4 \cot ^{-1}\left (a x^2\right ) \, dx &=\frac{1}{5} x^5 \cot ^{-1}\left (a x^2\right )+\frac{1}{5} (2 a) \int \frac{x^6}{1+a^2 x^4} \, dx\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left (a x^2\right )-\frac{2 \int \frac{x^2}{1+a^2 x^4} \, dx}{5 a}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left (a x^2\right )+\frac{\int \frac{1-a x^2}{1+a^2 x^4} \, dx}{5 a^2}-\frac{\int \frac{1+a x^2}{1+a^2 x^4} \, dx}{5 a^2}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left (a x^2\right )-\frac{\int \frac{1}{\frac{1}{a}-\frac{\sqrt{2} x}{\sqrt{a}}+x^2} \, dx}{10 a^3}-\frac{\int \frac{1}{\frac{1}{a}+\frac{\sqrt{2} x}{\sqrt{a}}+x^2} \, dx}{10 a^3}-\frac{\int \frac{\frac{\sqrt{2}}{\sqrt{a}}+2 x}{-\frac{1}{a}-\frac{\sqrt{2} x}{\sqrt{a}}-x^2} \, dx}{10 \sqrt{2} a^{5/2}}-\frac{\int \frac{\frac{\sqrt{2}}{\sqrt{a}}-2 x}{-\frac{1}{a}+\frac{\sqrt{2} x}{\sqrt{a}}-x^2} \, dx}{10 \sqrt{2} a^{5/2}}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left (a x^2\right )-\frac{\log \left (1-\sqrt{2} \sqrt{a} x+a x^2\right )}{10 \sqrt{2} a^{5/2}}+\frac{\log \left (1+\sqrt{2} \sqrt{a} x+a x^2\right )}{10 \sqrt{2} a^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{a} x\right )}{5 \sqrt{2} a^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{a} x\right )}{5 \sqrt{2} a^{5/2}}\\ &=\frac{2 x^3}{15 a}+\frac{1}{5} x^5 \cot ^{-1}\left (a x^2\right )+\frac{\tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )}{5 \sqrt{2} a^{5/2}}-\frac{\tan ^{-1}\left (1+\sqrt{2} \sqrt{a} x\right )}{5 \sqrt{2} a^{5/2}}-\frac{\log \left (1-\sqrt{2} \sqrt{a} x+a x^2\right )}{10 \sqrt{2} a^{5/2}}+\frac{\log \left (1+\sqrt{2} \sqrt{a} x+a x^2\right )}{10 \sqrt{2} a^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0414145, size = 136, normalized size = 0.89 \[ \frac{8 a^{3/2} x^3+12 a^{5/2} x^5 \cot ^{-1}\left (a x^2\right )-3 \sqrt{2} \log \left (a x^2-\sqrt{2} \sqrt{a} x+1\right )+3 \sqrt{2} \log \left (a x^2+\sqrt{2} \sqrt{a} x+1\right )+6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{a} x\right )-6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{a} x+1\right )}{60 a^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a^(3/2)*x^3 + 12*a^(5/2)*x^5*ArcCot[a*x^2] + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[a]*x] - 6*Sqrt[2]*ArcTan[1 +
Sqrt[2]*Sqrt[a]*x] - 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[a]*x + a*x^2] + 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[a]*x + a*x
^2])/(60*a^(5/2))

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Maple [A]  time = 0.061, size = 129, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}{\rm arccot} \left (a{x}^{2}\right )}{5}}+{\frac{2\,{x}^{3}}{15\,a}}-{\frac{\sqrt{2}}{20\,{a}^{3}}\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{\sqrt{2}}{10\,{a}^{3}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{-2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{a}^{-2}}}}}-{\frac{\sqrt{2}}{10\,{a}^{3}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{a}^{-2}}}}}-1 \right ){\frac{1}{\sqrt [4]{{a}^{-2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arccot(a*x^2)+2/15*x^3/a-1/20/a^3/(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)))-1/10/a^3/(1/a^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/a^2)^(1/4)*x+1)
-1/10/a^3/(1/a^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/a^2)^(1/4)*x-1)

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Maxima [B]  time = 1.45523, size = 362, normalized size = 2.38 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arccot}\left (a x^{2}\right ) + \frac{1}{60} \, a{\left (\frac{8 \, x^{3}}{a^{2}} + \frac{3 \,{\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 )}}{a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arccot(a*x^2) + 1/60*a*(8*x^3/a^2 + 3*(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 - sqrt(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)))/(s
qrt(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*s
qrt(a^2)*x + sqrt(2)*sqrt(-sqrt(a^2)) - sqrt(2)*(a^2)^(1/4)))/(sqrt(a^2)*sqrt(-sqrt(a^2))))/a^2)

