3.199 \(\int \frac {\tan ^{-1}(a x)}{x^4 (c+a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=183 \[ -\frac {10 a^3 \log (x)}{3 c^3}+\frac {35 a^3 \tan ^{-1}(a x)^2}{16 c^3}+\frac {3 a^2 \tan ^{-1}(a x)}{c^3 x}+\frac {11 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac {11 a^3}{16 c^3 \left (a^2 x^2+1\right )}+\frac {a^3}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 a^3 \log \left (a^2 x^2+1\right )}{3 c^3}-\frac {\tan ^{-1}(a x)}{3 c^3 x^3}-\frac {a}{6 c^3 x^2} \]

[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

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Rubi [A]  time = 0.69, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4966, 4918, 4852, 266, 44, 36, 29, 31, 4884, 4892, 261, 4896} \[ \frac {11 a^3}{16 c^3 \left (a^2 x^2+1\right )}+\frac {a^3}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac {5 a^3 \log \left (a^2 x^2+1\right )}{3 c^3}+\frac {11 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (a^2 x^2+1\right )}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {10 a^3 \log (x)}{3 c^3}+\frac {35 a^3 \tan ^{-1}(a x)^2}{16 c^3}+\frac {3 a^2 \tan ^{-1}(a x)}{c^3 x}-\frac {a}{6 c^3 x^2}-\frac {\tan ^{-1}(a x)}{3 c^3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a/(6*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 44

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] && L
tQ[m + n + 2, 0])

Rule 261

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 266

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 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4884

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 4892

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 4896

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 4918

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 4966

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 )^3} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)}{x^4 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=a^4 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac {a^2 \int \frac {\tan ^{-1}(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 \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\int \frac {\tan ^{-1}(a x)}{x^4} \, dx}{c^3}-\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}+\frac {\left (3 a^4\right ) \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^4 \int \frac {\tan ^{-1}(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 {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {3 a^3 \tan ^{-1}(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 {\tan ^{-1}(a x)}{x^2} \, dx}{c^3}+\frac {a^4 \int \frac {\tan ^{-1}(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 \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {a^3 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2} \, dx}{c^3}-\frac {a^4 \int \frac {\tan ^{-1}(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 {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^2}{16 c^3}+\frac {a \operatorname {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 \tan ^{-1}(a x)}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \tan ^{-1}(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 {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(a x)^2}{16 c^3}+\frac {a \operatorname {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 \operatorname {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 \tan ^{-1}(a x)}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^3 \operatorname {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 {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(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 \operatorname {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 \tan ^{-1}(a x)}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \tan ^{-1}(a x)^2}{4 c^3}+\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c^3}-\frac {a^5 \operatorname {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^3}\right )+\frac {a^5 \operatorname {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 {\tan ^{-1}(a x)}{3 c^3 x^3}+\frac {a^2 \tan ^{-1}(a x)}{c^3 x}+\frac {a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {3 a^4 x \tan ^{-1}(a x)}{8 c^3 \left (1+a^2 x^2\right )}+\frac {11 a^3 \tan ^{-1}(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 \tan ^{-1}(a x)}{c^3 x}-\frac {a^4 x \tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {3 a^3 \tan ^{-1}(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*}

