\(\int \frac {\arctan (a x)}{x^2 (c+a^2 c x^2)^3} \, dx\) [197]

Optimal result

Integrand size = 20, antiderivative size = 142 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {a}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {7 a}{16 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{c^3 x}-\frac {a^2 x \arctan (a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {7 a^2 x \arctan (a x)}{8 c^3 \left (1+a^2 x^2\right )}-\frac {15 a \arctan (a x)^2}{16 c^3}+\frac {a \log (x)}{c^3}-\frac {a \log \left (1+a^2 x^2\right )}{2 c^3} \] Output:

-1/16*a/c^3/(a^2*x^2+1)^2-7/16*a/c^3/(a^2*x^2+1)-arctan(a*x)/c^3/x-1/4*a^2 
*x*arctan(a*x)/c^3/(a^2*x^2+1)^2-7/8*a^2*x*arctan(a*x)/c^3/(a^2*x^2+1)-15/ 
16*a*arctan(a*x)^2/c^3+a*ln(x)/c^3-1/2*a*ln(a^2*x^2+1)/c^3
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.83 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\frac {-2 \left (8+25 a^2 x^2+15 a^4 x^4\right ) \arctan (a x)-15 a x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+a x \left (-8-7 a^2 x^2+16 \left (1+a^2 x^2\right )^2 \log (x)-8 \left (1+a^2 x^2\right )^2 \log \left (1+a^2 x^2\right )\right )}{16 c^3 x \left (1+a^2 x^2\right )^2} \] Input:

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

Output:

(-2*(8 + 25*a^2*x^2 + 15*a^4*x^4)*ArcTan[a*x] - 15*a*x*(1 + a^2*x^2)^2*Arc 
Tan[a*x]^2 + a*x*(-8 - 7*a^2*x^2 + 16*(1 + a^2*x^2)^2*Log[x] - 8*(1 + a^2* 
x^2)^2*Log[1 + a^2*x^2]))/(16*c^3*x*(1 + a^2*x^2)^2)
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.46, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5501, 27, 5431, 5427, 241, 5501, 5427, 241, 5453, 5361, 243, 47, 14, 16, 5419}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\arctan (a x)}{x^2 \left (a^2 c x^2+c\right )^3} \, dx\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\int \frac {\arctan (a x)}{c^2 x^2 \left (a^2 x^2+1\right )^2}dx}{c}-a^2 \int \frac {\arctan (a x)}{c^3 \left (a^2 x^2+1\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^3}dx}{c^3}\)

\(\Big \downarrow \) 5431

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3}{4} \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 5427

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {3}{4} \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 5427

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x^2 \left (a^2 x^2+1\right )}dx-a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\int \frac {\arctan (a x)}{x^2}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+a \int \frac {1}{x \left (a^2 x^2+1\right )}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx+\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )-\frac {\arctan (a x)}{x}}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {-a^2 \int \frac {\arctan (a x)}{a^2 x^2+1}dx-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\arctan (a x)}{x}}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

\(\Big \downarrow \) 5419

\(\displaystyle \frac {-\left (a^2 \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )\right )+\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {1}{2} a \arctan (a x)^2-\frac {\arctan (a x)}{x}}{c^3}-\frac {a^2 \left (\frac {x \arctan (a x)}{4 \left (a^2 x^2+1\right )^2}+\frac {3}{4} \left (\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}\right )+\frac {1}{16 a \left (a^2 x^2+1\right )^2}\right )}{c^3}\)

Input:

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

Output:

-((a^2*(1/(16*a*(1 + a^2*x^2)^2) + (x*ArcTan[a*x])/(4*(1 + a^2*x^2)^2) + ( 
3*(1/(4*a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x] 
^2/(4*a)))/4))/c^3) + (-(ArcTan[a*x]/x) - (a*ArcTan[a*x]^2)/2 - a^2*(1/(4* 
a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a) 
) + (a*(Log[x^2] - Log[1 + a^2*x^2]))/2)/c^3
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 47

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

rule 241

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 5361

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] - Simp[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 5419

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> 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 5427

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

rule 5431

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] + (-Simp[x*(d + e*x 
^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*( 
q + 1))   Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, 
 b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
 

rule 5453

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], 
 x] - Simp[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 5501

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c 
*x])^p, x], x] - Simp[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]
 

