\(\int \frac {(c+a^2 c x^2)^2 \arctan (a x)^2}{x^3} \, dx\) [272]

Optimal result

Integrand size = 22, antiderivative size = 207 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=-\frac {a c^2 \arctan (a x)}{x}-a^3 c^2 x \arctan (a x)-\frac {c^2 \arctan (a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \log (x)-2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-a^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+a^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \]

[Out]

-a*c^2*arctan(a*x)/x-a^3*c^2*x*arctan(a*x)-1/2*c^2*arctan(a*x)^2/x^2+1/2*a^4*c^2*x^2*arctan(a*x)^2-4*a^2*c^2*a
rctan(a*x)^2*arctanh(-1+2/(1+I*a*x))+a^2*c^2*ln(x)-2*I*a^2*c^2*arctan(a*x)*polylog(2,1-2/(1+I*a*x))+2*I*a^2*c^
2*arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-a^2*c^2*polylog(3,1-2/(1+I*a*x))+a^2*c^2*polylog(3,-1+2/(1+I*a*x))

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {5068, 4946, 5038, 272, 36, 29, 31, 5004, 4942, 5108, 5114, 6745, 5036, 4930, 266} \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2-a^3 c^2 x \arctan (a x)+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-a^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+a^2 c^2 \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )+a^2 c^2 \log (x)-\frac {c^2 \arctan (a x)^2}{2 x^2}-\frac {a c^2 \arctan (a x)}{x} \]

[In]

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

[Out]

-((a*c^2*ArcTan[a*x])/x) - a^3*c^2*x*ArcTan[a*x] - (c^2*ArcTan[a*x]^2)/(2*x^2) + (a^4*c^2*x^2*ArcTan[a*x]^2)/2
+ 4*a^2*c^2*ArcTan[a*x]^2*ArcTanh[1 - 2/(1 + I*a*x)] + a^2*c^2*Log[x] - (2*I)*a^2*c^2*ArcTan[a*x]*PolyLog[2,
1 - 2/(1 + I*a*x)] + (2*I)*a^2*c^2*ArcTan[a*x]*PolyLog[2, -1 + 2/(1 + I*a*x)] - a^2*c^2*PolyLog[3, 1 - 2/(1 +
I*a*x)] + a^2*c^2*PolyLog[3, -1 + 2/(1 + I*a*x)]

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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 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 4930

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

Rule 4942

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcTan[c*x])^p*ArcTanh[1 - 2/(1 +
I*c*x)], x] - Dist[2*b*c*p, Int[(a + b*ArcTan[c*x])^(p - 1)*(ArcTanh[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2)), x], x
] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]

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 5036

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

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 5068

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

Rule 5108

Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[L
og[1 + u]*((a + b*ArcTan[c*x])^p/(d + e*x^2)), x], x] - Dist[1/2, Int[Log[1 - u]*((a + b*ArcTan[c*x])^p/(d + e
*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c*x)))^
2, 0]

Rule 5114

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*p*(I/2), Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 -
u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2
*(I/(I - c*x)))^2, 0]

