Integrand size = 23, antiderivative size = 279 \[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\frac {2 (a+b \arctan (c+d x))^3 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right )}{d e}-\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 i b (a+b \arctan (c+d x))^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i (c+d x)}\right )}{2 d e}-\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 b^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i (c+d x)}\right )}{2 d e}+\frac {3 i b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i (c+d x)}\right )}{4 d e}-\frac {3 i b^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i (c+d x)}\right )}{4 d e} \]
-2*(a+b*arctan(d*x+c))^3*arctanh(-1+2/(1+I*(d*x+c)))/d/e-3/2*I*b*(a+b*arct an(d*x+c))^2*polylog(2,1-2/(1+I*(d*x+c)))/d/e+3/2*I*b*(a+b*arctan(d*x+c))^ 2*polylog(2,-1+2/(1+I*(d*x+c)))/d/e-3/2*b^2*(a+b*arctan(d*x+c))*polylog(3, 1-2/(1+I*(d*x+c)))/d/e+3/2*b^2*(a+b*arctan(d*x+c))*polylog(3,-1+2/(1+I*(d* x+c)))/d/e+3/4*I*b^3*polylog(4,1-2/(1+I*(d*x+c)))/d/e-3/4*I*b^3*polylog(4, -1+2/(1+I*(d*x+c)))/d/e
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(562\) vs. \(2(279)=558\).
Time = 0.55 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.01 \[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\frac {64 a^3 \log (c+d x)-24 i a^2 b \left (\pi ^2-4 \pi \arctan (c+d x)+8 \arctan (c+d x)^2-i \pi \log (16)+4 i \pi \log \left (1+e^{-2 i \arctan (c+d x)}\right )-8 i \arctan (c+d x) \log \left (1+e^{-2 i \arctan (c+d x)}\right )+8 i \arctan (c+d x) \log \left (1-e^{2 i \arctan (c+d x)}\right )+2 i \pi \log \left (1+c^2+2 c d x+d^2 x^2\right )+4 \operatorname {PolyLog}\left (2,-e^{-2 i \arctan (c+d x)}\right )+4 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c+d x)}\right )\right )+8 a b^2 \left (-i \pi ^3+16 i \arctan (c+d x)^3+24 \arctan (c+d x)^2 \log \left (1-e^{-2 i \arctan (c+d x)}\right )-24 \arctan (c+d x)^2 \log \left (1+e^{2 i \arctan (c+d x)}\right )+24 i \arctan (c+d x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c+d x)}\right )+24 i \arctan (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+12 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c+d x)}\right )-12 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )-i b^3 \left (\pi ^4-32 \arctan (c+d x)^4+64 i \arctan (c+d x)^3 \log \left (1-e^{-2 i \arctan (c+d x)}\right )-64 i \arctan (c+d x)^3 \log \left (1+e^{2 i \arctan (c+d x)}\right )-96 \arctan (c+d x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (c+d x)}\right )-96 \arctan (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+96 i \arctan (c+d x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (c+d x)}\right )-96 i \arctan (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (c+d x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (c+d x)}\right )\right )}{64 d e} \]
(64*a^3*Log[c + d*x] - (24*I)*a^2*b*(Pi^2 - 4*Pi*ArcTan[c + d*x] + 8*ArcTa n[c + d*x]^2 - I*Pi*Log[16] + (4*I)*Pi*Log[1 + E^((-2*I)*ArcTan[c + d*x])] - (8*I)*ArcTan[c + d*x]*Log[1 + E^((-2*I)*ArcTan[c + d*x])] + (8*I)*ArcTa n[c + d*x]*Log[1 - E^((2*I)*ArcTan[c + d*x])] + (2*I)*Pi*Log[1 + c^2 + 2*c *d*x + d^2*x^2] + 4*PolyLog[2, -E^((-2*I)*ArcTan[c + d*x])] + 4*PolyLog[2, E^((2*I)*ArcTan[c + d*x])]) + 8*a*b^2*((-I)*Pi^3 + (16*I)*ArcTan[c + d*x] ^3 + 24*ArcTan[c + d*x]^2*Log[1 - E^((-2*I)*ArcTan[c + d*x])] - 24*ArcTan[ c + d*x]^2*Log[1 + E^((2*I)*ArcTan[c + d*x])] + (24*I)*ArcTan[c + d*x]*Pol yLog[2, E^((-2*I)*ArcTan[c + d*x])] + (24*I)*ArcTan[c + d*x]*PolyLog[2, -E ^((2*I)*ArcTan[c + d*x])] + 