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Fricas [B]  time = 2.36893, size = 757, normalized size = 4.98 \begin{align*} \frac{12 \, a x^{5} \arctan \left (\frac{1}{a x^{2}}\right ) + 8 \, x^{3} + 12 \, \sqrt{2} a \frac{1}{a^{10}}^{\frac{1}{4}} \arctan \left (-\sqrt{2} a^{3} \frac{1}{a^{10}}^{\frac{1}{4}} x + \sqrt{2} \sqrt{\sqrt{2} a^{7} \frac{1}{a^{10}}^{\frac{3}{4}} x + a^{4} \sqrt{\frac{1}{a^{10}}} + x^{2}} a^{3} \frac{1}{a^{10}}^{\frac{1}{4}} - 1\right ) + 12 \, \sqrt{2} a \frac{1}{a^{10}}^{\frac{1}{4}} \arctan \left (-\sqrt{2} a^{3} \frac{1}{a^{10}}^{\frac{1}{4}} x + \sqrt{2} \sqrt{-\sqrt{2} a^{7} \frac{1}{a^{10}}^{\frac{3}{4}} x + a^{4} \sqrt{\frac{1}{a^{10}}} + x^{2}} a^{3} \frac{1}{a^{10}}^{\frac{1}{4}} + 1\right ) + 3 \, \sqrt{2} a \frac{1}{a^{10}}^{\frac{1}{4}} \log \left (\sqrt{2} a^{7} \frac{1}{a^{10}}^{\frac{3}{4}} x + a^{4} \sqrt{\frac{1}{a^{10}}} + x^{2}\right ) - 3 \, \sqrt{2} a \frac{1}{a^{10}}^{\frac{1}{4}} \log \left (-\sqrt{2} a^{7} \frac{1}{a^{10}}^{\frac{3}{4}} x + a^{4} \sqrt{\frac{1}{a^{10}}} + x^{2}\right )}{60 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*a*x^5*arctan(1/(a*x^2)) + 8*x^3 + 12*sqrt(2)*a*(a^(-10))^(1/4)*arctan(-sqrt(2)*a^3*(a^(-10))^(1/4)*x
+ sqrt(2)*sqrt(sqrt(2)*a^7*(a^(-10))^(3/4)*x + a^4*sqrt(a^(-10)) + x^2)*a^3*(a^(-10))^(1/4) - 1) + 12*sqrt(2)*
a*(a^(-10))^(1/4)*arctan(-sqrt(2)*a^3*(a^(-10))^(1/4)*x + sqrt(2)*sqrt(-sqrt(2)*a^7*(a^(-10))^(3/4)*x + a^4*sq
rt(a^(-10)) + x^2)*a^3*(a^(-10))^(1/4) + 1) + 3*sqrt(2)*a*(a^(-10))^(1/4)*log(sqrt(2)*a^7*(a^(-10))^(3/4)*x +
a^4*sqrt(a^(-10)) + x^2) - 3*sqrt(2)*a*(a^(-10))^(1/4)*log(-sqrt(2)*a^7*(a^(-10))^(3/4)*x + a^4*sqrt(a^(-10))
+ x^2))/a

