\(\int \frac {\arctan (a x)}{x^4 (c+a^2 c x^2)^3} \, dx\) [199]

Optimal result

Integrand size = 20, antiderivative size = 183 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {a}{6 c^3 x^2}+\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {11 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^3 x^3}+\frac {3 a^2 \arctan (a x)}{c^3 x}+\frac {a^4 x \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {11 a^4 x \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {35 a^3 \arctan (a x)^2}{16 c^3}-\frac {10 a^3 \log (x)}{3 c^3}+\frac {5 a^3 \log \left (1+a^2 x^2\right )}{3 c^3} \]

[Out]

-1/6*a/c^3/x^2+1/16*a^3/c^3/(a^2*x^2+1)^2+11/16*a^3/c^3/(a^2*x^2+1)-1/3*arctan(a*x)/c^3/x^3+3*a^2*arctan(a*x)/
c^3/x+1/4*a^4*x*arctan(a*x)/c^3/(a^2*x^2+1)^2+11/8*a^4*x*arctan(a*x)/c^3/(a^2*x^2+1)+35/16*a^3*arctan(a*x)^2/c
^3-10/3*a^3*ln(x)/c^3+5/3*a^3*ln(a^2*x^2+1)/c^3

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5086, 5038, 4946, 272, 46, 36, 29, 31, 5004, 5012, 267, 5016} \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {35 a^3 \arctan (a x)^2}{16 c^3}-\frac {10 a^3 \log (x)}{3 c^3}+\frac {3 a^2 \arctan (a x)}{c^3 x}+\frac {11 a^4 x \arctan (a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^4 x \arctan (a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {11 a^3}{16 c^3 \left (a^2 x^2+1\right )}+\frac {a^3}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 a^3 \log \left (a^2 x^2+1\right )}{3 c^3}-\frac {\arctan (a x)}{3 c^3 x^3}-\frac {a}{6 c^3 x^2} \]

[In]

Int[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^3),x]

[Out]

-1/6*a/(c^3*x^2) + a^3/(16*c^3*(1 + a^2*x^2)^2) + (11*a^3)/(16*c^3*(1 + a^2*x^2)) - ArcTan[a*x]/(3*c^3*x^3) +
(3*a^2*ArcTan[a*x])/(c^3*x) + (a^4*x*ArcTan[a*x])/(4*c^3*(1 + a^2*x^2)^2) + (11*a^4*x*ArcTan[a*x])/(8*c^3*(1 +
a^2*x^2)) + (35*a^3*ArcTan[a*x]^2)/(16*c^3) - (10*a^3*Log[x])/(3*c^3) + (5*a^3*Log[1 + a^2*x^2])/(3*c^3)

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 5016

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Simp[b*((d + e*x^2)^(q + 1)/(4
*c*d*(q + 1)^2)), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x] - Si
mp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*d*(q + 1))), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d
] && LtQ[q, -1] && NeQ[q, -3/2]

