Integrand size = 22, antiderivative size = 139 \[ \int \frac {(a+b \arctan (c x))^3}{d+i c d x} \, dx=\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {3 b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c d}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c d}+\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c d} \]
I*(a+b*arctan(c*x))^3*ln(2/(1+I*c*x))/c/d-3/2*b*(a+b*arctan(c*x))^2*polylo g(2,1-2/(1+I*c*x))/c/d+3/2*I*b^2*(a+b*arctan(c*x))*polylog(3,1-2/(1+I*c*x) )/c/d+3/4*b^3*polylog(4,1-2/(1+I*c*x))/c/d
Time = 0.06 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \arctan (c x))^3}{d+i c d x} \, dx=\frac {i \left (4 (a+b \arctan (c x))^3 \log \left (\frac {2 d}{d+i c d x}\right )+3 i b \left (2 (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )-b \left (2 i (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,\frac {i+c x}{-i+c x}\right )+b \operatorname {PolyLog}\left (4,\frac {i+c x}{-i+c x}\right )\right )\right )\right )}{4 c d} \]
((I/4)*(4*(a + b*ArcTan[c*x])^3*Log[(2*d)/(d + I*c*d*x)] + (3*I)*b*(2*(a + b*ArcTan[c*x])^2*PolyLog[2, (I + c*x)/(-I + c*x)] - b*((2*I)*(a + b*ArcTa n[c*x])*PolyLog[3, (I + c*x)/(-I + c*x)] + b*PolyLog[4, (I + c*x)/(-I + c* x)]))))/(c*d)
Time = 0.62 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5379, 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.
\(\displaystyle \int \frac {(a+b \arctan (c x))^3}{d+i c d x} \, dx\) |
\(\Big \downarrow \) 5379 |
\(\displaystyle \frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c d}-\frac {3 i b \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{d}\) |
\(\Big \downarrow \) 5529 |
\(\displaystyle \frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c d}-\frac {3 i b \left (i b \int \frac {(a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c}\right )}{d}\) |
\(\Big \downarrow \) 5533 |
\(\displaystyle \frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c d}-\frac {3 i b \left (i b \left (\frac {i \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx\right )-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c}\right )}{d}\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle \frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c d}-\frac {3 i b \left (i b \left (\frac {i \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c}+\frac {b \operatorname {PolyLog}\left (4,1-\frac {2}{i c x+1}\right )}{4 c}\right )-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c}\right )}{d}\) |
(I*(a + b*ArcTan[c*x])^3*Log[2/(1 + I*c*x)])/(c*d) - ((3*I)*b*(((-1/2*I)*( a + b*ArcTan[c*x])^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c + I*b*(((I/2)*(a + b *ArcTan[c*x])*PolyLog[3, 1 - 2/(1 + I*c*x)])/c + (b*PolyLog[4, 1 - 2/(1 + I*c*x)])/(4*c))))/d
3.2.23.3.1 Defintions of rubi rules used
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] + Simp[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 ]
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 = 1629, normalized size of antiderivative = 11.72
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1629\) |
default | \(\text {Expression too large to display}\) | \(1629\) |
parts | \(\text {Expression too large to display}\) | \(1640\) |
1/c*(-1/2*I*a^3/d*ln(c^2*x^2+1)+a^3/d*arctan(c*x)+b^3/d*(-I*ln(1+I*c*x)*ar ctan(c*x)^3+3*I*(1/3*arctan(c*x)^3*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))+1/6*I*P i*(csgn((1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3+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((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))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-c sgn(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+I*c*x)^2/(c^2*x^2+1)+1))^3+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+I*c*x)^2/(c^2*x^2+ 1)+1))+csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^2-1)*ar ctan(c*x)^3-1/2*I*arctan(c*x)^2*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/2*ar ctan(c*x)*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+1/4*I*polylog(4,-(1+I*c*x)^2 /(c^2*x^2+1))-1/6*I*arctan(c*x)^4))+3*a*b^2/d*(-I*ln(1+I*c*x)*arctan(c*x)^ 2+2*I*(1/2*arctan(c*x)^2*ln(2*I*(1+I*c*x)^2/(c^2*x^2+1))+1/4*I*Pi*(csgn((1 +I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1))^3+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((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)+...
\[ \int \frac {(a+b \arctan (c x))^3}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{i \, c d x + d} \,d x } \]
integral(-1/8*(b^3*log(-(c*x + I)/(c*x - I))^3 - 6*I*a*b^2*log(-(c*x + I)/ (c*x - I))^2 - 12*a^2*b*log(-(c*x + I)/(c*x - I)) + 8*I*a^3)/(c*d*x - I*d) , x)
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{d+i c d x} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \arctan (c x))^3}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{i \, c d x + d} \,d x } \]
-I*a^3*log(I*c*d*x + d)/(c*d) + 1/128*(16*b^3*arctan(c*x)^4 + 16*I*b^3*arc tan(c*x)^3*log(c^2*x^2 + 1) + 4*I*b^3*arctan(c*x)*log(c^2*x^2 + 1)^3 - b^3 *log(c^2*x^2 + 1)^4 + 16*(b^3*arctan(c*x)^4/(c*d) + 8*b^3*c*integrate(1/16 *x*log(c^2*x^2 + 1)^3/(c^2*d*x^2 + d), x) + 8*a*b^2*arctan(c*x)^3/(c*d) + 12*a^2*b*arctan(c*x)^2/(c*d))*c*d - 128*I*c*d*integrate(1/32*(40*b^3*c*x*a rctan(c*x)^3 + 6*b^3*c*x*arctan(c*x)*log(c^2*x^2 + 1)^2 + 96*a*b^2*c*x*arc tan(c*x)^2 + 96*a^2*b*c*x*arctan(c*x) + 12*b^3*arctan(c*x)^2*log(c^2*x^2 + 1) + b^3*log(c^2*x^2 + 1)^3)/(c^2*d*x^2 + d), x))/(c*d)
\[ \int \frac {(a+b \arctan (c x))^3}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3}}{i \, c d x + d} \,d x } \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{d+i c d x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]