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]

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

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Rubi [A]  time = 0.09, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, 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 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 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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [B]  time = 1.04, size = 269, normalized size = 1.79 \[ -\frac {4 \, \sqrt {2} {\left (a^{6}\right )}^{\frac {1}{4}} x^{3} \arctan \left (-\frac {\sqrt {2} {\left (a^{6}\right )}^{\frac {1}{4}} a^{5} x + a^{6} - \sqrt {2} \sqrt {a^{10} x^{2} + \sqrt {2} {\left (a^{6}\right )}^{\frac {3}{4}} a^{5} x + \sqrt {a^{6}} a^{6}} {\left (a^{6}\right )}^{\frac {1}{4}}}{a^{6}}\right ) + 4 \, \sqrt {2} {\left (a^{6}\right )}^{\frac {1}{4}} x^{3} \arctan \left (-\frac {\sqrt {2} {\left (a^{6}\right )}^{\frac {1}{4}} a^{5} x - a^{6} - \sqrt {2} \sqrt {a^{10} x^{2} - \sqrt {2} {\left (a^{6}\right )}^{\frac {3}{4}} a^{5} x + \sqrt {a^{6}} a^{6}} {\left (a^{6}\right )}^{\frac {1}{4}}}{a^{6}}\right ) + \sqrt {2} {\left (a^{6}\right )}^{\frac {1}{4}} x^{3} \log \left (a^{10} x^{2} + \sqrt {2} {\left (a^{6}\right )}^{\frac {3}{4}} a^{5} x + \sqrt {a^{6}} a^{6}\right ) - \sqrt {2} {\left (a^{6}\right )}^{\frac {1}{4}} x^{3} \log \left (a^{10} x^{2} - \sqrt {2} {\left (a^{6}\right )}^{\frac {3}{4}} a^{5} x + \sqrt {a^{6}} a^{6}\right ) - 8 \, a x^{2} + 4 \, \arctan \left (\frac {1}{a x^{2}}\right )}{12 \, x^{3}} \]

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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giac [A]  time = 0.13, size = 149, normalized size = 0.99 \[ \frac {1}{12} \, {\left (\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{{\left | a \right |}^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | a \right |}}}\right )} \sqrt {{\left | a \right |}}\right )}{{\left | a \right |}^{\frac {3}{2}}} - \sqrt {2} \sqrt {{\left | a \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right ) + \frac {\sqrt {2} a^{2} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | a \right |}}} + \frac {1}{{\left | a \right |}}\right )}{{\left | a \right |}^{\frac {3}{2}}} + \frac {8}{x}\right )} a - \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{3 \, x^{3}} \]

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

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maple [A]  time = 0.04, size = 121, normalized size = 0.81 \[ -\frac {\mathrm {arccot}\left (a \,x^{2}\right )}{3 x^{3}}+\frac {a \sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )}{12 \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+\frac {a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )}{6 \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+\frac {a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )}{6 \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+\frac {2 a}{3 x} \]

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 [A]  time = 0.41, size = 133, normalized size = 0.89 \[ \frac {1}{12} \, {\left (a^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x + \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, a x - \sqrt {2} \sqrt {a}\right )}}{2 \, \sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (a x^{2} + \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (a x^{2} - \sqrt {2} \sqrt {a} x + 1\right )}{a^{\frac {3}{2}}}\right )} + \frac {8}{x}\right )} a - \frac {\operatorname {arccot}\left (a x^{2}\right )}{3 \, x^{3}} \]

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*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*a*x + sqrt(2)*sqrt(a))/sqrt(a))/a^(3/2) + 2*sqrt(2)*arctan(1/2*sqrt
(2)*(2*a*x - sqrt(2)*sqrt(a))/sqrt(a))/a^(3/2) - sqrt(2)*log(a*x^2 + sqrt(2)*sqrt(a)*x + 1)/a^(3/2) + sqrt(2)*
log(a*x^2 - sqrt(2)*sqrt(a)*x + 1)/a^(3/2)) + 8/x)*a - 1/3*arccot(a*x^2)/x^3

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mupad [B]  time = 0.71, size = 52, normalized size = 0.35 \[ \frac {2\,a}{3\,x}-\frac {\mathrm {acot}\left (a\,x^2\right )}{3\,x^3}+\frac {{\left (-1\right )}^{1/4}\,a^{3/2}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\right )}{3}+\frac {{\left (-1\right )}^{1/4}\,a^{3/2}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {a}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a)/(3*x) - acot(a*x^2)/(3*x^3) + ((-1)^(1/4)*a^(3/2)*atan((-1)^(1/4)*a^(1/2)*x))/3 + ((-1)^(1/4)*a^(3/2)*at
an((-1)^(1/4)*a^(1/2)*x*1i)*1i)/3

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sympy [A]  time = 37.99, size = 162, normalized size = 1.08 \[ \begin {cases} \frac {\sqrt [4]{-1} a^{2} \sqrt [4]{\frac {1}{a^{2}}} \operatorname {acot}{\left (a x^{2} \right )}}{3} - \frac {\left (-1\right )^{\frac {3}{4}} a \log {\left (x - \sqrt [4]{-1} \sqrt [4]{\frac {1}{a^{2}}} \right )}}{3 \sqrt [4]{\frac {1}{a^{2}}}} + \frac {\left (-1\right )^{\frac {3}{4}} a \log {\left (x^{2} + i \sqrt {\frac {1}{a^{2}}} \right )}}{6 \sqrt [4]{\frac {1}{a^{2}}}} + \frac {\left (-1\right )^{\frac {3}{4}} a \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} x}{\sqrt [4]{\frac {1}{a^{2}}}} \right )}}{3 \sqrt [4]{\frac {1}{a^{2}}}} + \frac {2 a}{3 x} - \frac {\operatorname {acot}{\left (a x^{2} \right )}}{3 x^{3}} & \text {for}\: a \neq 0 \\- \frac {\pi }{6 x^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-1)**(1/4)*a**2*(a**(-2))**(1/4)*acot(a*x**2)/3 - (-1)**(3/4)*a*log(x - (-1)**(1/4)*(a**(-2))**(1/
4))/(3*(a**(-2))**(1/4)) + (-1)**(3/4)*a*log(x**2 + I*sqrt(a**(-2)))/(6*(a**(-2))**(1/4)) + (-1)**(3/4)*a*atan
((-1)**(3/4)*x/(a**(-2))**(1/4))/(3*(a**(-2))**(1/4)) + 2*a/(3*x) - acot(a*x**2)/(3*x**3), Ne(a, 0)), (-pi/(6*
x**3), True))

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