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*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = a^4 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx}{c} \\ & = \frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {a^4 x \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\int \frac {\arctan (a x)}{x^4} \, dx}{c^3}-\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}+\frac {\left (3 a^4\right ) \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (\frac {a^2 \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^4 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right ) \\ & = \frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {\arctan (a x)}{3 c^3 x^3}+\frac {a^4 x \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^3 \arctan (a x)^2}{16 c^3}+\frac {a \int \frac {1}{x^3 \left (1+a^2 x^2\right )} \, dx}{3 c^3}-\frac {a^2 \int \frac {\arctan (a x)}{x^2} \, dx}{c^3}+\frac {a^4 \int \frac {\arctan (a x)}{c+a^2 c x^2} \, dx}{c^2}-\frac {\left (3 a^5\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-2 \left (-\frac {a^4 x \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \arctan (a x)^2}{4 c^3}+\frac {a^2 \int \frac {\arctan (a x)}{x^2} \, dx}{c^3}-\frac {a^4 \int \frac {\arctan (a x)}{c+a^2 c x^2} \, dx}{c^2}+\frac {a^5 \int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right ) \\ & = \frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^3 x^3}+\frac {a^2 \arctan (a x)}{c^3 x}+\frac {a^4 x \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \arctan (a x)^2}{16 c^3}+\frac {a \text {Subst}\left (\int \frac {1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )}{6 c^3}-\frac {a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-2 \left (-\frac {a^3}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{c^3 x}-\frac {a^4 x \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \arctan (a x)^2}{4 c^3}+\frac {a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c^3}\right ) \\ & = \frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^3 x^3}+\frac {a^2 \arctan (a x)}{c^3 x}+\frac {a^4 x \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \arctan (a x)^2}{16 c^3}+\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^3}-\frac {a^3 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^3}-2 \left (-\frac {a^3}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{c^3 x}-\frac {a^4 x \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \arctan (a x)^2}{4 c^3}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^3}\right ) \\ & = -\frac {a}{6 c^3 x^2}+\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^3 x^3}+\frac {a^2 \arctan (a x)}{c^3 x}+\frac {a^4 x \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \arctan (a x)^2}{16 c^3}-\frac {a^3 \log (x)}{3 c^3}+\frac {a^3 \log \left (1+a^2 x^2\right )}{6 c^3}-\frac {a^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c^3}-2 \left (-\frac {a^3}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{c^3 x}-\frac {a^4 x \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \arctan (a x)^2}{4 c^3}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c^3}-\frac {a^5 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^3}\right )+\frac {a^5 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^3} \\ & = -\frac {a}{6 c^3 x^2}+\frac {a^3}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^3}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{3 c^3 x^3}+\frac {a^2 \arctan (a x)}{c^3 x}+\frac {a^4 x \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \arctan (a x)^2}{16 c^3}-\frac {4 a^3 \log (x)}{3 c^3}+\frac {2 a^3 \log \left (1+a^2 x^2\right )}{3 c^3}-2 \left (-\frac {a^3}{4 c^3 \left (1+a^2 x^2\right )}-\frac {a^2 \arctan (a x)}{c^3 x}-\frac {a^4 x \arctan (a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \arctan (a x)^2}{4 c^3}+\frac {a^3 \log (x)}{c^3}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.78 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {2 \left (-8+56 a^2 x^2+175 a^4 x^4+105 a^6 x^6\right ) \arctan (a x)+105 a^3 x^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2+a x \left (-8+20 a^2 x^2+25 a^4 x^4-160 \left (a x+a^3 x^3\right )^2 \log (x)+80 \left (a x+a^3 x^3\right )^2 \log \left (1+a^2 x^2\right )\right )}{48 c^3 x^3 \left (1+a^2 x^2\right )^2} \]

[In]

Integrate[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^3),x]

[Out]

(2*(-8 + 56*a^2*x^2 + 175*a^4*x^4 + 105*a^6*x^6)*ArcTan[a*x] + 105*a^3*x^3*(1 + a^2*x^2)^2*ArcTan[a*x]^2 + a*x
*(-8 + 20*a^2*x^2 + 25*a^4*x^4 - 160*(a*x + a^3*x^3)^2*Log[x] + 80*(a*x + a^3*x^3)^2*Log[1 + a^2*x^2]))/(48*c^
3*x^3*(1 + a^2*x^2)^2)

