Integrand size = 18, antiderivative size = 337 \[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=-\frac {3 i b f (a+b \arctan (c+d x))^2}{2 d^2}-\frac {3 b f (c+d x) (a+b \arctan (c+d x))^2}{2 d^2}+\frac {i (d e-c f) (a+b \arctan (c+d x))^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) (a+b \arctan (c+d x))^3}{2 d^2 f}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {3 b^2 f (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 b (d e-c f) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 i b^3 f \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}+\frac {3 i b^2 (d e-c f) (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2} \]
-3/2*I*b*f*(a+b*arctan(d*x+c))^2/d^2-3/2*b*f*(d*x+c)*(a+b*arctan(d*x+c))^2 /d^2+I*(-c*f+d*e)*(a+b*arctan(d*x+c))^3/d^2-1/2*(-c*f+d*e+f)*(d*e-(1+c)*f) *(a+b*arctan(d*x+c))^3/d^2/f+1/2*(f*x+e)^2*(a+b*arctan(d*x+c))^3/f-3*b^2*f *(a+b*arctan(d*x+c))*ln(2/(1+I*(d*x+c)))/d^2+3*b*(-c*f+d*e)*(a+b*arctan(d* x+c))^2*ln(2/(1+I*(d*x+c)))/d^2-3/2*I*b^3*f*polylog(2,1-2/(1+I*(d*x+c)))/d ^2+3*I*b^2*(-c*f+d*e)*(a+b*arctan(d*x+c))*polylog(2,1-2/(1+I*(d*x+c)))/d^2 +3/2*b^3*(-c*f+d*e)*polylog(3,1-2/(1+I*(d*x+c)))/d^2
Time = 5.07 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.76 \[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\frac {a^2 (2 a d e-3 b f-2 a c f) (c+d x)+a^3 f (c+d x)^2+3 a^2 b f \arctan (c+d x)-3 a^2 b (c+d x) (c f-d (2 e+f x)) \arctan (c+d x)+6 a b^2 f \left (-((c+d x) \arctan (c+d x))+\frac {1}{2} \left (1+(c+d x)^2\right ) \arctan (c+d x)^2-\log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )\right )-3 a^2 b (d e-c f) \log \left (1+(c+d x)^2\right )+6 a b^2 d e \left (\arctan (c+d x) \left ((-i+c+d x) \arctan (c+d x)+2 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )-6 a b^2 c f \left (\arctan (c+d x) \left ((-i+c+d x) \arctan (c+d x)+2 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )+b^3 f \left (\arctan (c+d x) \left (3 i \arctan (c+d x)-3 (c+d x) \arctan (c+d x)+\left (1+(c+d x)^2\right ) \arctan (c+d x)^2-6 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )+3 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )+2 b^3 d e \left (\arctan (c+d x)^2 \left ((-i+c+d x) \arctan (c+d x)+3 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-3 i \arctan (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )-2 b^3 c f \left (\arctan (c+d x)^2 \left ((-i+c+d x) \arctan (c+d x)+3 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-3 i \arctan (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )}{2 d^2} \]
(a^2*(2*a*d*e - 3*b*f - 2*a*c*f)*(c + d*x) + a^3*f*(c + d*x)^2 + 3*a^2*b*f *ArcTan[c + d*x] - 3*a^2*b*(c + d*x)*(c*f - d*(2*e + f*x))*ArcTan[c + d*x] + 6*a*b^2*f*(-((c + d*x)*ArcTan[c + d*x]) + ((1 + (c + d*x)^2)*ArcTan[c + d*x]^2)/2 - Log[1/Sqrt[1 + (c + d*x)^2]]) - 3*a^2*b*(d*e - c*f)*Log[1 + ( c + d*x)^2] + 6*a*b^2*d*e*(ArcTan[c + d*x]*((-I + c + d*x)*ArcTan[c + d*x] + 2*Log[1 + E^((2*I)*ArcTan[c + d*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]) - 6*a*b^2*c*f*(ArcTan[c + d*x]*((-I + c + d*x)*ArcTan[c + d*x] + 2*Log[1 + E^((2*I)*ArcTan[c + d*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]) + b^3*f*(ArcTan[c + d*x]*((3*I)*ArcTan[c + d*x] - 3*(c + d*x)*Ar cTan[c + d*x] + (1 + (c + d*x)^2)*ArcTan[c + d*x]^2 - 6*Log[1 + E^((2*I)*A rcTan[c + d*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]) + 2*b^3* d*e*(ArcTan[c + d*x]^2*((-I + c + d*x)*ArcTan[c + d*x] + 3*Log[1 + E^((2*I )*ArcTan[c + d*x])]) - (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/2) - 2*b^3*c*f*(Arc Tan[c + d*x]^2*((-I + c + d*x)*ArcTan[c + d*x] + 3*Log[1 + E^((2*I)*ArcTan [c + d*x])]) - (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x]) ] + (3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/2))/(2*d^2)
Time = 0.77 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5570, 27, 5389, 2009}
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.