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Mathematica [A]  time = 0.13, size = 142, normalized size = 0.78 \[ \frac {105 a^3 x^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+2 \left (105 a^6 x^6+175 a^4 x^4+56 a^2 x^2-8\right ) \tan ^{-1}(a x)+a x \left (25 a^4 x^4-160 \left (a^3 x^3+a x\right )^2 \log (x)+20 a^2 x^2+80 \left (a^3 x^3+a x\right )^2 \log \left (a^2 x^2+1\right )-8\right )}{48 c^3 x^3 \left (a^2 x^2+1\right )^2} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.68, size = 179, normalized size = 0.98 \[ \frac {25 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 105 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \arctan \left (a x\right )^{2} - 8 \, a x + 2 \, {\left (105 \, a^{6} x^{6} + 175 \, a^{4} x^{4} + 56 \, a^{2} x^{2} - 8\right )} \arctan \left (a x\right ) + 80 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right ) - 160 \, {\left (a^{7} x^{7} + 2 \, a^{5} x^{5} + a^{3} x^{3}\right )} \log \relax (x)}{48 \, {\left (a^{4} c^{3} x^{7} + 2 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.06, size = 170, normalized size = 0.93 \[ -\frac {\arctan \left (a x \right )}{3 c^{3} x^{3}}+\frac {3 a^{2} \arctan \left (a x \right )}{c^{3} x}+\frac {11 a^{6} \arctan \left (a x \right ) x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {13 a^{4} x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {35 a^{3} \arctan \left (a x \right )^{2}}{16 c^{3}}-\frac {a}{6 c^{3} x^{2}}-\frac {10 a^{3} \ln \left (a x \right )}{3 c^{3}}+\frac {5 a^{3} \ln \left (a^{2} x^{2}+1\right )}{3 c^{3}}+\frac {a^{3}}{16 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {11 a^{3}}{16 c^{3} \left (a^{2} x^{2}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*arctan(a*x)/c^3/x^3+3*a^2*arctan(a*x)/c^3/x+11/8*a^6/c^3*arctan(a*x)/(a^2*x^2+1)^2*x^3+13/8*a^4*x*arctan(
a*x)/c^3/(a^2*x^2+1)^2+35/16*a^3*arctan(a*x)^2/c^3-1/6*a/c^3/x^2-10/3*a^3/c^3*ln(a*x)+5/3*a^3*ln(a^2*x^2+1)/c^
3+1/16*a^3/c^3/(a^2*x^2+1)^2+11/16*a^3/c^3/(a^2*x^2+1)

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maxima [A]  time = 0.46, size = 223, normalized size = 1.22 \[ \frac {1}{24} \, {\left (\frac {105 \, a^{3} \arctan \left (a x\right )}{c^{3}} + \frac {105 \, a^{6} x^{6} + 175 \, a^{4} x^{4} + 56 \, a^{2} x^{2} - 8}{a^{4} c^{3} x^{7} + 2 \, a^{2} c^{3} x^{5} + c^{3} x^{3}}\right )} \arctan \left (a x\right ) + \frac {{\left (25 \, a^{4} x^{4} + 20 \, a^{2} x^{2} - 105 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 80 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right ) - 160 \, {\left (a^{6} x^{6} + 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \relax (x) - 8\right )} a}{48 \, {\left (a^{4} c^{3} x^{6} + 2 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.59, size = 163, normalized size = 0.89 \[ \frac {\frac {25\,a^5\,x^4}{2}+10\,a^3\,x^2-4\,a}{24\,a^4\,c^3\,x^6+48\,a^2\,c^3\,x^4+24\,c^3\,x^2}+\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {7\,x^2}{3\,c^3}-\frac {1}{3\,a^2\,c^3}+\frac {175\,a^2\,x^4}{24\,c^3}+\frac {35\,a^4\,x^6}{8\,c^3}\right )}{2\,x^5+\frac {x^3}{a^2}+a^2\,x^7}+\frac {5\,a^3\,\ln \left (a^2\,x^2+1\right )}{3\,c^3}-\frac {10\,a^3\,\ln \relax (x)}{3\,c^3}+\frac {35\,a^3\,{\mathrm {atan}\left (a\,x\right )}^2}{16\,c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [B]  time = 3.65, size = 722, normalized size = 3.95 \[ - \frac {160 a^{7} x^{7} \log {\relax (x )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {80 a^{7} x^{7} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {105 a^{7} x^{7} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {210 a^{6} x^{6} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {320 a^{5} x^{5} \log {\relax (x )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {160 a^{5} x^{5} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {210 a^{5} x^{5} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {25 a^{5} x^{5}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {350 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {160 a^{3} x^{3} \log {\relax (x )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {80 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {105 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {20 a^{3} x^{3}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} + \frac {112 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {8 a x}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} - \frac {16 \operatorname {atan}{\left (a x \right )}}{48 a^{4} c^{3} x^{7} + 96 a^{2} c^{3} x^{5} + 48 c^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 + 48*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*ata
n(a*x)/(48*a**4*c**3*x**7 + 96*a**2*c**3*x**5 + 48*c**3*x**3)

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