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(a \left (-\frac {7 \arctan \left (a x \right ) a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {9 a x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {15 \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{c^{3} a x}-\frac {-\frac {15 \arctan \left (a x \right )^{2}}{2}+\frac {7}{2 \left (a^{2} x^{2}+1\right )}+4 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 \left (a^{2} x^{2}+1\right )^{2}}-8 \ln \left (a x \right )}{8 c^{3}}\right )\)	\(136\)
default	\(a \left (-\frac {7 \arctan \left (a x \right ) a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {9 a x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {15 \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{c^{3} a x}-\frac {-\frac {15 \arctan \left (a x \right )^{2}}{2}+\frac {7}{2 \left (a^{2} x^{2}+1\right )}+4 \ln \left (a^{2} x^{2}+1\right )+\frac {1}{2 \left (a^{2} x^{2}+1\right )^{2}}-8 \ln \left (a x \right )}{8 c^{3}}\right )\)	\(136\)
parts	\(-\frac {7 \arctan \left (a x \right ) a^{4} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {9 a^{2} x \arctan \left (a x \right )}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {15 a \arctan \left (a x \right )^{2}}{8 c^{3}}-\frac {\arctan \left (a x \right )}{c^{3} x}-\frac {-\frac {15 a \arctan \left (a x \right )^{2}}{16}-\frac {a \left (-\frac {7}{2 \left (a^{2} x^{2}+1\right )}-4 \ln \left (a^{2} x^{2}+1\right )-\frac {1}{2 \left (a^{2} x^{2}+1\right )^{2}}+8 \ln \left (a x \right )\right )}{8}}{c^{3}}\)	\(139\)
parallelrisch	\(\frac {-15 a^{5} \arctan \left (a x \right )^{2} x^{5}+16 \ln \left (x \right ) a^{5} x^{5}-8 \ln \left (a^{2} x^{2}+1\right ) x^{5} a^{5}+8 a^{5} x^{5}-30 x^{4} \arctan \left (a x \right ) a^{4}-30 a^{3} \arctan \left (a x \right )^{2} x^{3}+32 \ln \left (x \right ) a^{3} x^{3}-16 a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}+9 a^{3} x^{3}-50 x^{2} a^{2} \arctan \left (a x \right )-15 a \arctan \left (a x \right )^{2} x +16 a x \ln \left (x \right )-8 a \ln \left (a^{2} x^{2}+1\right ) x -16 \arctan \left (a x \right )}{16 x \,c^{3} \left (a^{2} x^{2}+1\right )^{2}}\)	\(181\)
risch	\(\frac {7 i a^{2} \ln \left (-i a x +1\right ) x}{64 c^{3} \left (-i a x -1\right )}-\frac {i a^{2} \ln \left (-i a x +1\right ) x}{64 c^{3} \left (-i a x -1\right )^{2}}+\frac {a^{3} \ln \left (i a x +1\right ) x^{2}}{128 c^{3} \left (i a x -1\right )^{2}}+\frac {a^{3} \ln \left (-i a x +1\right ) x^{2}}{128 c^{3} \left (-i a x -1\right )^{2}}-\frac {7 a \ln \left (-i a x +1\right )}{64 c^{3} \left (-i a x -1\right )}+\frac {3 a \ln \left (-i a x +1\right )}{128 c^{3} \left (-i a x -1\right )^{2}}-\frac {i \ln \left (-i a x +1\right )}{2 c^{3} x}-\frac {7 a \ln \left (-i a x +1\right )}{32 c^{3} \left (-i a x +1\right )}-\frac {15 a \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{32 c^{3}}+\frac {15 a \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{16 c^{3}}-\frac {a \ln \left (i a x +1\right )}{32 c^{3} \left (i a x +1\right )^{2}}-\frac {7 a \ln \left (i a x +1\right )}{32 c^{3} \left (i a x +1\right )}-\frac {15 a \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{32 c^{3}}-\frac {7 a \ln \left (i a x +1\right )}{64 c^{3} \left (i a x -1\right )}+\frac {3 a \ln \left (i a x +1\right )}{128 c^{3} \left (i a x -1\right )^{2}}+\frac {i \ln \left (i a x +1\right )}{2 c^{3} x}-\frac {a \ln \left (-i a x +1\right )}{32 c^{3} \left (-i a x +1\right )^{2}}-\frac {a}{64 c^{3} \left (i a x +1\right )^{2}}+\frac {a \ln \left (i a x \right )}{2 c^{3}}-\frac {a \ln \left (i a x +1\right )}{2 c^{3}}-\frac {7 a}{32 c^{3} \left (i a x +1\right )}+\frac {15 a \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{32 c^{3}}+\frac {a}{64 c^{3} \left (i a x -1\right )}+\frac {15 a \ln \left (i a x +1\right )^{2}}{64 c^{3}}-\frac {a}{64 c^{3} \left (-i a x +1\right )^{2}}+\frac {a \ln \left (-i a x \right )}{2 c^{3}}-\frac {a \ln \left (-i a x +1\right )}{2 c^{3}}-\frac {7 a}{32 c^{3} \left (-i a x +1\right )}+\frac {15 a \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{32 c^{3}}+\frac {a}{64 c^{3} \left (-i a x -1\right )}+\frac {15 a \ln \left (-i a x +1\right )^{2}}{64 c^{3}}+\frac {15 a \ln \left (a^{2} x^{2}+1\right )}{128 c^{3}}-\frac {7 i a^{2} \ln \left (i a x +1\right ) x}{64 c^{3} \left (i a x -1\right )}+\frac {i a^{2} \ln \left (i a x +1\right ) x}{64 c^{3} \left (i a x -1\right )^{2}}\)	\(668\)

