Integrand size = 25, antiderivative size = 128 \[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=\frac {(a+b \arctan (c x))^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {3 i b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {3 b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i c x}\right )}{4 d} \]
(a+b*arctan(c*x))^3*ln(2-2/(1+I*c*x))/d+3/2*I*b*(a+b*arctan(c*x))^2*polylo g(2,-1+2/(1+I*c*x))/d+3/2*b^2*(a+b*arctan(c*x))*polylog(3,-1+2/(1+I*c*x))/ d-3/4*I*b^3*polylog(4,-1+2/(1+I*c*x))/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(268\) vs. \(2(128)=256\).
Time = 0.61 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.09 \[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=-\frac {i \left (8 a b^2 \pi ^3+b^3 \pi ^4+64 a^3 \arctan (c x)+192 a^2 b \arctan (c x)^2+192 i a b^2 \arctan (c x)^2 \log \left (1-e^{-2 i \arctan (c x)}\right )+64 i b^3 \arctan (c x)^3 \log \left (1-e^{-2 i \arctan (c x)}\right )+192 i a^2 b \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )+64 i a^3 \log (c x)-32 i a^3 \log \left (1+c^2 x^2\right )-96 b^2 \arctan (c x) (2 a+b \arctan (c x)) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c x)}\right )+96 a^2 b \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+96 i a b^2 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+96 i b^3 \arctan (c x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c x)}\right )+48 b^3 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (c x)}\right )\right )}{64 d} \]
((-1/64*I)*(8*a*b^2*Pi^3 + b^3*Pi^4 + 64*a^3*ArcTan[c*x] + 192*a^2*b*ArcTa n[c*x]^2 + (192*I)*a*b^2*ArcTan[c*x]^2*Log[1 - E^((-2*I)*ArcTan[c*x])] + ( 64*I)*b^3*ArcTan[c*x]^3*Log[1 - E^((-2*I)*ArcTan[c*x])] + (192*I)*a^2*b*Ar cTan[c*x]*Log[1 - E^((2*I)*ArcTan[c*x])] + (64*I)*a^3*Log[c*x] - (32*I)*a^ 3*Log[1 + c^2*x^2] - 96*b^2*ArcTan[c*x]*(2*a + b*ArcTan[c*x])*PolyLog[2, E ^((-2*I)*ArcTan[c*x])] + 96*a^2*b*PolyLog[2, E^((2*I)*ArcTan[c*x])] + (96* I)*a*b^2*PolyLog[3, E^((-2*I)*ArcTan[c*x])] + (96*I)*b^3*ArcTan[c*x]*PolyL og[3, E^((-2*I)*ArcTan[c*x])] + 48*b^3*PolyLog[4, E^((-2*I)*ArcTan[c*x])]) )/d
Time = 0.64 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5403, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {\log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{d}-\frac {3 b c \int \frac {(a+b \arctan (c x))^2 \log \left (2-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{d}\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle \frac {\log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{d}-\frac {3 b c \left (i b \int \frac {(a+b \arctan (c x)) \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right )}{c^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))^2}{2 c}\right )}{d}\) |
| \(\Big \downarrow \) 5533 |
| \(\displaystyle \frac {\log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{d}-\frac {3 b c \left (i b \left (\frac {i \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{2 c}-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right )}{c^2 x^2+1}dx\right )-\frac {i \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))^2}{2 c}\right )}{d}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {\log \left (2-\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{d}-\frac {3 b c \left (i b \left (\frac {i \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))}{2 c}+\frac {b \operatorname {PolyLog}\left (4,\frac {2}{i c x+1}-1\right )}{4 c}\right )-\frac {i \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) (a+b \arctan (c x))^2}{2 c}\right )}{d}\) |
((a + b*ArcTan[c*x])^3*Log[2 - 2/(1 + I*c*x)])/d - (3*b*c*(((-1/2*I)*(a + b*ArcTan[c*x])^2*PolyLog[2, -1 + 2/(1 + I*c*x)])/c + I*b*(((I/2)*(a + b*Ar cTan[c*x])*PolyLog[3, -1 + 2/(1 + I*c*x)])/c + (b*PolyLog[4, -1 + 2/(1 + I *c*x)])/(4*c))))/d
3.2.30.3.1 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[(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 = 2.54 (sec) , antiderivative size = 2795, normalized size of antiderivative = 21.84
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(2795\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(2797\) |
| default | \(\text {Expression too large to display}\) | \(2797\) |
-1/2*a^3/d*ln(c^2*x^2+1)-I*a^3/d*arctan(c*x)+a^3/d*ln(x)+b^3/d*(arctan(c*x )^3*ln(c*x)-ln(c*x-I)*arctan(c*x)^3+arctan(c*x)^3*ln(2*I*(1+I*c*x)^2/(c^2* x^2+1))-arctan(c*x)^3*ln((1+I*c*x)^2/(c^2*x^2+1)-1)+arctan(c*x)^3*ln(1-(1+ I*c*x)/(c^2*x^2+1)^(1/2))-3*I*arctan(c*x)^2*polylog(2,(1+I*c*x)/(c^2*x^2+1 )^(1/2))+6*arctan(c*x)*polylog(3,(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*I*polylog( 4,(1+I*c*x)/(c^2*x^2+1)^(1/2))+arctan(c*x)^3*ln(1+(1+I*c*x)/(c^2*x^2+1)^(1 /2))-3*I*arctan(c*x)^2*polylog(2,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*arctan(c* x)*polylog(3,-(1+I*c*x)/(c^2*x^2+1)^(1/2))+6*I*polylog(4,-(1+I*c*x)/(c^2*x ^2+1)^(1/2))+1/2*I*Pi*(csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^ 2*x^2+1)+1))*csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1)) -csgn(((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-csgn(I/(( 1+I*c*x)^2/(c^2*x^2+1)+1))*csgn((1+I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/ (c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))+csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1 ))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2+csgn(I/((1+ I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1))*csgn(I*((1+I* c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1))-csgn(I/((1+I*c*x)^2/(c^ 2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)-1)/((1+I*c*x)^2/(c^2*x^2+1)+1 ))^2-2*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-csgn((1 +I*c*x)^2/(c^2*x^2+1))*csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+ 1)+1))^2-csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3-cs...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (109) = 218\).
