3.4.93 \(\int \frac {\text {ArcTan}(a x)^3}{x^2 (c+a^2 c x^2)} \, dx\) [393]

Optimal. Leaf size=122 \[ -\frac {i a \text {ArcTan}(a x)^3}{c}-\frac {\text {ArcTan}(a x)^3}{c x}-\frac {a \text {ArcTan}(a x)^4}{4 c}+\frac {3 a \text {ArcTan}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {3 i a \text {ArcTan}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}+\frac {3 a \text {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c} \]

[Out]

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

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Rubi [A]
time = 0.21, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5038, 4946, 5044, 4988, 5004, 5112, 6745} \begin {gather*} -\frac {3 i a \text {ArcTan}(a x) \text {Li}_2\left (\frac {2}{1-i a x}-1\right )}{c}-\frac {a \text {ArcTan}(a x)^4}{4 c}-\frac {i a \text {ArcTan}(a x)^3}{c}-\frac {\text {ArcTan}(a x)^3}{c x}+\frac {3 a \text {ArcTan}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}+\frac {3 a \text {Li}_3\left (\frac {2}{1-i a x}-1\right )}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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^2 \left (c+a^2 c x^2\right )} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^3}{x^2} \, dx}{c}\\ &=-\frac {\tan ^{-1}(a x)^3}{c x}-\frac {a \tan ^{-1}(a x)^4}{4 c}+\frac {(3 a) \int \frac {\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c}\\ &=-\frac {i a \tan ^{-1}(a x)^3}{c}-\frac {\tan ^{-1}(a x)^3}{c x}-\frac {a \tan ^{-1}(a x)^4}{4 c}+\frac {(3 i a) \int \frac {\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c}\\ &=-\frac {i a \tan ^{-1}(a x)^3}{c}-\frac {\tan ^{-1}(a x)^3}{c x}-\frac {a \tan ^{-1}(a x)^4}{4 c}+\frac {3 a \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {\left (6 a^2\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 {i a \tan ^{-1}(a x)^3}{c}-\frac {\tan ^{-1}(a x)^3}{c x}-\frac {a \tan ^{-1}(a x)^4}{4 c}+\frac {3 a \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {3 i a \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c}+\frac {\left (3 i a^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c}\\ &=-\frac {i a \tan ^{-1}(a x)^3}{c}-\frac {\tan ^{-1}(a x)^3}{c x}-\frac {a \tan ^{-1}(a x)^4}{4 c}+\frac {3 a \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {3 i a \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{c}+\frac {3 a \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )}{2 c}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 108, normalized size = 0.89 \begin {gather*} \frac {a \left (-\frac {i \pi ^3}{8}+i \text {ArcTan}(a x)^3-\frac {\text {ArcTan}(a x)^3}{a x}-\frac {1}{4} \text {ArcTan}(a x)^4+3 \text {ArcTan}(a x)^2 \log \left (1-e^{-2 i \text {ArcTan}(a x)}\right )+3 i \text {ArcTan}(a x) \text {PolyLog}\left (2,e^{-2 i \text {ArcTan}(a x)}\right )+\frac {3}{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^2*(c + a^2*c*x^2)),x]

[Out]

(a*((-1/8*I)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x) - ArcTan[a*x]^4/4 + 3*ArcTan[a*x]^2*Log[1 - E^((-2*I
)*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])]
)/2))/c

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 40.05, size = 1722, normalized size = 14.11

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1722\)
default	\(\text {Expression too large to display}\)	\(1722\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1024*(80*a*x*arctan(a*x)^4 - 3*a*x*log(a^2*x^2 + 1)^4 - (48*a*arctan(a*x)^4/c - 12288*a^3*integrate(1/128*x
^3*arctan(a*x)^2*log(a^2*x^2 + 1)/(a^2*c*x^4 + c*x^2), x) - 3*a*log(a^2*x^2 + 1)^4/c + 6144*a^2*integrate(1/12
8*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*c*x^4 + c*x^2), x) - 49152*a^2*integrate(1/128*x^2*arctan(a*x)*log(a
^2*x^2 + 1)/(a^2*c*x^4 + c*x^2), x) + 49152*a*integrate(1/128*x*arctan(a*x)^2/(a^2*c*x^4 + c*x^2), x) - 12288*
a*integrate(1/128*x*log(a^2*x^2 + 1)^2/(a^2*c*x^4 + c*x^2), x) + 114688*integrate(1/128*arctan(a*x)^3/(a^2*c*x
^4 + c*x^2), x) + 12288*integrate(1/128*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*c*x^4 + c*x^2), x))*c*x + 128*arct
an(a*x)^3 - 24*(a*x*arctan(a*x)^2 + 4*arctan(a*x))*log(a^2*x^2 + 1)^2)/(c*x)

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

[Out]

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

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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^{4} + x^{2}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^2\,\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^2*(c + a^2*c*x^2)),x)

[Out]

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

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