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Sympy [A]  time = 58.1177, size = 1086, normalized size = 7.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x**5*acot((-sqrt(2)/2 - sqrt(2)*I/2)**(-2))/5, Eq(a, -1/(x**2*(-sqrt(2)/2 - sqrt(2)*I/2)**2))), (-
x**5*acot((-sqrt(2)/2 + sqrt(2)*I/2)**(-2))/5, Eq(a, -1/(x**2*(-sqrt(2)/2 + sqrt(2)*I/2)**2))), (-x**5*acot((s
qrt(2)/2 - sqrt(2)*I/2)**(-2))/5, Eq(a, -1/(x**2*(sqrt(2)/2 - sqrt(2)*I/2)**2))), (-x**5*acot((sqrt(2)/2 + sqr
t(2)*I/2)**(-2))/5, Eq(a, -1/(x**2*(sqrt(2)/2 + sqrt(2)*I/2)**2))), (pi*x**5/10, Eq(a, 0)), (-6*(-1)**(3/4)*a*
*10*x**9*(a**(-2))**(15/4)*acot(a*x**2)/(-30*(-1)**(3/4)*a**10*x**4*(a**(-2))**(15/4) - 30*(-1)**(3/4)*a**8*(a
**(-2))**(15/4)) - 4*(-1)**(3/4)*a**9*x**7*(a**(-2))**(15/4)/(-30*(-1)**(3/4)*a**10*x**4*(a**(-2))**(15/4) - 3
0*(-1)**(3/4)*a**8*(a**(-2))**(15/4)) - 6*(-1)**(3/4)*a**8*x**5*(a**(-2))**(15/4)*acot(a*x**2)/(-30*(-1)**(3/4
)*a**10*x**4*(a**(-2))**(15/4) - 30*(-1)**(3/4)*a**8*(a**(-2))**(15/4)) - 4*(-1)**(3/4)*a**7*x**3*(a**(-2))**(
15/4)/(-30*(-1)**(3/4)*a**10*x**4*(a**(-2))**(15/4) - 30*(-1)**(3/4)*a**8*(a**(-2))**(15/4)) + 6*I*a**5*x**4*(
a**(-2))**(5/2)*log(x - (-1)**(1/4)*(a**(-2))**(1/4))/(-30*(-1)**(3/4)*a**10*x**4*(a**(-2))**(15/4) - 30*(-1)*
*(3/4)*a**8*(a**(-2))**(15/4)) - 3*I*a**5*x**4*(a**(-2))**(5/2)*log(x**2 + I*sqrt(a**(-2)))/(-30*(-1)**(3/4)*a
**10*x**4*(a**(-2))**(15/4) - 30*(-1)**(3/4)*a**8*(a**(-2))**(15/4)) - 6*I*a**5*x**4*(a**(-2))**(5/2)*atan((-1
)**(3/4)*x/(a**(-2))**(1/4))/(-30*(-1)**(3/4)*a**10*x**4*(a**(-2))**(15/4) - 30*(-1)**(3/4)*a**8*(a**(-2))**(1
5/4)) + 6*I*a**3*(a**(-2))**(5/2)*log(x - (-1)**(1/4)*(a**(-2))**(1/4))/(-30*(-1)**(3/4)*a**10*x**4*(a**(-2))*
*(15/4) - 30*(-1)**(3/4)*a**8*(a**(-2))**(15/4)) - 3*I*a**3*(a**(-2))**(5/2)*log(x**2 + I*sqrt(a**(-2)))/(-30*
(-1)**(3/4)*a**10*x**4*(a**(-2))**(15/4) - 30*(-1)**(3/4)*a**8*(a**(-2))**(15/4)) - 6*I*a**3*(a**(-2))**(5/2)*
atan((-1)**(3/4)*x/(a**(-2))**(1/4))/(-30*(-1)**(3/4)*a**10*x**4*(a**(-2))**(15/4) - 30*(-1)**(3/4)*a**8*(a**(
-2))**(15/4)) - 6*x**4*acot(a*x**2)/(-30*(-1)**(3/4)*a**10*x**4*(a**(-2))**(15/4) - 30*(-1)**(3/4)*a**8*(a**(-
2))**(15/4)) - 6*acot(a*x**2)/(-30*(-1)**(3/4)*a**12*x**4*(a**(-2))**(15/4) - 30*(-1)**(3/4)*a**10*(a**(-2))**
(15/4)), True))

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Giac [A]  time = 1.1401, size = 211, normalized size = 1.39 \begin{align*} \frac{1}{5} \, x^{5} \arctan \left (\frac{1}{a x^{2}}\right ) + \frac{1}{60} \, a{\left (\frac{8 \, x^{3}}{a^{2}} - \frac{6 \, \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 )}{a^{4}} - \frac{6 \, \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 )}{a^{4}} + \frac{3 \, \sqrt{2} \sqrt{{\left | a \right |}} \log \left (x^{2} + \frac{\sqrt{2} x}{\sqrt{{\left | a \right |}}} + \frac{1}{{\left | a \right |}}\right )}{a^{4}} - \frac{3 \, \sqrt{2} \sqrt{{\left | a \right |}} \log \left (x^{2} - \frac{\sqrt{2} x}{\sqrt{{\left | a \right |}}} + \frac{1}{{\left | a \right |}}\right )}{a^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arctan(1/(a*x^2)) + 1/60*a*(8*x^3/a^2 - 6*sqrt(2)*sqrt(abs(a))*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(
abs(a)))*sqrt(abs(a)))/a^4 - 6*sqrt(2)*sqrt(abs(a))*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(a)))*sqrt(abs(a
)))/a^4 + 3*sqrt(2)*sqrt(abs(a))*log(x^2 + sqrt(2)*x/sqrt(abs(a)) + 1/abs(a))/a^4 - 3*sqrt(2)*sqrt(abs(a))*log
(x^2 - sqrt(2)*x/sqrt(abs(a)) + 1/abs(a))/a^4)