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(a^{3} \left (\frac {11 \arctan \left (a x \right ) a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{3 c^{3} a^{3} x^{3}}+\frac {3 \arctan \left (a x \right )}{c^{3} a x}-\frac {-40 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {33}{2 \left (a^{2} x^{2}+1\right )}+\frac {4}{a^{2} x^{2}}+80 \ln \left (a x \right )+\frac {105 \arctan \left (a x \right )^{2}}{2}}{24 c^{3}}\right )\)	\(161\)
default	\(a^{3} \left (\frac {11 \arctan \left (a x \right ) a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{3 c^{3} a^{3} x^{3}}+\frac {3 \arctan \left (a x \right )}{c^{3} a x}-\frac {-40 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {33}{2 \left (a^{2} x^{2}+1\right )}+\frac {4}{a^{2} x^{2}}+80 \ln \left (a x \right )+\frac {105 \arctan \left (a x \right )^{2}}{2}}{24 c^{3}}\right )\)	\(161\)
parts	\(\frac {11 \arctan \left (a x \right ) a^{6} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a^{4} x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 a^{3} \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{3 c^{3} x^{3}}+\frac {3 a^{2} \arctan \left (a x \right )}{c^{3} x}-\frac {\frac {105 a^{3} \arctan \left (a x \right )^{2}}{16}+\frac {a^{3} \left (-40 \ln \left (a^{2} x^{2}+1\right )-\frac {3}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {33}{2 \left (a^{2} x^{2}+1\right )}+\frac {4}{a^{2} x^{2}}+80 \ln \left (a x \right )\right )}{8}}{3 c^{3}}\)	\(168\)
parallelrisch	\(-\frac {-105 a^{7} \arctan \left (a x \right )^{2} x^{7}+160 \ln \left (x \right ) a^{7} x^{7}-80 \ln \left (a^{2} x^{2}+1\right ) x^{7} a^{7}+4 a^{7} x^{7}-210 a^{6} \arctan \left (a x \right ) x^{6}-210 a^{5} \arctan \left (a x \right )^{2} x^{5}+320 \ln \left (x \right ) x^{5} a^{5}-160 \ln \left (a^{2} x^{2}+1\right ) x^{5} a^{5}-17 a^{5} x^{5}-350 \arctan \left (a x \right ) a^{4} x^{4}-105 a^{3} \arctan \left (a x \right )^{2} x^{3}+160 \ln \left (x \right ) a^{3} x^{3}-80 a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}-16 a^{3} x^{3}-112 a^{2} \arctan \left (a x \right ) x^{2}+8 a x +16 \arctan \left (a x \right )}{48 x^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}\)	\(217\)
risch	\(-\frac {35 a^{3} \ln \left (i a x +1\right )^{2}}{64 c^{3}}+\frac {\left (105 a^{7} x^{7} \ln \left (-i a x +1\right )-210 i a^{6} x^{6}+210 a^{5} x^{5} \ln \left (-i a x +1\right )-350 i a^{4} x^{4}+105 a^{3} x^{3} \ln \left (-i a x +1\right )-112 i a^{2} x^{2}+16 i\right ) \ln \left (i a x +1\right )}{96 x^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {105 a^{7} x^{7} \ln \left (-i a x +1\right )^{2}+640 \ln \left (x \right ) a^{7} x^{7}-320 \ln \left (3 a^{2} x^{2}+3\right ) a^{7} x^{7}-224 i a^{2} x^{2} \ln \left (-i a x +1\right )+210 a^{5} x^{5} \ln \left (-i a x +1\right )^{2}+1280 \ln \left (x \right ) x^{5} a^{5}-640 \ln \left (3 a^{2} x^{2}+3\right ) a^{5} x^{5}-420 i a^{6} x^{6} \ln \left (-i a x +1\right )-100 a^{5} x^{5}+105 a^{3} \ln \left (-i a x +1\right )^{2} x^{3}+640 \ln \left (x \right ) a^{3} x^{3}-320 \ln \left (3 a^{2} x^{2}+3\right ) a^{3} x^{3}-700 i a^{4} \ln \left (-i a x +1\right ) x^{4}-80 a^{3} x^{3}+32 i \ln \left (-i a x +1\right )+32 a x}{192 x^{3} \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2}}\)	\(373\)

[In]

int(arctan(a*x)/x^4/(a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

a^3*(11/8/c^3*arctan(a*x)/(a^2*x^2+1)^2*a^3*x^3+13/8*a*x*arctan(a*x)/c^3/(a^2*x^2+1)^2+35/8*arctan(a*x)^2/c^3-
1/3/c^3*arctan(a*x)/a^3/x^3+3/c^3*arctan(a*x)/a/x-1/24/c^3*(-40*ln(a^2*x^2+1)-3/2/(a^2*x^2+1)^2-33/2/(a^2*x^2+
1)+4/a^2/x^2+80*ln(a*x)+105/2*arctan(a*x)^2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.98 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {25 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 105 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \arctan \left (a x\right )^{2} - 8 \, a x + 2 \, {\left (105 \, a^{6} x^{6} + 175 \, a^{4} x^{4} + 56 \, a^{2} x^{2} - 8\right )} \arctan \left (a x\right ) + 80 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right ) - 160 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (x\right )}{48 \, {\left (a^{4} c^{3} x^{7} + 2 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )}} \]

[In]

integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/48*(25*a^5*x^5 + 20*a^3*x^3 + 105*(a^7*x^7 + 2*a^5*x^5 + a^3*x^3)*arctan(a*x)^2 - 8*a*x + 2*(105*a^6*x^6 + 1
75*a^4*x^4 + 56*a^2*x^2 - 8)*arctan(a*x) + 80*(a^7*x^7 + 2*a^5*x^5 + a^3*x^3)*log(a^2*x^2 + 1) - 160*(a^7*x^7
+ 2*a^5*x^5 + a^3*x^3)*log(x))/(a^4*c^3*x^7 + 2*a^2*c^3*x^5 + c^3*x^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 724 vs. \(2 (177) = 354\).