Rule 6745

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^2 \arctan (a x)^2}{x^3}+\frac {2 a^2 c^2 \arctan (a x)^2}{x}+a^4 c^2 x \arctan (a x)^2\right ) \, dx \\ & = c^2 \int \frac {\arctan (a x)^2}{x^3} \, dx+\left (2 a^2 c^2\right ) \int \frac {\arctan (a x)^2}{x} \, dx+\left (a^4 c^2\right ) \int x \arctan (a x)^2 \, dx \\ & = -\frac {c^2 \arctan (a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\left (a c^2\right ) \int \frac {\arctan (a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (8 a^3 c^2\right ) \int \frac {\arctan (a x) \text {arctanh}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (a^5 c^2\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {c^2 \arctan (a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\left (a c^2\right ) \int \frac {\arctan (a x)}{x^2} \, dx-\left (a^3 c^2\right ) \int \arctan (a x) \, dx+\left (4 a^3 c^2\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (4 a^3 c^2\right ) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {a c^2 \arctan (a x)}{x}-a^3 c^2 x \arctan (a x)-\frac {c^2 \arctan (a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\left (a^2 c^2\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx+\left (2 i a^3 c^2\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 i a^3 c^2\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^4 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx \\ & = -\frac {a c^2 \arctan (a x)}{x}-a^3 c^2 x \arctan (a x)-\frac {c^2 \arctan (a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c^2 \log \left (1+a^2 x^2\right )-2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-a^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+a^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {a c^2 \arctan (a x)}{x}-a^3 c^2 x \arctan (a x)-\frac {c^2 \arctan (a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} a^2 c^2 \log \left (1+a^2 x^2\right )-2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-a^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+a^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^4 c^2\right ) \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {a c^2 \arctan (a x)}{x}-a^3 c^2 x \arctan (a x)-\frac {c^2 \arctan (a x)^2}{2 x^2}+\frac {1}{2} a^4 c^2 x^2 \arctan (a x)^2+4 a^2 c^2 \arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+a^2 c^2 \log (x)-2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+2 i a^2 c^2 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-a^2 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+a^2 c^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=a^2 c^2 \left (-\frac {i \pi ^3}{12}-\frac {\arctan (a x)}{a x}-a x \arctan (a x)-\frac {\arctan (a x)^2}{2 a^2 x^2}+\frac {1}{2} a^2 x^2 \arctan (a x)^2+\frac {4}{3} i \arctan (a x)^3+2 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )-2 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+\log \left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+\frac {1}{2} \log \left (1+a^2 x^2\right )+2 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+2 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right ) \]

[In]

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

[Out]

a^2*c^2*((-1/12*I)*Pi^3 - ArcTan[a*x]/(a*x) - a*x*ArcTan[a*x] - ArcTan[a*x]^2/(2*a^2*x^2) + (a^2*x^2*ArcTan[a*
x]^2)/2 + ((4*I)/3)*ArcTan[a*x]^3 + 2*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] - 2*ArcTan[a*x]^2*Log[1 +
E^((2*I)*ArcTan[a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] + Log[1 + a^2*x^2]/2 + (2*I)*ArcTan[a*x]*PolyLog[2, E^((
-2*I)*ArcTan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + PolyLog[3, E^((-2*I)*ArcTan[a*x])
] - PolyLog[3, -E^((2*I)*ArcTan[a*x])])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 58.06 (sec) , antiderivative size = 1184, normalized size of antiderivative = 5.72

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1184\)
default	\(\text {Expression too large to display}\)	\(1184\)
parts	\(\text {Expression too large to display}\)	\(1614\)

[In]

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

[Out]

