3.2.91 \(\int \frac {\text {ArcTan}(a x)}{x^4 (c+a^2 c x^2)^2} \, dx\) [191]

Optimal. Leaf size=136 \[ -\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\text {ArcTan}(a x)}{3 c^2 x^3}+\frac {2 a^2 \text {ArcTan}(a x)}{c^2 x}+\frac {a^4 x \text {ArcTan}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {5 a^3 \text {ArcTan}(a x)^2}{4 c^2}-\frac {7 a^3 \log (x)}{3 c^2}+\frac {7 a^3 \log \left (1+a^2 x^2\right )}{6 c^2} \]

[Out]

-1/6*a/c^2/x^2+1/4*a^3/c^2/(a^2*x^2+1)-1/3*arctan(a*x)/c^2/x^3+2*a^2*arctan(a*x)/c^2/x+1/2*a^4*x*arctan(a*x)/c
^2/(a^2*x^2+1)+5/4*a^3*arctan(a*x)^2/c^2-7/3*a^3*ln(x)/c^2+7/6*a^3*ln(a^2*x^2+1)/c^2

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5086, 5038, 4946, 272, 46, 36, 29, 31, 5004, 5012, 267} \begin {gather*} \frac {5 a^3 \text {ArcTan}(a x)^2}{4 c^2}-\frac {7 a^3 \log (x)}{3 c^2}+\frac {2 a^2 \text {ArcTan}(a x)}{c^2 x}+\frac {a^4 x \text {ArcTan}(a x)}{2 c^2 \left (a^2 x^2+1\right )}+\frac {a^3}{4 c^2 \left (a^2 x^2+1\right )}+\frac {7 a^3 \log \left (a^2 x^2+1\right )}{6 c^2}-\frac {\text {ArcTan}(a x)}{3 c^2 x^3}-\frac {a}{6 c^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^2),x]

[Out]

-1/6*a/(c^2*x^2) + a^3/(4*c^2*(1 + a^2*x^2)) - ArcTan[a*x]/(3*c^2*x^3) + (2*a^2*ArcTan[a*x])/(c^2*x) + (a^4*x*
ArcTan[a*x])/(2*c^2*(1 + a^2*x^2)) + (5*a^3*ArcTan[a*x]^2)/(4*c^2) - (7*a^3*Log[x])/(3*c^2) + (7*a^3*Log[1 + a
^2*x^2])/(6*c^2)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[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]

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 5012

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Simp[x*((a + b*ArcTan[c*x])
^p/(2*d*(d + e*x^2))), x] + (-Dist[b*c*(p/2), Int[x*((a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x] + Simp
[(a + b*ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p,
0]

Rule 5038

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f^2), Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x
^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rule 5086

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int[
x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] &
& NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=a^4 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {\int \frac {\tan ^{-1}(a x)}{x^4} \, dx}{c^2}-2 \frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)^2}{4 c^2}-\frac {1}{2} a^5 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx+\frac {a \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^2}-2 \left (\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2} \, dx}{c^2}-\frac {a^4 \int \frac {\tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{c}\right )\\ &=\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)^2}{4 c^2}+\frac {a \text {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )}{6 c^2}-2 \left (-\frac {a^2 \tan ^{-1}(a x)}{c^2 x}-\frac {a^3 \tan ^{-1}(a x)^2}{2 c^2}+\frac {a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c^2}\right )\\ &=\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)^2}{4 c^2}+\frac {a \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {a^2}{x}+\frac {a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )}{6 c^2}-2 \left (-\frac {a^2 \tan ^{-1}(a x)}{c^2 x}-\frac {a^3 \tan ^{-1}(a x)^2}{2 c^2}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^2}\right )\\ &=-\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)^2}{4 c^2}-\frac {a^3 \log (x)}{3 c^2}+\frac {a^3 \log \left (1+a^2 x^2\right )}{6 c^2}-2 \left (-\frac {a^2 \tan ^{-1}(a x)}{c^2 x}-\frac {a^3 \tan ^{-1}(a x)^2}{2 c^2}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c^2}-\frac {a^5 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^2}\right )\\ &=-\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\tan ^{-1}(a x)}{3 c^2 x^3}+\frac {a^4 x \tan ^{-1}(a x)}{2 c^2 \left (1+a^2 x^2\right )}+\frac {a^3 \tan ^{-1}(a x)^2}{4 c^2}-\frac {a^3 \log (x)}{3 c^2}+\frac {a^3 \log \left (1+a^2 x^2\right )}{6 c^2}-2 \left (-\frac {a^2 \tan ^{-1}(a x)}{c^2 x}-\frac {a^3 \tan ^{-1}(a x)^2}{2 c^2}+\frac {a^3 \log (x)}{c^2}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c^2}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 124, normalized size = 0.91 \begin {gather*} -\frac {a}{6 c^2 x^2}+\frac {a^3}{4 c^2 \left (1+a^2 x^2\right )}+\frac {\left (-2+10 a^2 x^2+15 a^4 x^4\right ) \text {ArcTan}(a x)}{6 c^2 x^3 \left (1+a^2 x^2\right )}+\frac {5 a^3 \text {ArcTan}(a x)^2}{4 c^2}-\frac {7 a^3 \log (x)}{3 c^2}+\frac {7 a^3 \log \left (1+a^2 x^2\right )}{6 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^2),x]

