\(\int \frac {x^4 \arctan (a x)^3}{c+a^2 c x^2} \, dx\) [387]

Optimal result

Integrand size = 22, antiderivative size = 217 \[ \int \frac {x^4 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {x \arctan (a x)}{a^4 c}-\frac {\arctan (a x)^2}{2 a^5 c}-\frac {x^2 \arctan (a x)^2}{2 a^3 c}-\frac {4 i \arctan (a x)^3}{3 a^5 c}-\frac {x \arctan (a x)^3}{a^4 c}+\frac {x^3 \arctan (a x)^3}{3 a^2 c}+\frac {\arctan (a x)^4}{4 a^5 c}-\frac {4 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^5 c}-\frac {\log \left (1+a^2 x^2\right )}{2 a^5 c}-\frac {4 i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^5 c}-\frac {2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{a^5 c} \]

[Out]

x*arctan(a*x)/a^4/c-1/2*arctan(a*x)^2/a^5/c-1/2*x^2*arctan(a*x)^2/a^3/c-4/3*I*arctan(a*x)^3/a^5/c-x*arctan(a*x
)^3/a^4/c+1/3*x^3*arctan(a*x)^3/a^2/c+1/4*arctan(a*x)^4/a^5/c-4*arctan(a*x)^2*ln(2/(1+I*a*x))/a^5/c-1/2*ln(a^2
*x^2+1)/a^5/c-4*I*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a^5/c-2*polylog(3,1-2/(1+I*a*x))/a^5/c

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5036, 4946, 4930, 266, 5004, 5040, 4964, 5114, 6745} \[ \int \frac {x^4 \arctan (a x)^3}{c+a^2 c x^2} \, dx=-\frac {4 i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^5 c}+\frac {\arctan (a x)^4}{4 a^5 c}-\frac {4 i \arctan (a x)^3}{3 a^5 c}-\frac {\arctan (a x)^2}{2 a^5 c}-\frac {4 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^5 c}-\frac {2 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{a^5 c}-\frac {x \arctan (a x)^3}{a^4 c}+\frac {x \arctan (a x)}{a^4 c}-\frac {x^2 \arctan (a x)^2}{2 a^3 c}+\frac {x^3 \arctan (a x)^3}{3 a^2 c}-\frac {\log \left (a^2 x^2+1\right )}{2 a^5 c} \]

[In]

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

[Out]

(x*ArcTan[a*x])/(a^4*c) - ArcTan[a*x]^2/(2*a^5*c) - (x^2*ArcTan[a*x]^2)/(2*a^3*c) - (((4*I)/3)*ArcTan[a*x]^3)/
(a^5*c) - (x*ArcTan[a*x]^3)/(a^4*c) + (x^3*ArcTan[a*x]^3)/(3*a^2*c) + ArcTan[a*x]^4/(4*a^5*c) - (4*ArcTan[a*x]
^2*Log[2/(1 + I*a*x)])/(a^5*c) - Log[1 + a^2*x^2]/(2*a^5*c) - ((4*I)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)]
)/(a^5*c) - (2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(a^5*c)

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 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 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 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[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 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 5040

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*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 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}& = -\frac {\int \frac {x^2 \arctan (a x)^3}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x^2 \arctan (a x)^3 \, dx}{a^2 c} \\ & = \frac {x^3 \arctan (a x)^3}{3 a^2 c}+\frac {\int \frac {\arctan (a x)^3}{c+a^2 c x^2} \, dx}{a^4}-\frac {\int \arctan (a x)^3 \, dx}{a^4 c}-\frac {\int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx}{a c} \\ & = -\frac {x \arctan (a x)^3}{a^4 c}+\frac {x^3 \arctan (a x)^3}{3 a^2 c}+\frac {\arctan (a x)^4}{4 a^5 c}-\frac {\int x \arctan (a x)^2 \, dx}{a^3 c}+\frac {\int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{a^3 c}+\frac {3 \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx}{a^3 c} \\ & = -\frac {x^2 \arctan (a x)^2}{2 a^3 c}-\frac {4 i \arctan (a x)^3}{3 a^5 c}-\frac {x \arctan (a x)^3}{a^4 c}+\frac {x^3 \arctan (a x)^3}{3 a^2 c}+\frac {\arctan (a x)^4}{4 a^5 c}-\frac {\int \frac {\arctan (a x)^2}{i-a x} \, dx}{a^4 c}-\frac {3 \int \frac {\arctan (a x)^2}{i-a x} \, dx}{a^4 c}+\frac {\int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx}{a^2 c} \\ & = -\frac {x^2 \arctan (a x)^2}{2 a^3 c}-\frac {4 i \arctan (a x)^3}{3 a^5 c}-\frac {x \arctan (a x)^3}{a^4 c}+\frac {x^3 \arctan (a x)^3}{3 a^2 c}+\frac {\arctan (a x)^4}{4 a^5 c}-\frac {4 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^5 c}+\frac {\int \arctan (a x) \, dx}{a^4 c}-\frac {\int \frac {\arctan (a x)}{1+a^2 x^2} \, dx}{a^4 c}+\frac {2 \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^4 c}+\frac {6 \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^4 c} \\ & = \frac {x \arctan (a x)}{a^4 c}-\frac {\arctan (a x)^2}{2 a^5 c}-\frac {x^2 \arctan (a x)^2}{2 a^3 c}-\frac {4 i \arctan (a x)^3}{3 a^5 c}-\frac {x \arctan (a x)^3}{a^4 c}+\frac {x^3 \arctan (a x)^3}{3 a^2 c}+\frac {\arctan (a x)^4}{4 a^5 c}-\frac {4 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^5 c}-\frac {4 i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^5 c}+\frac {i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^4 c}+\frac {(3 i) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^4 c}-\frac {\int \frac {x}{1+a^2 x^2} \, dx}{a^3 c} \\ & = \frac {x \arctan (a x)}{a^4 c}-\frac {\arctan (a x)^2}{2 a^5 c}-\frac {x^2 \arctan (a x)^2}{2 a^3 c}-\frac {4 i \arctan (a x)^3}{3 a^5 c}-\frac {x \arctan (a x)^3}{a^4 c}+\frac {x^3 \arctan (a x)^3}{3 a^2 c}+\frac {\arctan (a x)^4}{4 a^5 c}-\frac {4 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^5 c}-\frac {\log \left (1+a^2 x^2\right )}{2 a^5 c}-\frac {4 i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^5 c}-\frac {2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{a^5 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.71 \[ \int \frac {x^4 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {12 a x \arctan (a x)-6 \arctan (a x)^2-6 a^2 x^2 \arctan (a x)^2+16 i \arctan (a x)^3-12 a x \arctan (a x)^3+4 a^3 x^3 \arctan (a x)^3+3 \arctan (a x)^4-48 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-6 \log \left (1+a^2 x^2\right )+48 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-24 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )}{12 a^5 c} \]