Time = 0.28 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=\frac {-3 i \, b^{3} {\rm Li}_2\left (-\frac {2 \, c x}{c x - i} + 1\right ) \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 12 \, a b^{2} {\rm Li}_2\left (-\frac {2 \, c x}{c x - i} + 1\right ) \log \left (-\frac {c x + i}{c x - i}\right ) - 12 i \, a^{2} b {\rm Li}_2\left (\frac {c x + i}{c x - i} + 1\right ) + 8 \, a^{3} \log \left (x\right ) - 8 \, a^{3} \log \left (\frac {c x - i}{c}\right ) - 6 i \, b^{3} {\rm polylog}\left (4, -\frac {c x + i}{c x - i}\right ) + {\left (-i \, b^{3} \log \left (-\frac {c x + i}{c x - i}\right )^{3} - 6 \, a b^{2} \log \left (-\frac {c x + i}{c x - i}\right )^{2}\right )} \log \left (\frac {2 \, c x}{c x - i}\right ) - 6 \, {\left (-i \, b^{3} \log \left (-\frac {c x + i}{c x - i}\right ) - 2 \, a b^{2}\right )} {\rm polylog}\left (3, -\frac {c x + i}{c x - i}\right )}{8 \, d} \]
1/8*(-3*I*b^3*dilog(-2*c*x/(c*x - I) + 1)*log(-(c*x + I)/(c*x - I))^2 - 12 *a*b^2*dilog(-2*c*x/(c*x - I) + 1)*log(-(c*x + I)/(c*x - I)) - 12*I*a^2*b* dilog((c*x + I)/(c*x - I) + 1) + 8*a^3*log(x) - 8*a^3*log((c*x - I)/c) - 6 *I*b^3*polylog(4, -(c*x + I)/(c*x - I)) + (-I*b^3*log(-(c*x + I)/(c*x - I) )^3 - 6*a*b^2*log(-(c*x + I)/(c*x - I))^2)*log(2*c*x/(c*x - I)) - 6*(-I*b^ 3*log(-(c*x + I)/(c*x - I)) - 2*a*b^2)*polylog(3, -(c*x + I)/(c*x - I)))/d
\[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=- \frac {i \left (\int \frac {a^{3}}{c x^{2} - i x}\, dx + \int \frac {b^{3} \operatorname {atan}^{3}{\left (c x \right )}}{c x^{2} - i x}\, dx + \int \frac {3 a b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{c x^{2} - i x}\, dx + \int \frac {3 a^{2} b \operatorname {atan}{\left (c x \right )}}{c x^{2} - i x}\, dx\right )}{d} \]
-I*(Integral(a**3/(c*x**2 - I*x), x) + Integral(b**3*atan(c*x)**3/(c*x**2 - I*x), x) + Integral(3*a*b**2*atan(c*x)**2/(c*x**2 - I*x), x) + Integral( 3*a**2*b*atan(c*x)/(c*x**2 - I*x), x))/d
\[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (i \, c d x + d\right )} x} \,d x } \]
-a^3*(log(I*c*x + 1)/d - log(x)/d) + 1/512*(-64*I*b^3*arctan(c*x)^4 + 64*b ^3*arctan(c*x)^3*log(c^2*x^2 + 1) + 16*b^3*arctan(c*x)*log(c^2*x^2 + 1)^3 + 4*I*b^3*log(c^2*x^2 + 1)^4 - I*(64*b^3*arctan(c*x)^4/d + 6144*b^3*c^2*in tegrate(1/64*x^2*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^3 + d*x), x) + 3* b^3*log(c^2*x^2 + 1)^4/d + 512*a*b^2*arctan(c*x)^3/d + 768*a^2*b*arctan(c* x)^2/d + 6144*b^3*integrate(1/64*arctan(c*x)^2*log(c^2*x^2 + 1)/(c^2*d*x^3 + d*x), x) - 512*b^3*integrate(1/64*log(c^2*x^2 + 1)^3/(c^2*d*x^3 + d*x), x))*d - 512*d*integrate(1/32*(12*b^3*c*x*arctan(c*x)^2*log(c^2*x^2 + 1) + b^3*c*x*log(c^2*x^2 + 1)^3 - 96*a*b^2*arctan(c*x)^2 - 96*a^2*b*arctan(c*x ) + 4*(3*b^3*c^2*x^2 - 7*b^3)*arctan(c*x)^3 + 3*(b^3*c^2*x^2 - b^3)*arctan (c*x)*log(c^2*x^2 + 1)^2)/(c^2*d*x^3 + d*x), x))/d
\[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (i \, c d x + d\right )} x} \,d x } \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{x (d+i c d x)} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{x\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )} \,d x \]