a^2*(1/2*a^2*c^2*x^2*arctan(a*x)^2+2*c^2*arctan(a*x)^2*ln(a*x)-1/2*c^2*arctan(a*x)^2/a^2/x^2-c^2*(2*arctan(a*x
)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-2*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-I*Pi*csgn(((1+I*a*x)^2/(a^
2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^3*arctan(a*x)^2-4*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*arctan(a*
x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1
)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2-4*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/2*arctan(a*x)*(I*
a*x-(a^2*x^2+1)^(1/2)+1)/a/x+polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+ln((1+I*a*x)^2/(a^2*x^2+1)+1)-ln((1+I*a*x)/(a
^2*x^2+1)^(1/2)+1)-ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2
*x^2+1)+1))^3*arctan(a*x)^2+1/2*arctan(a*x)*(I*a*x+(a^2*x^2+1)^(1/2)+1)/a/x-2*I*arctan(a*x)*polylog(2,-(1+I*a*
x)^2/(a^2*x^2+1))+arctan(a*x)*(a*x-I)+4*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*Pi*csgn(((1+I*
a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2-I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1
+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2+4*I*arct
an(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*Pi*arctan(a*x)^2+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+
I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x)^2+I*Pi*cs
gn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2*arctan(a*x
)^2-I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+
1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*arctan(a*x)^2))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/32*(2*a^4*c^2*x^4 - 4*a^4*c^2*x^2*integrate(4*x*arctan(a*x)^2 + x*log(a^2*x^2 + 1)^2, x) - 8*a^3*c^2*x^2*in
tegrate(-1/4*(12*(a^2*x^2 + 1)*a*x*arctan(a*x)^2 - 3*(a^2*x^2 + 1)*a*x*log(a^2*x^2 + 1)^2 + 12*(a^2*x^2 + 1)*a
rctan(a*x)*log(a^2*x^2 + 1) + (4*(a^2*x^2 + 1)^2*arctan(a*x)*cos(3*arctan(a*x))*log(a^2*x^2 + 1) - 12*(a^2*x^2
+ 1)^(3/2)*arctan(a*x)*cos(2*arctan(a*x))*log(a^2*x^2 + 1) - 4*sqrt(a^2*x^2 + 1)*arctan(a*x)*log(a^2*x^2 + 1)
+ (4*(a^2*x^2 + 1)^2*arctan(a*x)^2 - (a^2*x^2 + 1)^2*log(a^2*x^2 + 1)^2)*sin(3*arctan(a*x)) - 3*(4*(a^2*x^2 +
1)^(3/2)*arctan(a*x)^2 - (a^2*x^2 + 1)^(3/2)*log(a^2*x^2 + 1)^2)*sin(2*arctan(a*x)))*sqrt(a^2*x^2 + 1))/((a^2
*x^2 + 1)^4*cos(3*arctan(a*x))^2 + (a^2*x^2 + 1)^4*sin(3*arctan(a*x))^2 - 6*(a^2*x^2 + 1)^(7/2)*sin(3*arctan(a
*x))*sin(2*arctan(a*x)) + 9*(a^2*x^2 + 1)^3*cos(2*arctan(a*x))^2 + 9*(a^2*x^2 + 1)^3*sin(2*arctan(a*x))^2 + a^
2*x^2 + 6*(a^2*x^2 + 1)^2*cos(2*arctan(a*x)) + 9*(a^2*x^2 + 1)^2 - 2*(3*(a^2*x^2 + 1)^(7/2)*cos(2*arctan(a*x))
+ (a^2*x^2 + 1)^(5/2))*cos(3*arctan(a*x)) + 6*((a^2*x^2 + 1)^2*a*x*sin(3*arctan(a*x)) - 3*(a^2*x^2 + 1)^(3/2)
*a*x*sin(2*arctan(a*x)) + (a^2*x^2 + 1)^2*cos(3*arctan(a*x)) - 3*(a^2*x^2 + 1)^(3/2)*cos(2*arctan(a*x)) - sqrt
(a^2*x^2 + 1))*sqrt(a^2*x^2 + 1) + 1), x) - 8*a^3*c^2*x^2*integrate(1/4*(4*(a^2*x^2 + 1)*arctan(a*x)*log(a^2*x
^2 + 1) - (4*(a^2*x^2 + 1)*a*x*arctan(a*x)^2 - (a^2*x^2 + 1)*a*x*log(a^2*x^2 + 1)^2 + 4*(a^2*x^2 + 1)*arctan(a
*x)*log(a^2*x^2 + 1))*cos(2*arctan(a*x)) - (4*(a^2*x^2 + 1)*a*x*arctan(a*x)*log(a^2*x^2 + 1) - 4*(a^2*x^2 + 1)
*arctan(a*x)^2 + (a^2*x^2 + 1)*log(a^2*x^2 + 1)^2)*sin(2*arctan(a*x)))/(a^2*x^2 + 1), x) - 8*a^2*c^2*x^2*integ
rate((4*arctan(a*x)^2 + log(a^2*x^2 + 1)^2)/x, x) + 4*a^2*c^2*x^2*integrate(-(4*arctan(a*x)^2 - log(a^2*x^2 +
1)^2)/x, x) + 4*a^2*c^2*x^2*integrate(log(a^2*x^2 + 1)/x, x) - 8*a^2*c^2*x^2*log(x) + 8*a^2*c^2*x^2 - 4*c^2*x^
2*integrate((4*arctan(a*x)^2 + log(a^2*x^2 + 1)^2)/x^3, x) - 4*(a^4*c^2*x^4 - c^2)*arctan(a*x)^2 + (a^4*c^2*x^
4 - c^2)*log(a^2*x^2 + 1)^2 + 8*(a^3*c^2*x^3 + a*c^2*x)*arctan(a*x) - 2*(a^4*c^2*x^4 + a^2*c^2*x^2)*log(a^2*x^
2 + 1))/x^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^2 \arctan (a x)^2}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^2}{x^3} \,d x \]

[In]

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

[Out]

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