3.1.84 \(\int \frac {\cot ^{-1}(a x^2)}{x^4} \, dx\) [84]

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

[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 = 0.07, 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 = {4947, 331, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {a^{3/2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {a} x\right )}{3 \sqrt {2}}+\frac {a^{3/2} \text {ArcTan}\left (\sqrt {2} \sqrt {a} x+1\right )}{3 \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} \log \left (a x^2+\sqrt {2} \sqrt {a} x+1\right )}{6 \sqrt {2}}-\frac {\cot ^{-1}\left (a x^2\right )}{3 x^3}+\frac {2 a}{3 x} \end {gather*}

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

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 331

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 631

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 642

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 1176

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 1179

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 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && 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} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {a} x\right )}{3 \sqrt {2}}-\frac {a^{3/2} \text {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 = 0.04, size = 146, normalized size = 0.97 \begin {gather*} \frac {-4 \cot ^{-1}\left (a x^2\right )+a x^2 \left (8-2 \sqrt {2} \sqrt {a} x \text {ArcTan}\left (1-\sqrt {2} \sqrt {a} x\right )+2 \sqrt {2} \sqrt {a} x \text {ArcTan}\left (1+\sqrt {2} \sqrt {a} x\right )+\sqrt {2} \sqrt {a} x \log \left (1-\sqrt {2} \sqrt {a} x+a x^2\right )-\sqrt {2} \sqrt {a} x \log \left (1+\sqrt {2} \sqrt {a} x+a x^2\right )\right )}{12 x^3} \end {gather*}

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 = 0.07, size = 106, normalized size = 0.71

method	result	size
default	\(-\frac {\mathrm {arccot}\left (a \,x^{2}\right )}{3 x^{3}}-\frac {2 a \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}{x^{2}+\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{a^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {1}{a^{2}}\right )^{\frac {1}{4}}}-\frac {1}{x}\right )}{3}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*arccot(a*x^2)/x^3-2/3*a*(-1/8/(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)))+2*arctan(2^(1/2)/(1/a^2)^(1/4)*x+1)+2*arctan(2^(1/2)/(1/a^2)^(1/4)*x-1
))-1/x)

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Maxima [A]
time = 0.46, size = 133, normalized size = 0.89 \begin {gather*} \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}} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (105) = 210\).
time = 3.48, size = 269, normalized size = 1.79 \begin {gather*} -\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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 16.95, size = 459, normalized size = 3.06 \begin {gather*} \begin {cases} - \frac {\pi }{6 x^{3}} & \text {for}\: a = 0 \\- \frac {\infty i}{x^{3}} & \text {for}\: a = - \frac {i}{x^{2}} \\\frac {\infty i}{x^{3}} & \text {for}\: a = \frac {i}{x^{2}} \\- \frac {2 a^{3} x^{7} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{4}} \log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} + \frac {a^{3} x^{7} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{4}} \log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} - \frac {2 a^{3} x^{7} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} + \frac {2 a^{2} x^{7} \sqrt [4]{- \frac {1}{a^{2}}} \operatorname {acot}{\left (a x^{2} \right )}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} + \frac {4 a x^{6}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} - \frac {2 a x^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{4}} \log {\left (x - \sqrt [4]{- \frac {1}{a^{2}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} + \frac {a x^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{4}} \log {\left (x^{2} + \sqrt {- \frac {1}{a^{2}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} - \frac {2 a x^{3} \left (- \frac {1}{a^{2}}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {x}{\sqrt [4]{- \frac {1}{a^{2}}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} - \frac {2 x^{4} \operatorname {acot}{\left (a x^{2} \right )}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} + \frac {2 x^{3} \sqrt [4]{- \frac {1}{a^{2}}} \operatorname {acot}{\left (a x^{2} \right )}}{6 x^{7} + \frac {6 x^{3}}{a^{2}}} + \frac {4 x^{2}}{6 a x^{7} + \frac {6 x^{3}}{a}} - \frac {2 \operatorname {acot}{\left (a x^{2} \right )}}{6 a^{2} x^{7} + 6 x^{3}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-pi/(6*x**3), Eq(a, 0)), (-oo*I/x**3, Eq(a, -I/x**2)), (oo*I/x**3, Eq(a, I/x**2)), (-2*a**3*x**7*(-
1/a**2)**(3/4)*log(x - (-1/a**2)**(1/4))/(6*x**7 + 6*x**3/a**2) + a**3*x**7*(-1/a**2)**(3/4)*log(x**2 + sqrt(-
1/a**2))/(6*x**7 + 6*x**3/a**2) - 2*a**3*x**7*(-1/a**2)**(3/4)*atan(x/(-1/a**2)**(1/4))/(6*x**7 + 6*x**3/a**2)
+ 2*a**2*x**7*(-1/a**2)**(1/4)*acot(a*x**2)/(6*x**7 + 6*x**3/a**2) + 4*a*x**6/(6*x**7 + 6*x**3/a**2) - 2*a*x*
*3*(-1/a**2)**(3/4)*log(x - (-1/a**2)**(1/4))/(6*x**7 + 6*x**3/a**2) + a*x**3*(-1/a**2)**(3/4)*log(x**2 + sqrt
(-1/a**2))/(6*x**7 + 6*x**3/a**2) - 2*a*x**3*(-1/a**2)**(3/4)*atan(x/(-1/a**2)**(1/4))/(6*x**7 + 6*x**3/a**2)
- 2*x**4*acot(a*x**2)/(6*x**7 + 6*x**3/a**2) + 2*x**3*(-1/a**2)**(1/4)*acot(a*x**2)/(6*x**7 + 6*x**3/a**2) + 4
*x**2/(6*a*x**7 + 6*x**3/a) - 2*acot(a*x**2)/(6*a**2*x**7 + 6*x**3), True))

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Giac [A]
time = 0.42, size = 149, normalized size = 0.99 \begin {gather*} \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}} \end {gather*}

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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Mupad [B]
time = 0.71, size = 52, normalized size = 0.35 \begin {gather*} \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} \end {gather*}

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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