12*PolyLog[3, E^((-2*I)*ArcTan[c + d*x])] - 12 *PolyLog[3, -E^((2*I)*ArcTan[c + d*x])]) - I*b^3*(Pi^4 - 32*ArcTan[c + d*x ]^4 + (64*I)*ArcTan[c + d*x]^3*Log[1 - E^((-2*I)*ArcTan[c + d*x])] - (64*I )*ArcTan[c + d*x]^3*Log[1 + E^((2*I)*ArcTan[c + d*x])] - 96*ArcTan[c + d*x ]^2*PolyLog[2, E^((-2*I)*ArcTan[c + d*x])] - 96*ArcTan[c + d*x]^2*PolyLog[ 2, -E^((2*I)*ArcTan[c + d*x])] + (96*I)*ArcTan[c + d*x]*PolyLog[3, E^((-2* I)*ArcTan[c + d*x])] - (96*I)*ArcTan[c + d*x]*PolyLog[3, -E^((2*I)*ArcTan[ c + d*x])] + 48*PolyLog[4, E^((-2*I)*ArcTan[c + d*x])] + 48*PolyLog[4, -E^ ((2*I)*ArcTan[c + d*x])]))/(64*d*e)
Time = 0.99 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5566, 27, 5357, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx\) |
| \(\Big \downarrow \) 5566 |
| \(\displaystyle \frac {\int \frac {(a+b \arctan (c+d x))^3}{e (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arctan (c+d x))^3}{c+d x}d(c+d x)}{d e}\) |
| \(\Big \downarrow \) 5357 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^3-6 b \int \frac {(a+b \arctan (c+d x))^2 \text {arctanh}\left (1-\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)}{d e}\) |
| \(\Big \downarrow \) 5523 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^3-6 b \left (\frac {1}{2} \int \frac {(a+b \arctan (c+d x))^2 \log \left (2-\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)-\frac {1}{2} \int \frac {(a+b \arctan (c+d x))^2 \log \left (\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)\right )}{d e}\) |
| \(\Big \downarrow \) 5529 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^3-6 b \left (\frac {1}{2} \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))^2-i b \int \frac {(a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)\right )+\frac {1}{2} \left (i b \int \frac {(a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,\frac {2}{i (c+d x)+1}-1\right )}{(c+d x)^2+1}d(c+d x)-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {2}{i (c+d x)+1}-1\right ) (a+b \arctan (c+d x))^2\right )\right )}{d e}\) |
| \(\Big \downarrow \) 5533 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^3-6 b \left (\frac {1}{2} \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)\right )\right )+\frac {1}{2} \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (3,\frac {2}{i (c+d x)+1}-1\right ) (a+b \arctan (c+d x))-\frac {1}{2} i b \int \frac {\operatorname {PolyLog}\left (3,\frac {2}{i (c+d x)+1}-1\right )}{(c+d x)^2+1}d(c+d x)\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {2}{i (c+d x)+1}-1\right ) (a+b \arctan (c+d x))^2\right )\right )}{d e}\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle \frac {2 \text {arctanh}\left (1-\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^3-6 b \left (\frac {1}{2} \left (\frac {1}{2} i \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))^2-i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))+\frac {1}{4} b \operatorname {PolyLog}\left (4,1-\frac {2}{i (c+d x)+1}\right )\right )\right )+\frac {1}{2} \left (i b \left (\frac {1}{2} i \operatorname {PolyLog}\left (3,\frac {2}{i (c+d x)+1}-1\right ) (a+b \arctan (c+d x))+\frac {1}{4} b \operatorname {PolyLog}\left (4,\frac {2}{i (c+d x)+1}-1\right )\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,\frac {2}{i (c+d x)+1}-1\right ) (a+b \arctan (c+d x))^2\right )\right )}{d e}\) |
(2*(a + b*ArcTan[c + d*x])^3*ArcTanh[1 - 2/(1 + I*(c + d*x))] - 6*b*(((I/2 )*(a + b*ArcTan[c + d*x])^2*PolyLog[2, 1 - 2/(1 + I*(c + d*x))] - I*b*((I/ 2)*(a + b*ArcTan[c + d*x])*PolyLog[3, 1 - 2/(1 + I*(c + d*x))] + (b*PolyLo g[4, 1 - 2/(1 + I*(c + d*x))])/4))/2 + ((-1/2*I)*(a + b*ArcTan[c + d*x])^2 *PolyLog[2, -1 + 2/(1 + I*(c + d*x))] + I*b*((I/2)*(a + b*ArcTan[c + d*x]) *PolyLog[3, -1 + 2/(1 + I*(c + d*x))] + (b*PolyLog[4, -1 + 2/(1 + I*(c + d *x))])/4))/2))/(d*e)