[In]

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

[Out]

(12*a*x*ArcTan[a*x] - 6*ArcTan[a*x]^2 - 6*a^2*x^2*ArcTan[a*x]^2 + (16*I)*ArcTan[a*x]^3 - 12*a*x*ArcTan[a*x]^3
+ 4*a^3*x^3*ArcTan[a*x]^3 + 3*ArcTan[a*x]^4 - 48*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 6*Log[1 + a^2*
x^2] + (48*I)*ArcTan[a*x]*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - 24*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/(12*a^5*
c)

Maple [C] (warning: unable to verify)

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

Time = 18.16 (sec) , antiderivative size = 888, normalized size of antiderivative = 4.09

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(888\)
default	\(\text {Expression too large to display}\)	\(888\)
parts	\(\text {Expression too large to display}\)	\(898\)

[In]

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

[Out]

1/a^5*(1/3/c*arctan(a*x)^3*a^3*x^3-1/c*arctan(a*x)^3*a*x+1/c*arctan(a*x)^4-1/c*(1/2*x^2*arctan(a*x)^2*a^2-2*ar
ctan(a*x)^2*ln(a^2*x^2+1)+4*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))-1/6*I*arctan(a*x)*(6*arctan(a*x)*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))-12*arctan(a*x)*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+6*arctan(a*x)*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3+6*arctan(a*
x)*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)*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)^2)-6*arctan(a*x)*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)^2*c
sgn(I*(1+I*a*x)^2/(a^2*x^2+1))-6*arctan(a*x)*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+12*arctan(a*x)*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))-6*
arctan(a*x)*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3-6*arctan(a*x)*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)^2*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+6*arctan(a*x)*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+8*arctan(a*x)^2+24*I*arctan(a*x)*ln(2)+3*I*arctan(a*x)-6-6*I*a*x)-ln((1+I*a
*x)^2/(a^2*x^2+1)+1)-4*I*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))+2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))
+3/4*arctan(a*x)^4))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^4 \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{4} \operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/3072*(48*(7168*a^4*integrate(1/128*x^4*arctan(a*x)^3/(a^6*c*x^2 + a^4*c), x) + 768*a^4*integrate(1/128*x^4*a
rctan(a*x)*log(a^2*x^2 + 1)^2/(a^6*c*x^2 + a^4*c), x) + 1024*a^4*integrate(1/128*x^4*arctan(a*x)*log(a^2*x^2 +
1)/(a^6*c*x^2 + a^4*c), x) - 1024*a^3*integrate(1/128*x^3*arctan(a*x)^2/(a^6*c*x^2 + a^4*c), x) + 256*a^3*int
egrate(1/128*x^3*log(a^2*x^2 + 1)^2/(a^6*c*x^2 + a^4*c), x) - 3072*a^2*integrate(1/128*x^2*arctan(a*x)*log(a^2
*x^2 + 1)/(a^6*c*x^2 + a^4*c), x) + 768*a*integrate(1/128*x*arctan(a*x)^2*log(a^2*x^2 + 1)/(a^6*c*x^2 + a^4*c)
, x) + 192*a*integrate(1/128*x*log(a^2*x^2 + 1)^3/(a^6*c*x^2 + a^4*c), x) + 3072*a*integrate(1/128*x*arctan(a*
x)^2/(a^6*c*x^2 + a^4*c), x) - 768*a*integrate(1/128*x*log(a^2*x^2 + 1)^2/(a^6*c*x^2 + a^4*c), x) - 3*arctan(a
*x)^4/(a^5*c) - 384*integrate(1/128*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^6*c*x^2 + a^4*c), x))*a^5*c + 128*(a^3*x
^3 - 3*a*x)*arctan(a*x)^3 + 240*arctan(a*x)^4 - 9*log(a^2*x^2 + 1)^4 - 24*(4*(a^3*x^3 - 3*a*x)*arctan(a*x) + 3
*arctan(a*x)^2)*log(a^2*x^2 + 1)^2)/(a^5*c)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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