[Out]

-1/6*a/(c^2*x^2) + a^3/(4*c^2*(1 + a^2*x^2)) + ((-2 + 10*a^2*x^2 + 15*a^4*x^4)*ArcTan[a*x])/(6*c^2*x^3*(1 + a^
2*x^2)) + (5*a^3*ArcTan[a*x]^2)/(4*c^2) - (7*a^3*Log[x])/(3*c^2) + (7*a^3*Log[1 + a^2*x^2])/(6*c^2)

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 121, normalized size = 0.89

method	result	size
derivativedivides	\(a^{3} \left (\frac {a x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )}{c^{2} a x}-\frac {-\frac {3}{2 \left (a^{2} x^{2}+1\right )}-7 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{a^{2} x^{2}}+14 \ln \left (a x \right )+\frac {15 \arctan \left (a x \right )^{2}}{2}}{6 c^{2}}\right )\)	\(121\)
default	\(a^{3} \left (\frac {a x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {5 \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{3 c^{2} a^{3} x^{3}}+\frac {2 \arctan \left (a x \right )}{c^{2} a x}-\frac {-\frac {3}{2 \left (a^{2} x^{2}+1\right )}-7 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{a^{2} x^{2}}+14 \ln \left (a x \right )+\frac {15 \arctan \left (a x \right )^{2}}{2}}{6 c^{2}}\right )\)	\(121\)
risch	\(-\frac {i a^{4} \ln \left (-i a x +1\right ) x}{16 c^{2} \left (-i a x -1\right )}-\frac {i \ln \left (-i a x +1\right )}{6 c^{2} x^{3}}+\frac {i \ln \left (i a x +1\right )}{6 c^{2} x^{3}}-\frac {a}{6 c^{2} x^{2}}+\frac {a^{3} \ln \left (i a x +1\right )}{8 c^{2} \left (i a x +1\right )}+\frac {5 a^{3} \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{8 c^{2}}+\frac {a^{3} \ln \left (-i a x +1\right )}{16 c^{2} \left (-i a x -1\right )}+\frac {a^{3} \ln \left (-i a x +1\right )}{8 c^{2} \left (-i a x +1\right )}-\frac {5 a^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c^{2}}+\frac {5 a^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{8 c^{2}}+\frac {a^{3} \ln \left (i a x +1\right )}{16 c^{2} \left (i a x -1\right )}+\frac {a^{3}}{8 c^{2} \left (i a x +1\right )}-\frac {5 a^{3} \dilog \left (\frac {1}{2}+\frac {i a x}{2}\right )}{8 c^{2}}-\frac {7 a^{3} \ln \left (i a x \right )}{6 c^{2}}-\frac {5 a^{3} \ln \left (i a x +1\right )^{2}}{16 c^{2}}+\frac {a^{3}}{8 c^{2} \left (-i a x +1\right )}-\frac {5 a^{3} \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{8 c^{2}}-\frac {7 a^{3} \ln \left (-i a x \right )}{6 c^{2}}-\frac {5 a^{3} \ln \left (-i a x +1\right )^{2}}{16 c^{2}}+\frac {i a^{4} \ln \left (i a x +1\right ) x}{16 c^{2} \left (i a x -1\right )}-\frac {i a^{2} \ln \left (i a x +1\right )}{c^{2} x}+\frac {i a^{2} \ln \left (-i a x +1\right )}{c^{2} x}+\frac {53 a^{3} \ln \left (a^{2} x^{2}+1\right )}{48 c^{2}}\)	\(459\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)/x^4/(a^2*c*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

a^3*(1/2*a*x*arctan(a*x)/c^2/(a^2*x^2+1)+5/2*arctan(a*x)^2/c^2-1/3/c^2*arctan(a*x)/a^3/x^3+2/c^2*arctan(a*x)/a
/x-1/6/c^2*(-3/2/(a^2*x^2+1)-7*ln(a^2*x^2+1)+1/a^2/x^2+14*ln(a*x)+15/2*arctan(a*x)^2))

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 160, normalized size = 1.18 \begin {gather*} \frac {1}{6} \, {\left (\frac {15 \, a^{3} \arctan \left (a x\right )}{c^{2}} + \frac {15 \, a^{4} x^{4} + 10 \, a^{2} x^{2} - 2}{a^{2} c^{2} x^{5} + c^{2} x^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} + a^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 14 \, {\left (a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right ) - 28 \, {\left (a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (x\right ) - 2\right )} a}{12 \, {\left (a^{2} c^{2} x^{4} + c^{2} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