3.1.17.3.1 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_)]*(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[(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[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[(f*(x/d))^m*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 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 = 1.81 (sec) , antiderivative size = 2313, normalized size of antiderivative = 8.29
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2313\) |
| default | \(\text {Expression too large to display}\) | \(2313\) |
| parts | \(\text {Expression too large to display}\) | \(2321\) |
1/d*(a^3/e*ln(d*x+c)+b^3/e*(ln(d*x+c)*arctan(d*x+c)^3-arctan(d*x+c)^3*ln(( 1+I*(d*x+c))^2/(1+(d*x+c)^2)-1)+arctan(d*x+c)^3*ln(1-(1+I*(d*x+c))/(1+(d*x +c)^2)^(1/2))-3*I*arctan(d*x+c)^2*polylog(2,(1+I*(d*x+c))/(1+(d*x+c)^2)^(1 /2))+6*arctan(d*x+c)*polylog(3,(1+I*(d*x+c))/(1+(d*x+c)^2)^(1/2))+6*I*poly log(4,(1+I*(d*x+c))/(1+(d*x+c)^2)^(1/2))+arctan(d*x+c)^3*ln(1+(1+I*(d*x+c) )/(1+(d*x+c)^2)^(1/2))-3*I*arctan(d*x+c)^2*polylog(2,-(1+I*(d*x+c))/(1+(d* x+c)^2)^(1/2))+6*arctan(d*x+c)*polylog(3,-(1+I*(d*x+c))/(1+(d*x+c)^2)^(1/2 ))+6*I*polylog(4,-(1+I*(d*x+c))/(1+(d*x+c)^2)^(1/2))+1/2*I*Pi*(csgn(I*((1+ I*(d*x+c))^2/(1+(d*x+c)^2)-1)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))*csgn(((1+ I*(d*x+c))^2/(1+(d*x+c)^2)-1)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))-csgn(((1+ I*(d*x+c))^2/(1+(d*x+c)^2)-1)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))^2+csgn(I* ((1+I*(d*x+c))^2/(1+(d*x+c)^2)-1))*csgn(I/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2) ))*csgn(I*((1+I*(d*x+c))^2/(1+(d*x+c)^2)-1)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^ 2)))-csgn(I*((1+I*(d*x+c))^2/(1+(d*x+c)^2)-1))*csgn(I*((1+I*(d*x+c))^2/(1+ (d*x+c)^2)-1)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))^2-csgn(I/(1+(1+I*(d*x+c)) ^2/(1+(d*x+c)^2)))*csgn(I*((1+I*(d*x+c))^2/(1+(d*x+c)^2)-1)/(1+(1+I*(d*x+c ))^2/(1+(d*x+c)^2)))^2+csgn(I*((1+I*(d*x+c))^2/(1+(d*x+c)^2)-1)/(1+(1+I*(d *x+c))^2/(1+(d*x+c)^2)))^3-csgn(I*((1+I*(d*x+c))^2/(1+(d*x+c)^2)-1)/(1+(1+ I*(d*x+c))^2/(1+(d*x+c)^2)))*csgn(((1+I*(d*x+c))^2/(1+(d*x+c)^2)-1)/(1+(1+ I*(d*x+c))^2/(1+(d*x+c)^2)))^2+csgn(((1+I*(d*x+c))^2/(1+(d*x+c)^2)-1)/(...
\[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
integral((b^3*arctan(d*x + c)^3 + 3*a*b^2*arctan(d*x + c)^2 + 3*a^2*b*arct an(d*x + c) + a^3)/(d*e*x + c*e), x)
\[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\frac {\int \frac {a^{3}}{c + d x}\, dx + \int \frac {b^{3} \operatorname {atan}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a b^{2} \operatorname {atan}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {3 a^{2} b \operatorname {atan}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
(Integral(a**3/(c + d*x), x) + Integral(b**3*atan(c + d*x)**3/(c + d*x), x ) + Integral(3*a*b**2*atan(c + d*x)**2/(c + d*x), x) + Integral(3*a**2*b*a tan(c + d*x)/(c + d*x), x))/e
\[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{3}}{d e x + c e} \,d x } \]
a^3*log(d*e*x + c*e)/(d*e) + integrate(1/32*(28*b^3*arctan(d*x + c)^3 + 3* b^3*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 96*a*b^2*arctan(d *x + c)^2 + 96*a^2*b*arctan(d*x + c))/(d*e*x + c*e), x)
Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3}{c\,e+d\,e\,x} \,d x \]