Integrand size = 22, antiderivative size = 122 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx=-\frac {i a \arctan (a x)^3}{c}-\frac {\arctan (a x)^3}{c x}-\frac {a \arctan (a x)^4}{4 c}+\frac {3 a \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {3 i a \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c}+\frac {3 a \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c} \]
-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*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c+3/2 *a*polylog(3,-1+2/(1-I*a*x))/c
Time = 0.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89 \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx=\frac {a \left (-\frac {i \pi ^3}{8}+i \arctan (a x)^3-\frac {\arctan (a x)^3}{a x}-\frac {1}{4} \arctan (a x)^4+3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )\right )}{c} \]
(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]*Po lyLog[2, E^((-2*I)*ArcTan[a*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/ 2))/c
Time = 0.87 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5453, 27, 5361, 5419, 5459, 5403, 5527, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3}{x^2 \left (a^2 c x^2+c\right )} \, dx\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2}dx}{c}-a^2 \int \frac {\arctan (a x)^3}{c \left (a^2 x^2+1\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^3}{x^2}dx}{c}-\frac {a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx}{c}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{x}}{c}-\frac {a^2 \int \frac {\arctan (a x)^3}{a^2 x^2+1}dx}{c}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{x}}{c}-\frac {a \arctan (a x)^4}{4 c}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle -\frac {a \arctan (a x)^4}{4 c}+\frac {-\frac {\arctan (a x)^3}{x}+3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )}{c}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle -\frac {a \arctan (a x)^4}{4 c}+\frac {-\frac {\arctan (a x)^3}{x}+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )}{c}\) |
| \(\Big \downarrow \) 5527 |
| \(\displaystyle -\frac {a \arctan (a x)^4}{4 c}+\frac {-\frac {\arctan (a x)^3}{x}+3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )}{c}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle -\frac {a \arctan (a x)^4}{4 c}+\frac {-\frac {\arctan (a x)^3}{x}+3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )}{c}\) |
-1/4*(a*ArcTan[a*x]^4)/c + (-(ArcTan[a*x]^3/x) + 3*a*((-1/3*I)*ArcTan[a*x] ^3 + I*((-I)*ArcTan[a*x]^2*Log[2 - 2/(1 - I*a*x)] + (2*I)*a*(((I/2)*ArcTan [a*x]*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - PolyLog[3, -1 + 2/(1 - I*a*x)]/( 4*a)))))/c
3.4.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(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[(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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 66.13 (sec) , antiderivative size = 1609, normalized size of antiderivative = 13.19
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1609\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1611\) |
| default | \(\text {Expression too large to display}\) | \(1611\) |
-a*arctan(a*x)^4/c-arctan(a*x)^3/c/x-3/c*(-1/4*a*arctan(a*x)^4-a*(arctan(a *x)^2*ln(a*x)-1/2*arctan(a*x)^2*ln(a^2*x^2+1)+arctan(a*x)^2*ln((1+I*a*x)/( a^2*x^2+1)^(1/2))-arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)-1/3*I*arctan (a*x)^3+1/4*(-I*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))-I*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)^3-2*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))^2-2*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))^2-2*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+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+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+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))+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))+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3+I*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+I*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)+2*I*Pi-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)^2)^2+I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*cs...
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{2}} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{4} + x^{2}}\, dx}{c} \]
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{2}} \,d x } \]
-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(1/128* x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*c*x^4 + c*x^2), x) - 49152*a^2*int egrate(1/128*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*c*x^4 + c*x^2), x) + 49 152*a*integrate(1/128*x*arctan(a*x)^2/(a^2*c*x^4 + c*x^2), x) - 12288*a*in tegrate(1/128*x*log(a^2*x^2 + 1)^2/(a^2*c*x^4 + c*x^2), x) + 114688*integr ate(1/128*arctan(a*x)^3/(a^2*c*x^4 + c*x^2), x) + 12288*integrate(1/128*ar ctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*c*x^4 + c*x^2), x))*c*x + 128*arctan(a*x )^3 - 24*(a*x*arctan(a*x)^2 + 4*arctan(a*x))*log(a^2*x^2 + 1)^2)/(c*x)
\[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]