Time = 1.72 (sec) , antiderivative size = 724, normalized size of antiderivative = 3.96 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\begin {cases} - \frac {160 a^{7} x^{7} \log {\left (x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {80 a^{7} x^{7} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {105 a^{7} x^{7} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {210 a^{6} x^{6} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {320 a^{5} x^{5} \log {\left (x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {160 a^{5} x^{5} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {210 a^{5} x^{5} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {25 a^{5} x^{5}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {350 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {160 a^{3} x^{3} \log {\left (x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {80 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {105 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {20 a^{3} x^{3}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {112 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {8 a x}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {16 \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(atan(a*x)/x**4/(a**2*c*x**2+c)**3,x)

[Out]

Piecewise((-160*a**7*x**7*log(x)/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 80*a**7*x**7*log(x**
2 + a**(-2))/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 105*a**7*x**7*atan(a*x)**2/(48*a**4*c**3
*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 210*a**6*x**6*atan(a*x)/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 4
8*c**3*x**3) - 320*a**5*x**5*log(x)/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 160*a**5*x**5*log
(x**2 + a**(-2))/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 210*a**5*x**5*atan(a*x)**2/(48*a**4*
c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 25*a**5*x**5/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*
x**3) + 350*a**4*x**4*atan(a*x)/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) - 160*a**3*x**3*log(x)/
(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 80*a**3*x**3*log(x**2 + a**(-2))/(48*a**4*c**3*x**7 +
96*a**2*c**3*x**5 + 48*c**3*x**3) + 105*a**3*x**3*atan(a*x)**2/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c*
*3*x**3) + 20*a**3*x**3/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) + 112*a**2*x**2*atan(a*x)/(48*a
**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3) - 8*a*x/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**
3) - 16*atan(a*x)/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3), Ne(a, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.22 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {1}{24} \, {\left (\frac {105 \, a^{3} \arctan \left (a x\right )}{c^{3}} + \frac {105 \, a^{6} x^{6} + 175 \, a^{4} x^{4} + 56 \, a^{2} x^{2} - 8}{a^{4} c^{3} x^{7} + 2 \, a^{2} c^{3} x^{5} + c^{3} x^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (25 \, a^{4} x^{4} + 20 \, a^{2} x^{2} - 105 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 80 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right ) - 160 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (x\right ) - 8\right )} a}{48 \, {\left (a^{4} c^{3} x^{6} + 2 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )}} \]

[In]

integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^3,x, algorithm="maxima")

[Out]

1/24*(105*a^3*arctan(a*x)/c^3 + (105*a^6*x^6 + 175*a^4*x^4 + 56*a^2*x^2 - 8)/(a^4*c^3*x^7 + 2*a^2*c^3*x^5 + c^
3*x^3))*arctan(a*x) + 1/48*(25*a^4*x^4 + 20*a^2*x^2 - 105*(a^6*x^6 + 2*a^4*x^4 + a^2*x^2)*arctan(a*x)^2 + 80*(
a^6*x^6 + 2*a^4*x^4 + a^2*x^2)*log(a^2*x^2 + 1) - 160*(a^6*x^6 + 2*a^4*x^4 + a^2*x^2)*log(x) - 8)*a/(a^4*c^3*x
^6 + 2*a^2*c^3*x^4 + c^3*x^2)

Giac [F]

\[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{4}} \,d x } \]

[In]

integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.89 \[ \int \frac {\arctan (a x)}{x^4 \left (c+a^2 c x^2\right )^3} \, dx=\frac {\frac {25\,a^5\,x^4}{2}+10\,a^3\,x^2-4\,a}{24\,a^4\,c^3\,x^6+48\,a^2\,c^3\,x^4+24\,c^3\,x^2}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {7\,x^2}{3\,c^3}-\frac {1}{3\,a^2\,c^3}+\frac {175\,a^2\,x^4}{24\,c^3}+\frac {35\,a^4\,x^6}{8\,c^3}\right )}{2\,x^5+\frac {x^3}{a^2}+a^2\,x^7}+\frac {5\,a^3\,\ln \left (a^2\,x^2+1\right )}{3\,c^3}-\frac {10\,a^3\,\ln \left (x\right )}{3\,c^3}+\frac {35\,a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{16\,c^3} \]

[In]

int(atan(a*x)/(x^4*(c + a^2*c*x^2)^3),x)

[Out]

(10*a^3*x^2 - 4*a + (25*a^5*x^4)/2)/(24*c^3*x^2 + 48*a^2*c^3*x^4 + 24*a^4*c^3*x^6) + (atan(a*x)*((7*x^2)/(3*c^
3) - 1/(3*a^2*c^3) + (175*a^2*x^4)/(24*c^3) + (35*a^4*x^6)/(8*c^3)))/(2*x^5 + x^3/a^2 + a^2*x^7) + (5*a^3*log(
a^2*x^2 + 1))/(3*c^3) - (10*a^3*log(x))/(3*c^3) + (35*a^3*atan(a*x)^2)/(16*c^3)