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

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

[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)

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Rubi [A]  time = 0.68997, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, 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 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]

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 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 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 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 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 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 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 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 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]

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*}

Mathematica [A]  time = 0.129204, size = 142, normalized size = 0.78 \[ \frac{a x \left (25 a^4 x^4+20 a^2 x^2-160 \left (a^3 x^3+a x\right )^2 \log (x)+80 \left (a^3 x^3+a x\right )^2 \log \left (a^2 x^2+1\right )-8\right )+105 a^3 x^3 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2+2 \left (105 a^6 x^6+175 a^4 x^4+56 a^2 x^2-8\right ) \tan ^{-1}(a x)}{48 c^3 x^3 \left (a^2 x^2+1\right )^2} \]

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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Maple [A]  time = 0.051, size = 170, normalized size = 0.9 \begin{align*}{\frac{11\,{a}^{6}\arctan \left ( ax \right ){x}^{3}}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{13\,{a}^{4}x\arctan \left ( ax \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{35\,{a}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{16\,{c}^{3}}}-{\frac{\arctan \left ( ax \right ) }{3\,{c}^{3}{x}^{3}}}+3\,{\frac{{a}^{2}\arctan \left ( ax \right ) }{{c}^{3}x}}+{\frac{{a}^{3}}{16\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{2}}}+{\frac{5\,{a}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3\,{c}^{3}}}+{\frac{11\,{a}^{3}}{16\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }}-{\frac{a}{6\,{c}^{3}{x}^{2}}}-{\frac{10\,{a}^{3}\ln \left ( ax \right ) }{3\,{c}^{3}}} \end{align*}

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]

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

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Maxima [A]  time = 1.68526, size = 301, normalized size = 1.64 \begin{align*} \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 )}} \end{align*}

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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Fricas [A]  time = 1.78557, size = 394, normalized size = 2.15 \begin{align*} \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 )}} \end{align*}

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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Sympy [B]  time = 6.95688, size = 763, normalized size = 4.17 \begin{align*} - \frac{640 a^{7} x^{7} \log{\left (x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{320 a^{7} x^{7} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{420 a^{7} x^{7} \operatorname{atan}^{2}{\left (a x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} - \frac{25 a^{7} x^{7}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{840 a^{6} x^{6} \operatorname{atan}{\left (a x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} - \frac{1280 a^{5} x^{5} \log{\left (x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{640 a^{5} x^{5} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{840 a^{5} x^{5} \operatorname{atan}^{2}{\left (a x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{50 a^{5} x^{5}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{1400 a^{4} x^{4} \operatorname{atan}{\left (a x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} - \frac{640 a^{3} x^{3} \log{\left (x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{320 a^{3} x^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{420 a^{3} x^{3} \operatorname{atan}^{2}{\left (a x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{55 a^{3} x^{3}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} + \frac{448 a^{2} x^{2} \operatorname{atan}{\left (a x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} - \frac{32 a x}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} - \frac{64 \operatorname{atan}{\left (a x \right )}}{192 a^{4} c^{3} x^{7} + 384 a^{2} c^{3} x^{5} + 192 c^{3} x^{3}} \end{align*}

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]

-640*a**7*x**7*log(x)/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) + 320*a**7*x**7*log(x**2 + a**
(-2))/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) + 420*a**7*x**7*atan(a*x)**2/(192*a**4*c**3*x*
*7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) - 25*a**7*x**7/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x*
*3) + 840*a**6*x**6*atan(a*x)/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) - 1280*a**5*x**5*log(x
)/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) + 640*a**5*x**5*log(x**2 + a**(-2))/(192*a**4*c**3
*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) + 840*a**5*x**5*atan(a*x)**2/(192*a**4*c**3*x**7 + 384*a**2*c**3*x
**5 + 192*c**3*x**3) + 50*a**5*x**5/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) + 1400*a**4*x**4
*atan(a*x)/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) - 640*a**3*x**3*log(x)/(192*a**4*c**3*x**
7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) + 320*a**3*x**3*log(x**2 + a**(-2))/(192*a**4*c**3*x**7 + 384*a**2*c**
3*x**5 + 192*c**3*x**3) + 420*a**3*x**3*atan(a*x)**2/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3)
+ 55*a**3*x**3/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) + 448*a**2*x**2*atan(a*x)/(192*a**4*
c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3) - 32*a*x/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x
**3) - 64*atan(a*x)/(192*a**4*c**3*x**7 + 384*a**2*c**3*x**5 + 192*c**3*x**3)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{4}}\,{d x} \end{align*}

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]

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