Integrand size = 20, antiderivative size = 310 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^3} \, dx=-\frac {3}{2} i a^2 c \arctan (a x)^2-\frac {3 a c \arctan (a x)^2}{2 x}-\frac {1}{2} a^2 c \arctan (a x)^3-\frac {c \arctan (a x)^3}{2 x^2}+2 a^2 c \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+3 a^2 c \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i a^2 c \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} a^2 c \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a^2 c \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i a^2 c \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i a^2 c \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]
-3/2*I*a^2*c*arctan(a*x)^2-3/2*a*c*arctan(a*x)^2/x-1/2*a^2*c*arctan(a*x)^3 -1/2*c*arctan(a*x)^3/x^2-2*a^2*c*arctan(a*x)^3*arctanh(-1+2/(1+I*a*x))+3*a ^2*c*arctan(a*x)*ln(2-2/(1-I*a*x))-3/2*I*a^2*c*polylog(2,-1+2/(1-I*a*x))-3 /2*I*a^2*c*arctan(a*x)^2*polylog(2,1-2/(1+I*a*x))+3/2*I*a^2*c*arctan(a*x)^ 2*polylog(2,-1+2/(1+I*a*x))-3/2*a^2*c*arctan(a*x)*polylog(3,1-2/(1+I*a*x)) +3/2*a^2*c*arctan(a*x)*polylog(3,-1+2/(1+I*a*x))+3/4*I*a^2*c*polylog(4,1-2 /(1+I*a*x))-3/4*I*a^2*c*polylog(4,-1+2/(1+I*a*x))
Time = 0.30 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^3} \, dx=\frac {1}{2} a^2 c \arctan (a x)^3+\frac {c \left (-1-a^2 x^2\right ) \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^2 c \left (-\frac {1}{3} \arctan (a x) \left (\frac {3 \arctan (a x)}{a x}+\arctan (a x) (3 i+\arctan (a x))-6 \log \left (1-e^{2 i \arctan (a x)}\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )-\frac {1}{64} i a^2 c \left (\pi ^4-32 \arctan (a x)^4+64 i \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )-64 i \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )-96 \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-96 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+96 i \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-96 i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right ) \]
(a^2*c*ArcTan[a*x]^3)/2 + (c*(-1 - a^2*x^2)*ArcTan[a*x]^3)/(2*x^2) + (3*a^ 2*c*(-1/3*(ArcTan[a*x]*((3*ArcTan[a*x])/(a*x) + ArcTan[a*x]*(3*I + ArcTan[ a*x]) - 6*Log[1 - E^((2*I)*ArcTan[a*x])])) - I*PolyLog[2, E^((2*I)*ArcTan[ a*x])]))/2 - (I/64)*a^2*c*(Pi^4 - 32*ArcTan[a*x]^4 + (64*I)*ArcTan[a*x]^3* Log[1 - E^((-2*I)*ArcTan[a*x])] - (64*I)*ArcTan[a*x]^3*Log[1 + E^((2*I)*Ar cTan[a*x])] - 96*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] - 96*Arc Tan[a*x]^2*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + (96*I)*ArcTan[a*x]*PolyLog [3, E^((-2*I)*ArcTan[a*x])] - (96*I)*ArcTan[a*x]*PolyLog[3, -E^((2*I)*ArcT an[a*x])] + 48*PolyLog[4, E^((-2*I)*ArcTan[a*x])] + 48*PolyLog[4, -E^((2*I )*ArcTan[a*x])])
Time = 2.06 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {5485, 5357, 5361, 5453, 5361, 5419, 5459, 5403, 2897, 5523, 5529, 5533, 7164}
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.
| \(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int \frac {\arctan (a x)^3}{x}dx+c \int \frac {\arctan (a x)^3}{x^3}dx\) |
| \(\Big \downarrow \) 5357 |
| \(\displaystyle c \int \frac {\arctan (a x)^3}{x^3}dx+a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle c \left (\frac {3}{2} a \int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle c \left (\frac {3}{2} a \left (\int \frac {\arctan (a x)^2}{x^2}dx-a^2 \int \frac {\arctan (a x)^2}{a^2 x^2+1}dx\right )-\frac {\arctan (a x)^3}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle c \left (\frac {3}{2} a \left (a^2 \left (-\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx\right )+2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{x}\right )-\frac {\arctan (a x)^3}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle c \left (\frac {3}{2} a \left (2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )-\frac {\arctan (a x)^3}{2 x^2}\right )+a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+c \left (-\frac {\arctan (a x)^3}{2 x^2}+\frac {3}{2} a \left (2 a \left (i \int \frac {\arctan (a x)}{x (a x+i)}dx-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+c \left (-\frac {\arctan (a x)^3}{2 x^2}+\frac {3}{2} a \left (2 a \left (i \left (i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+c \left (-\frac {\arctan (a x)^3}{2 x^2}+\frac {3}{2} a \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )\) |
| \(\Big \downarrow \) 5523 |
| \(\displaystyle a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \left (\frac {1}{2} \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {1}{2} \int \frac {\arctan (a x)^2 \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\right )+c \left (-\frac {\arctan (a x)^3}{2 x^2}+\frac {3}{2} a \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \left (\frac {1}{2} \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-i \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )+\frac {1}{2} \left (i \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{2 a}\right )\right )\right )+c \left (-\frac {\arctan (a x)^3}{2 x^2}+\frac {3}{2} a \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )\) |
| \(\Big \downarrow \) 5533 |