| \(\displaystyle \int (e+f x) (a+b \arctan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 5570 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right ) (a+b \arctan (c+d x))^3}{d}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x)) (a+b \arctan (c+d x))^3d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {3 b \int \left (f^2 (a+b \arctan (c+d x))^2+\frac {((d e-c f+f) (d e-(c+1) f)+2 f (d e-c f) (c+d x)) (a+b \arctan (c+d x))^2}{(c+d x)^2+1}\right )d(c+d x)}{2 f}}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {3 b \left (-2 i b f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))-\frac {2 i f (d e-c f) (a+b \arctan (c+d x))^3}{3 b}+\frac {(-c f+d e+f) (d e-(c+1) f) (a+b \arctan (c+d x))^3}{3 b}-2 f (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^2+i f^2 (a+b \arctan (c+d x))^2+f^2 (c+d x) (a+b \arctan (c+d x))^2+2 b f^2 \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))-b^2 f (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )+i b^2 f^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )\right )}{2 f}}{d^2}\) |
(((d*e - c*f + f*(c + d*x))^2*(a + b*ArcTan[c + d*x])^3)/(2*f) - (3*b*(I*f ^2*(a + b*ArcTan[c + d*x])^2 + f^2*(c + d*x)*(a + b*ArcTan[c + d*x])^2 - ( ((2*I)/3)*f*(d*e - c*f)*(a + b*ArcTan[c + d*x])^3)/b + ((d*e + f - c*f)*(d *e - (1 + c)*f)*(a + b*ArcTan[c + d*x])^3)/(3*b) + 2*b*f^2*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))] - 2*f*(d*e - c*f)*(a + b*ArcTan[c + d*x] )^2*Log[2/(1 + I*(c + d*x))] + I*b^2*f^2*PolyLog[2, 1 - 2/(1 + I*(c + d*x) )] - (2*I)*b*f*(d*e - c*f)*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 + I *(c + d*x))] - b^2*f*(d*e - c*f)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))]))/(2* f))/d^2
3.1.37.3.1 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_)*((d_) + (e_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])^p/(e*(q + 1))), x] - S imp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p - 1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && I GtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.33 (sec) , antiderivative size = 8267, normalized size of antiderivative = 24.53
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(8267\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(8269\) |
| default | \(\text {Expression too large to display}\) | \(8269\) |
\[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int { {\left (f x + e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
integral(a^3*f*x + a^3*e + (b^3*f*x + b^3*e)*arctan(d*x + c)^3 + 3*(a*b^2* f*x + a*b^2*e)*arctan(d*x + c)^2 + 3*(a^2*b*f*x + a^2*b*e)*arctan(d*x + c) , x)
\[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int \left (a + b \operatorname {atan}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )\, dx \]
\[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int { {\left (f x + e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
7/8*b^3*c^2*e*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 3*a*b^2*c^2*e* arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - (3*arctan(d*x + c)*arctan((d ^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a*b^2*c^2*e - 7/32*(6*ar ctan(d*x + c)^2*arctan((d^2*x + c*d)/d)^2/d - 4*arctan(d*x + c)*arctan((d^ 2*x + c*d)/d)^3/d + arctan((d^2*x + c*d)/d)^4/d)*b^3*c^2*e + 7/8*b^3*e*arc tan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 56*b^3*d^2*f*integrate(1/64*x^3 *arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*d^2*f*integra te(1/64*x^3*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*d^2*f*integrate(1/64*x^3*arctan(d*x + c )^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*d^2*e*integrate(1/64*x^2*ar ctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 112*b^3*c*d*f*integrat e(1/64*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*d^ 2*f*integrate(1/64*x^3*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d ^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*d^2*e*integrate(1/64*x^2*arctan(d* x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x ) + 12*b^3*c*d*f*integrate(1/64*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*d^2*e*integrate (1/64*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 384*a*b^2* c*d*f*integrate(1/64*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 112*b^3*c*d*e*integrate(1/64*x*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*...
\[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int { {\left (f x + e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x \]