3.4.95 \(\int \frac {\text {ArcTan}(a x)^3}{x^4 (c+a^2 c x^2)} \, dx\) [395]

Optimal. Leaf size=227 \[ -\frac {a^2 \text {ArcTan}(a x)}{c x}-\frac {a^3 \text {ArcTan}(a x)^2}{2 c}-\frac {a \text {ArcTan}(a x)^2}{2 c x^2}+\frac {4 i a^3 \text {ArcTan}(a x)^3}{3 c}-\frac {\text {ArcTan}(a x)^3}{3 c x^3}+\frac {a^2 \text {ArcTan}(a x)^3}{c x}+\frac {a^3 \text {ArcTan}(a x)^4}{4 c}+\frac {a^3 \log (x)}{c}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c}-\frac {4 a^3 \text {ArcTan}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \text {ArcTan}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \text {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.51, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5038, 4946, 272, 36, 29, 31, 5004, 5044, 4988, 5112, 6745} \begin {gather*} \frac {4 i a^3 \text {ArcTan}(a x) \text {Li}_2\left (\frac {2}{1-i a x}-1\right )}{c}+\frac {a^3 \text {ArcTan}(a x)^4}{4 c}+\frac {4 i a^3 \text {ArcTan}(a x)^3}{3 c}-\frac {a^3 \text {ArcTan}(a x)^2}{2 c}-\frac {4 a^3 \text {ArcTan}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \text {Li}_3\left (\frac {2}{1-i a x}-1\right )}{c}+\frac {a^3 \log (x)}{c}+\frac {a^2 \text {ArcTan}(a x)^3}{c x}-\frac {a^2 \text {ArcTan}(a x)}{c x}-\frac {a^3 \log \left (a^2 x^2+1\right )}{2 c}-\frac {\text {ArcTan}(a x)^3}{3 c x^3}-\frac {a \text {ArcTan}(a x)^2}{2 c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 4988

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

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 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 5044