| \(\displaystyle a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \left (\frac {1}{2} \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx\right )\right )+\frac {1}{2} \left (i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )}{a^2 x^2+1}dx\right )-\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{2 a}\right )\right )\right )+c \left (-\frac {\arctan (a x)^3}{2 x^2}+\frac {3}{2} a \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle a^2 c \left (2 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-6 a \left (\frac {1}{2} \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )}{4 a}\right )\right )+\frac {1}{2} \left (i \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (4,\frac {2}{i a x+1}-1\right )}{4 a}\right )-\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )}{2 a}\right )\right )\right )+c \left (-\frac {\arctan (a x)^3}{2 x^2}+\frac {3}{2} a \left (2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}\right )\right )\) |
c*(-1/2*ArcTan[a*x]^3/x^2 + (3*a*(-(ArcTan[a*x]^2/x) - (a*ArcTan[a*x]^3)/3 + 2*a*((-1/2*I)*ArcTan[a*x]^2 + I*((-I)*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x) ] - PolyLog[2, -1 + 2/(1 - I*a*x)]/2))))/2) + a^2*c*(2*ArcTan[a*x]^3*ArcTa nh[1 - 2/(1 + I*a*x)] - 6*a*((((I/2)*ArcTan[a*x]^2*PolyLog[2, 1 - 2/(1 + I *a*x)])/a - I*(((I/2)*ArcTan[a*x]*PolyLog[3, 1 - 2/(1 + I*a*x)])/a + PolyL og[4, 1 - 2/(1 + I*a*x)]/(4*a)))/2 + (((-1/2*I)*ArcTan[a*x]^2*PolyLog[2, - 1 + 2/(1 + I*a*x)])/a + I*(((I/2)*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 + I*a*x )])/a + PolyLog[4, -1 + 2/(1 + I*a*x)]/(4*a)))/2))
3.4.69.3.1 Defintions of rubi rules used
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
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] - Simp[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]
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] - Simp[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]
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] - Si mp[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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> 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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1))), x] + Si mp[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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Int[(ArcTanh[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x _)^2), x_Symbol] :> Simp[1/2 Int[Log[1 + u]*((a + b*ArcTan[c*x])^p/(d + e *x^2)), x], x] - Simp[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]
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_. )*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcTan[c*x])^p*(PolyLog[k + 1, u]/(2* c*d)), x] - Simp[b*p*(I/2) Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[k + 1 , u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c*x)))^2, 0]
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]
Time = 45.81 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.69
| method | result | size |
| derivativedivides | \(a^{2} \left (-\frac {c \arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 a^{2} x^{2}}-3 i c \arctan \left (a x \right )^{2}+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 i c \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+\frac {3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+c \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+3 c \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-c \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+6 i c \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )\) | \(525\) |
| default | \(a^{2} \left (-\frac {c \arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )-3 i a x +x \arctan \left (a x \right ) a \right ) \left (a x +i\right )}{2 a^{2} x^{2}}-3 i c \arctan \left (a x \right )^{2}+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 i c \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+\frac {3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+c \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+3 c \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-c \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )+6 i c \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )\) | \(525\) |
a^2*(-1/2*c*arctan(a*x)^2*(-I*arctan(a*x)-3*I*a*x+x*arctan(a*x)*a)*(I+a*x) /a^2/x^2-3*I*c*arctan(a*x)^2+6*c*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2 +1)^(1/2))+6*I*c*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*c*arctan(a*x)*l n((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-3*I*c*arctan(a*x)^2*polylog(2,(1+I*a*x)/( a^2*x^2+1)^(1/2))+6*c*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3 *I*c*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+c*arctan(a*x)^3*ln((1+I*a*x)/( a^2*x^2+1)^(1/2)+1)-3*I*c*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-3/2*c*ar ctan(a*x)*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))+3/2*I*c*arctan(a*x)^2*polylo g(2,-(1+I*a*x)^2/(a^2*x^2+1))+c*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^( 1/2))-3/4*I*c*polylog(4,-(1+I*a*x)^2/(a^2*x^2+1))+3*c*arctan(a*x)*ln(1-(1+ I*a*x)/(a^2*x^2+1)^(1/2))-3*I*c*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^ 2+1)^(1/2))-c*arctan(a*x)^3*ln((1+I*a*x)^2/(a^2*x^2+1)+1)+6*I*c*polylog(4, (1+I*a*x)/(a^2*x^2+1)^(1/2)))
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^3} \, dx=c \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx\right ) \]
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
-1/64*(4*c*arctan(a*x)^3 - 3*c*arctan(a*x)*log(a^2*x^2 + 1)^2 - 64*x^2*int egrate(-1/64*(12*a^2*c*x^2*arctan(a*x)*log(a^2*x^2 + 1) - 12*a*c*x*arctan( a*x)^2 - 56*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x)^3 + 3*(a*c*x - 2*(a^ 4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x))*log(a^2*x^2 + 1)^2)/(a^2*x^5 + x^3 ), x))/x^2
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )}{x^3} \,d x \]