1/6*(15*a^3*arctan(a*x)/c^2 + (15*a^4*x^4 + 10*a^2*x^2 - 2)/(a^2*c^2*x^5 + c^2*x^3))*arctan(a*x) + 1/12*(a^2*x
^2 - 15*(a^4*x^4 + a^2*x^2)*arctan(a*x)^2 + 14*(a^4*x^4 + a^2*x^2)*log(a^2*x^2 + 1) - 28*(a^4*x^4 + a^2*x^2)*l
og(x) - 2)*a/(a^2*c^2*x^4 + c^2*x^2)

________________________________________________________________________________________

Fricas [A]
time = 1.73, size = 127, normalized size = 0.93 \begin {gather*} \frac {a^{3} x^{3} + 15 \, {\left (a^{5} x^{5} + a^{3} x^{3}\right )} \arctan \left (a x\right )^{2} - 2 \, a x + 2 \, {\left (15 \, a^{4} x^{4} + 10 \, a^{2} x^{2} - 2\right )} \arctan \left (a x\right ) + 14 \, {\left (a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right ) - 28 \, {\left (a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (x\right )}{12 \, {\left (a^{2} c^{2} x^{5} + c^{2} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

1/12*(a^3*x^3 + 15*(a^5*x^5 + a^3*x^3)*arctan(a*x)^2 - 2*a*x + 2*(15*a^4*x^4 + 10*a^2*x^2 - 2)*arctan(a*x) + 1
4*(a^5*x^5 + a^3*x^3)*log(a^2*x^2 + 1) - 28*(a^5*x^5 + a^3*x^3)*log(x))/(a^2*c^2*x^5 + c^2*x^3)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (129) = 258\).
time = 1.18, size = 362, normalized size = 2.66 \begin {gather*} \begin {cases} - \frac {28 a^{5} x^{5} \log {\left (x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {14 a^{5} x^{5} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {15 a^{5} x^{5} \operatorname {atan}^{2}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {30 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} - \frac {28 a^{3} x^{3} \log {\left (x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {14 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {15 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {a^{3} x^{3}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} + \frac {20 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} - \frac {2 a x}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} - \frac {4 \operatorname {atan}{\left (a x \right )}}{12 a^{2} c^{2} x^{5} + 12 c^{2} x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)/x**4/(a**2*c*x**2+c)**2,x)

[Out]

Piecewise((-28*a**5*x**5*log(x)/(12*a**2*c**2*x**5 + 12*c**2*x**3) + 14*a**5*x**5*log(x**2 + a**(-2))/(12*a**2
*c**2*x**5 + 12*c**2*x**3) + 15*a**5*x**5*atan(a*x)**2/(12*a**2*c**2*x**5 + 12*c**2*x**3) + 30*a**4*x**4*atan(
a*x)/(12*a**2*c**2*x**5 + 12*c**2*x**3) - 28*a**3*x**3*log(x)/(12*a**2*c**2*x**5 + 12*c**2*x**3) + 14*a**3*x**
3*log(x**2 + a**(-2))/(12*a**2*c**2*x**5 + 12*c**2*x**3) + 15*a**3*x**3*atan(a*x)**2/(12*a**2*c**2*x**5 + 12*c
**2*x**3) + a**3*x**3/(12*a**2*c**2*x**5 + 12*c**2*x**3) + 20*a**2*x**2*atan(a*x)/(12*a**2*c**2*x**5 + 12*c**2
*x**3) - 2*a*x/(12*a**2*c**2*x**5 + 12*c**2*x**3) - 4*atan(a*x)/(12*a**2*c**2*x**5 + 12*c**2*x**3), Ne(a, 0)),
(0, True))

________________________________________________________________________________________

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(arctan(a*x)/x^4/(a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

Mupad [B]
time = 0.55, size = 123, normalized size = 0.90 \begin {gather*} \frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {5\,x^2}{3\,c^2}-\frac {1}{3\,a^2\,c^2}+\frac {5\,a^2\,x^4}{2\,c^2}\right )}{x^5+\frac {x^3}{a^2}}-\frac {a-\frac {a^3\,x^2}{2}}{6\,a^2\,c^2\,x^4+6\,c^2\,x^2}+\frac {7\,a^3\,\ln \left (a^2\,x^2+1\right )}{6\,c^2}-\frac {7\,a^3\,\ln \left (x\right )}{3\,c^2}+\frac {5\,a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{4\,c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)/(x^4*(c + a^2*c*x^2)^2),x)

[Out]

(atan(a*x)*((5*x^2)/(3*c^2) - 1/(3*a^2*c^2) + (5*a^2*x^4)/(2*c^2)))/(x^5 + x^3/a^2) - (a - (a^3*x^2)/2)/(6*c^2
*x^2 + 6*a^2*c^2*x^4) + (7*a^3*log(a^2*x^2 + 1))/(6*c^2) - (7*a^3*log(x))/(3*c^2) + (5*a^3*atan(a*x)^2)/(4*c^2
)

________________________________________________________________________________________