Integrand size = 22, antiderivative size = 124 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx=-\frac {i \arctan (a x)^4}{4 c}+\frac {\arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )}{c}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c}+\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )}{2 c}+\frac {3 i \operatorname {PolyLog}\left (4,-1+\frac {2}{1-i a x}\right )}{4 c} \]
-1/4*I*arctan(a*x)^4/c+arctan(a*x)^3*ln(2-2/(1-I*a*x))/c-3/2*I*arctan(a*x) ^2*polylog(2,-1+2/(1-I*a*x))/c+3/2*arctan(a*x)*polylog(3,-1+2/(1-I*a*x))/c +3/4*I*polylog(4,-1+2/(1-I*a*x))/c
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81 \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx=-\frac {i \left (\pi ^4-16 \arctan (a x)^4+64 i \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )-96 \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+96 i \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )\right )}{64 c} \]
((-1/64*I)*(Pi^4 - 16*ArcTan[a*x]^4 + (64*I)*ArcTan[a*x]^3*Log[1 - E^((-2* I)*ArcTan[a*x])] - 96*ArcTan[a*x]^2*PolyLog[2, E^((-2*I)*ArcTan[a*x])] + ( 96*I)*ArcTan[a*x]*PolyLog[3, E^((-2*I)*ArcTan[a*x])] + 48*PolyLog[4, E^((- 2*I)*ArcTan[a*x])]))/c
Time = 0.66 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5459, 5403, 5527, 5531, 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.
\(\displaystyle \int \frac {\arctan (a x)^3}{x \left (a^2 c x^2+c\right )} \, dx\) |
\(\Big \downarrow \) 5459 |
\(\displaystyle \frac {i \int \frac {\arctan (a x)^3}{x (a x+i)}dx}{c}-\frac {i \arctan (a x)^4}{4 c}\) |
\(\Big \downarrow \) 5403 |
\(\displaystyle \frac {i \left (3 i a \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )}{c}-\frac {i \arctan (a x)^4}{4 c}\) |
\(\Big \downarrow \) 5527 |
\(\displaystyle \frac {i \left (3 i a \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-i \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )}{c}-\frac {i \arctan (a x)^4}{4 c}\) |
\(\Big \downarrow \) 5531 |
\(\displaystyle \frac {i \left (3 i a \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-i \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 a}\right )\right )-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )}{c}-\frac {i \arctan (a x)^4}{4 c}\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {i \left (3 i a \left (\frac {i \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-i \left (\frac {\operatorname {PolyLog}\left (4,\frac {2}{1-i a x}-1\right )}{4 a}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{2 a}\right )\right )-i \arctan (a x)^3 \log \left (2-\frac {2}{1-i a x}\right )\right )}{c}-\frac {i \arctan (a x)^4}{4 c}\) |
((-1/4*I)*ArcTan[a*x]^4)/c + (I*((-I)*ArcTan[a*x]^3*Log[2 - 2/(1 - I*a*x)] + (3*I)*a*(((I/2)*ArcTan[a*x]^2*PolyLog[2, -1 + 2/(1 - I*a*x)])/a - I*((( -1/2*I)*ArcTan[a*x]*PolyLog[3, -1 + 2/(1 - I*a*x)])/a + PolyLog[4, -1 + 2/ (1 - I*a*x)]/(4*a)))))/c
3.4.92.3.1 Defintions of rubi rules used
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_.)/((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[(((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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 68.93 (sec) , antiderivative size = 1640, normalized size of antiderivative = 13.23
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1640\) |
default | \(\text {Expression too large to display}\) | \(1640\) |
parts | \(\text {Expression too large to display}\) | \(2057\) |
1/c*arctan(a*x)^3*ln(a*x)-1/2/c*ln(a^2*x^2+1)*arctan(a*x)^3-3/2/c*(-2/3*ar ctan(a*x)^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+1/6*I*arctan(a*x)^4-1/6*(-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*P i*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*c sgn(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))*c sgn(((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)*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*I*Pi*csgn(I*((1+I*a*x)^2/(a^...
\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx=\frac {\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{3} + x}\, dx}{c} \]
\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )} x} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x\,\left (c\,a^2\,x^2+c\right )} \,d x \]