Input:

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

Output:

a*(-7/8/c^3*arctan(a*x)/(a^2*x^2+1)^2*a^3*x^3-9/8*a*x*arctan(a*x)/c^3/(a^2 
*x^2+1)^2-15/8*arctan(a*x)^2/c^3-1/c^3*arctan(a*x)/a/x-1/8/c^3*(-15/2*arct 
an(a*x)^2+7/2/(a^2*x^2+1)+4*ln(a^2*x^2+1)+1/2/(a^2*x^2+1)^2-8*ln(a*x)))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.05 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {7 \, a^{3} x^{3} + 15 \, {\left (a^{5} x^{5} + 2 \, a^{3} x^{3} + a x\right )} \arctan \left (a x\right )^{2} + 8 \, a x + 2 \, {\left (15 \, a^{4} x^{4} + 25 \, a^{2} x^{2} + 8\right )} \arctan \left (a x\right ) + 8 \, {\left (a^{5} x^{5} + 2 \, a^{3} x^{3} + a x\right )} \log \left (a^{2} x^{2} + 1\right ) - 16 \, {\left (a^{5} x^{5} + 2 \, a^{3} x^{3} + a x\right )} \log \left (x\right )}{16 \, {\left (a^{4} c^{3} x^{5} + 2 \, a^{2} c^{3} x^{3} + c^{3} x\right )}} \] Input:

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

Output:

-1/16*(7*a^3*x^3 + 15*(a^5*x^5 + 2*a^3*x^3 + a*x)*arctan(a*x)^2 + 8*a*x + 
2*(15*a^4*x^4 + 25*a^2*x^2 + 8)*arctan(a*x) + 8*(a^5*x^5 + 2*a^3*x^3 + a*x 
)*log(a^2*x^2 + 1) - 16*(a^5*x^5 + 2*a^3*x^3 + a*x)*log(x))/(a^4*c^3*x^5 + 
 2*a^2*c^3*x^3 + c^3*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (134) = 268\).

Time = 1.22 (sec) , antiderivative size = 604, normalized size of antiderivative = 4.25 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\begin {cases} \frac {16 a^{5} x^{5} \log {\left (x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {8 a^{5} x^{5} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {15 a^{5} x^{5} \operatorname {atan}^{2}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {30 a^{4} x^{4} \operatorname {atan}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} + \frac {32 a^{3} x^{3} \log {\left (x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {16 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {30 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {7 a^{3} x^{3}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {50 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} + \frac {16 a x \log {\left (x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {8 a x \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {15 a x \operatorname {atan}^{2}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {8 a x}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} - \frac {16 \operatorname {atan}{\left (a x \right )}}{16 a^{4} c^{3} x^{5} + 32 a^{2} c^{3} x^{3} + 16 c^{3} x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((16*a**5*x**5*log(x)/(16*a**4*c**3*x**5 + 32*a**2*c**3*x**3 + 16 
*c**3*x) - 8*a**5*x**5*log(x**2 + a**(-2))/(16*a**4*c**3*x**5 + 32*a**2*c* 
*3*x**3 + 16*c**3*x) - 15*a**5*x**5*atan(a*x)**2/(16*a**4*c**3*x**5 + 32*a 
**2*c**3*x**3 + 16*c**3*x) - 30*a**4*x**4*atan(a*x)/(16*a**4*c**3*x**5 + 3 
2*a**2*c**3*x**3 + 16*c**3*x) + 32*a**3*x**3*log(x)/(16*a**4*c**3*x**5 + 3 
2*a**2*c**3*x**3 + 16*c**3*x) - 16*a**3*x**3*log(x**2 + a**(-2))/(16*a**4* 
c**3*x**5 + 32*a**2*c**3*x**3 + 16*c**3*x) - 30*a**3*x**3*atan(a*x)**2/(16 
*a**4*c**3*x**5 + 32*a**2*c**3*x**3 + 16*c**3*x) - 7*a**3*x**3/(16*a**4*c* 
*3*x**5 + 32*a**2*c**3*x**3 + 16*c**3*x) - 50*a**2*x**2*atan(a*x)/(16*a**4 
*c**3*x**5 + 32*a**2*c**3*x**3 + 16*c**3*x) + 16*a*x*log(x)/(16*a**4*c**3* 
x**5 + 32*a**2*c**3*x**3 + 16*c**3*x) - 8*a*x*log(x**2 + a**(-2))/(16*a**4 
*c**3*x**5 + 32*a**2*c**3*x**3 + 16*c**3*x) - 15*a*x*atan(a*x)**2/(16*a**4 
*c**3*x**5 + 32*a**2*c**3*x**3 + 16*c**3*x) - 8*a*x/(16*a**4*c**3*x**5 + 3 
2*a**2*c**3*x**3 + 16*c**3*x) - 16*atan(a*x)/(16*a**4*c**3*x**5 + 32*a**2* 
c**3*x**3 + 16*c**3*x), Ne(a, 0)), (0, True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.27 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=-\frac {1}{8} \, {\left (\frac {15 \, a^{4} x^{4} + 25 \, a^{2} x^{2} + 8}{a^{4} c^{3} x^{5} + 2 \, a^{2} c^{3} x^{3} + c^{3} x} + \frac {15 \, a \arctan \left (a x\right )}{c^{3}}\right )} \arctan \left (a x\right ) - \frac {{\left (7 \, a^{2} x^{2} - 15 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \log \left (a^{2} x^{2} + 1\right ) - 16 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \log \left (x\right ) + 8\right )} a}{16 \, {\left (a^{4} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{2} + c^{3}\right )}} \] Input:

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

Output:

-1/8*((15*a^4*x^4 + 25*a^2*x^2 + 8)/(a^4*c^3*x^5 + 2*a^2*c^3*x^3 + c^3*x) 
+ 15*a*arctan(a*x)/c^3)*arctan(a*x) - 1/16*(7*a^2*x^2 - 15*(a^4*x^4 + 2*a^ 
2*x^2 + 1)*arctan(a*x)^2 + 8*(a^4*x^4 + 2*a^2*x^2 + 1)*log(a^2*x^2 + 1) - 
16*(a^4*x^4 + 2*a^2*x^2 + 1)*log(x) + 8)*a/(a^4*c^3*x^4 + 2*a^2*c^3*x^2 + 
c^3)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.94 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\frac {a\,\ln \left (x\right )}{c^3}-\frac {a\,\ln \left (a^2\,x^2+1\right )}{2\,c^3}-\frac {\frac {7\,a^3\,x^2}{2}+4\,a}{8\,a^4\,c^3\,x^4+16\,a^2\,c^3\,x^2+8\,c^3}-\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{a^2\,c^3}+\frac {25\,x^2}{8\,c^3}+\frac {15\,a^2\,x^4}{8\,c^3}\right )}{\frac {x}{a^2}+2\,x^3+a^2\,x^5}-\frac {15\,a\,{\mathrm {atan}\left (a\,x\right )}^2}{16\,c^3} \] Input:

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

Output:

(a*log(x))/c^3 - (a*log(a^2*x^2 + 1))/(2*c^3) - (4*a + (7*a^3*x^2)/2)/(8*c 
^3 + 16*a^2*c^3*x^2 + 8*a^4*c^3*x^4) - (atan(a*x)*(1/(a^2*c^3) + (25*x^2)/ 
(8*c^3) + (15*a^2*x^4)/(8*c^3)))/(x/a^2 + 2*x^3 + a^2*x^5) - (15*a*atan(a* 
x)^2)/(16*c^3)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.30 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^3} \, dx=\frac {-30 \mathit {atan} \left (a x \right )^{2} a^{5} x^{5}-60 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}-30 \mathit {atan} \left (a x \right )^{2} a x -60 \mathit {atan} \left (a x \right ) a^{4} x^{4}-100 \mathit {atan} \left (a x \right ) a^{2} x^{2}-32 \mathit {atan} \left (a x \right )-16 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{5} x^{5}-32 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{3} x^{3}-16 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a x +32 \,\mathrm {log}\left (x \right ) a^{5} x^{5}+64 \,\mathrm {log}\left (x \right ) a^{3} x^{3}+32 \,\mathrm {log}\left (x \right ) a x +7 a^{5} x^{5}-9 a x}{32 c^{3} x \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 30*atan(a*x)**2*a**5*x**5 - 60*atan(a*x)**2*a**3*x**3 - 30*atan(a*x)** 
2*a*x - 60*atan(a*x)*a**4*x**4 - 100*atan(a*x)*a**2*x**2 - 32*atan(a*x) - 
16*log(a**2*x**2 + 1)*a**5*x**5 - 32*log(a**2*x**2 + 1)*a**3*x**3 - 16*log 
(a**2*x**2 + 1)*a*x + 32*log(x)*a**5*x**5 + 64*log(x)*a**3*x**3 + 32*log(x 
)*a*x + 7*a**5*x**5 - 9*a*x)/(32*c**3*x*(a**4*x**4 + 2*a**2*x**2 + 1))