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

Rule 5112

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTa
n[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*} \int \frac {\tan ^{-1}(a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^3}{x^4} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)^3}{3 c x^3}+a^4 \int \frac {\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx+\frac {a \int \frac {\tan ^{-1}(a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx}{c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^3}{x^2} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)^3}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^3}{c x}+\frac {a^3 \tan ^{-1}(a x)^4}{4 c}+\frac {a \int \frac {\tan ^{-1}(a x)^2}{x^3} \, dx}{c}-\frac {a^3 \int \frac {\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c}-\frac {\left (3 a^3\right ) \int \frac {\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c}\\ &=-\frac {a \tan ^{-1}(a x)^2}{2 c x^2}+\frac {4 i a^3 \tan ^{-1}(a x)^3}{3 c}-\frac {\tan ^{-1}(a x)^3}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^3}{c x}+\frac {a^3 \tan ^{-1}(a x)^4}{4 c}+\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx}{c}-\frac {\left (i a^3\right ) \int \frac {\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c}-\frac {\left (3 i a^3\right ) \int \frac {\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c}\\ &=-\frac {a \tan ^{-1}(a x)^2}{2 c x^2}+\frac {4 i a^3 \tan ^{-1}(a x)^3}{3 c}-\frac {\tan ^{-1}(a x)^3}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^3}{c x}+\frac {a^3 \tan ^{-1}(a x)^4}{4 c}-\frac {4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {a^2 \int \frac {\tan ^{-1}(a x)}{x^2} \, dx}{c}-\frac {a^4 \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{c}+\frac {\left (2 a^4\right ) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}+\frac {\left (6 a^4\right ) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}\\ &=-\frac {a^2 \tan ^{-1}(a x)}{c x}-\frac {a^3 \tan ^{-1}(a x)^2}{2 c}-\frac {a \tan ^{-1}(a x)^2}{2 c x^2}+\frac {4 i a^3 \tan ^{-1}(a x)^3}{3 c}-\frac {\tan ^{-1}(a x)^3}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^3}{c x}+\frac {a^3 \tan ^{-1}(a x)^4}{4 c}-\frac {4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c}+\frac {a^3 \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx}{c}-\frac {\left (i a^4\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}-\frac {\left (3 i a^4\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}\\ &=-\frac {a^2 \tan ^{-1}(a x)}{c x}-\frac {a^3 \tan ^{-1}(a x)^2}{2 c}-\frac {a \tan ^{-1}(a x)^2}{2 c x^2}+\frac {4 i a^3 \tan ^{-1}(a x)^3}{3 c}-\frac {\tan ^{-1}(a x)^3}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^3}{c x}+\frac {a^3 \tan ^{-1}(a x)^4}{4 c}-\frac {4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{c}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {a^2 \tan ^{-1}(a x)}{c x}-\frac {a^3 \tan ^{-1}(a x)^2}{2 c}-\frac {a \tan ^{-1}(a x)^2}{2 c x^2}+\frac {4 i a^3 \tan ^{-1}(a x)^3}{3 c}-\frac {\tan ^{-1}(a x)^3}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^3}{c x}+\frac {a^3 \tan ^{-1}(a x)^4}{4 c}-\frac {4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{c}+\frac {a^3 \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 c}-\frac {a^5 \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {a^2 \tan ^{-1}(a x)}{c x}-\frac {a^3 \tan ^{-1}(a x)^2}{2 c}-\frac {a \tan ^{-1}(a x)^2}{2 c x^2}+\frac {4 i a^3 \tan ^{-1}(a x)^3}{3 c}-\frac {\tan ^{-1}(a x)^3}{3 c x^3}+\frac {a^2 \tan ^{-1}(a x)^3}{c x}+\frac {a^3 \tan ^{-1}(a x)^4}{4 c}+\frac {a^3 \log (x)}{c}-\frac {a^3 \log \left (1+a^2 x^2\right )}{2 c}-\frac {4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {4 i a^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c}-\frac {2 a^3 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{c}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.37, size = 180, normalized size = 0.79 \begin {gather*} \frac {a^3 \left (\frac {i \pi ^3}{6}-\frac {\text {ArcTan}(a x)}{a x}-\frac {1}{2} \text {ArcTan}(a x)^2-\frac {\text {ArcTan}(a x)^2}{2 a^2 x^2}-\frac {4}{3} i \text {ArcTan}(a x)^3-\frac {\text {ArcTan}(a x)^3}{3 a^3 x^3}+\frac {\text {ArcTan}(a x)^3}{a x}+\frac {1}{4} \text {ArcTan}(a x)^4-4 \text {ArcTan}(a x)^2 \log \left (1-e^{-2 i \text {ArcTan}(a x)}\right )+\log \left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-4 i \text {ArcTan}(a x) \text {PolyLog}\left (2,e^{-2 i \text {ArcTan}(a x)}\right )-2 \text {PolyLog}\left (3,e^{-2 i \text {ArcTan}(a x)}\right )\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*((I/6)*Pi^3 - ArcTan[a*x]/(a*x) - ArcTan[a*x]^2/2 - ArcTan[a*x]^2/(2*a^2*x^2) - ((4*I)/3)*ArcTan[a*x]^3 -
ArcTan[a*x]^3/(3*a^3*x^3) + ArcTan[a*x]^3/(a*x) + ArcTan[a*x]^4/4 - 4*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[
a*x])] + Log[(a*x)/Sqrt[1 + a^2*x^2]] - (4*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - 2*PolyLog[3, E^
((-2*I)*ArcTan[a*x])]))/c

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 127.36, size = 5082, normalized size = 22.39

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(5082\)
default	\(\text {Expression too large to display}\)	\(5082\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arctan(a*x)^3/(a^2*c*x^6 + c*x^4), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{6} + x^{4}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atan(a*x)**3/(a**2*x**6 + x**4), x)/c

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________