3.6.57 \(\int \frac {1}{x^3 (c+a^2 c x^2)^2 \arctan (a x)^2} \, dx\) [557]

3.6.57.1 Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=-\frac {1}{a c^2 x^3 \arctan (a x)}+\frac {a}{c^2 x \arctan (a x)}-\frac {a^3 x}{c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {a^2 \operatorname {CosIntegral}(2 \arctan (a x))}{c^2}-\frac {3 \text {Int}\left (\frac {1}{x^4 \arctan (a x)},x\right )}{a c^2}+\frac {a \text {Int}\left (\frac {1}{x^2 \arctan (a x)},x\right )}{c^2} \]

output

-1/a/c^2/x^3/arctan(a*x)+a/c^2/x/arctan(a*x)-a^3*x/c^2/(a^2*x^2+1)/arctan( 
a*x)+a^2*Ci(2*arctan(a*x))/c^2-3*Unintegrable(1/x^4/arctan(a*x),x)/a/c^2+a 
*Unintegrable(1/x^2/arctan(a*x),x)/c^2
 

3.6.57.2 Mathematica [N/A]

Not integrable

Time = 2.82 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx \]

input

Integrate[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]
 

output

Integrate[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2), x]
 

3.6.57.3 Rubi [N/A]

Not integrable

Time = 1.63 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5501, 27, 5461, 5377, 5501, 5461, 5377, 5503, 5439, 3042, 3793, 2009, 5505, 3042, 3793, 2009}

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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5461

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

\(\Big \downarrow \) 5377

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 5461

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

\(\Big \downarrow \) 5377

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

\(\Big \downarrow \) 5503

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

\(\Big \downarrow \) 5439

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

input

Int[1/(x^3*(c + a^2*c*x^2)^2*ArcTan[a*x]^2),x]
 

output

$Aborted
 

3.6.57.3.1 Defintions of rubi rules used

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 5377

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Sy 
mbol] :> Unintegrable[(d*x)^m*(a + b*ArcTan[c*x^n])^p, x] /; FreeQ[{a, b, c 
, d, m, n, p}, x]
 

rule 5439

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[d^q/c   Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar 
cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( 
q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
 

rule 5461

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

rule 5503

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

rule 5505

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1)   Subst[Int[(a + b*x)^p*(Sin[x]^m/ 
Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p 
}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q 
] || GtQ[d, 0])
 

3.6.57.4 Maple [N/A] (verified)

Not integrable

Time = 27.78 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{2}}d x\]

input

int(1/x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x)
 

output

int(1/x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x)
 

3.6.57.5 Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]

input

integrate(1/x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="fricas")
 

output

integral(1/((a^4*c^2*x^7 + 2*a^2*c^2*x^5 + c^2*x^3)*arctan(a*x)^2), x)
 

3.6.57.6 Sympy [N/A]

Not integrable

Time = 1.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\frac {\int \frac {1}{a^{4} x^{7} \operatorname {atan}^{2}{\left (a x \right )} + 2 a^{2} x^{5} \operatorname {atan}^{2}{\left (a x \right )} + x^{3} \operatorname {atan}^{2}{\left (a x \right )}}\, dx}{c^{2}} \]

input

integrate(1/x**3/(a**2*c*x**2+c)**2/atan(a*x)**2,x)
 

output

Integral(1/(a**4*x**7*atan(a*x)**2 + 2*a**2*x**5*atan(a*x)**2 + x**3*atan( 
a*x)**2), x)/c**2
 

3.6.57.7 Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.82 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]

input

integrate(1/x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="maxima")
 

output

-((a^3*c^2*x^5 + a*c^2*x^3)*arctan(a*x)*integrate((5*a^2*x^2 + 3)/((a^5*c^ 
2*x^8 + 2*a^3*c^2*x^6 + a*c^2*x^4)*arctan(a*x)), x) + 1)/((a^3*c^2*x^5 + a 
*c^2*x^3)*arctan(a*x))
 

3.6.57.8 Giac [N/A]

Not integrable

Time = 89.28 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]

input

integrate(1/x^3/(a^2*c*x^2+c)^2/arctan(a*x)^2,x, algorithm="giac")
 

output

sage0*x
 

3.6.57.9 Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

input

int(1/(x^3*atan(a*x)^2*(c + a^2*c*x^2)^2),x)
 

output

int(1/(x^3*atan(a*x)^2*(c + a^2*c*x^2)^2), x)