Integrand size = 20, antiderivative size = 564 \[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \arctan (c+d x)}{d^3}-\frac {b f^2 (a+b \arctan (c+d x))^2}{2 d^3}-\frac {3 i b f (d e-c f) (a+b \arctan (c+d x))^2}{d^3}-\frac {3 b f (d e-c f) (c+d x) (a+b \arctan (c+d x))^2}{d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) (a+b \arctan (c+d x))^3}{3 d^3 f}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {6 b^2 f (d e-c f) (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}-\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^3} \]
a*b^2*f^2*x/d^2+b^3*f^2*(d*x+c)*arctan(d*x+c)/d^3-1/2*b*f^2*(a+b*arctan(d* x+c))^2/d^3-3*I*b*f*(-c*f+d*e)*(a+b*arctan(d*x+c))^2/d^3-3*b*f*(-c*f+d*e)* (d*x+c)*(a+b*arctan(d*x+c))^2/d^3-1/2*b*f^2*(d*x+c)^2*(a+b*arctan(d*x+c))^ 2/d^3+1/3*I*(3*d^2*e^2-6*c*d*e*f-(-3*c^2+1)*f^2)*(a+b*arctan(d*x+c))^3/d^3 -1/3*(-c*f+d*e)*(d^2*e^2-2*c*d*e*f-(-c^2+3)*f^2)*(a+b*arctan(d*x+c))^3/d^3 /f+1/3*(f*x+e)^3*(a+b*arctan(d*x+c))^3/f-6*b^2*f*(-c*f+d*e)*(a+b*arctan(d* x+c))*ln(2/(1+I*(d*x+c)))/d^3+b*(3*d^2*e^2-6*c*d*e*f-(-3*c^2+1)*f^2)*(a+b* arctan(d*x+c))^2*ln(2/(1+I*(d*x+c)))/d^3-1/2*b^3*f^2*ln(1+(d*x+c)^2)/d^3-3 *I*b^3*f*(-c*f+d*e)*polylog(2,1-2/(1+I*(d*x+c)))/d^3+I*b^2*(3*d^2*e^2-6*c* d*e*f-(-3*c^2+1)*f^2)*(a+b*arctan(d*x+c))*polylog(2,1-2/(1+I*(d*x+c)))/d^3 +1/2*b^3*(3*d^2*e^2-6*c*d*e*f-(-3*c^2+1)*f^2)*polylog(3,1-2/(1+I*(d*x+c))) /d^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1844\) vs. \(2(564)=1128\).
Time = 13.86 (sec) , antiderivative size = 1844, normalized size of antiderivative = 3.27 \[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx =\text {Too large to display} \]
(a^2*(a*d^2*e^2 - 3*b*d*e*f + 2*b*c*f^2)*x)/d^2 - (a^2*f*(-2*a*d*e + b*f)* x^2)/(2*d) + (a^3*f^2*x^3)/3 + ((3*a^2*b*c*d^2*e^2 + 3*a^2*b*d*e*f - 3*a^2 *b*c^2*d*e*f - 3*a^2*b*c*f^2 + a^2*b*c^3*f^2)*ArcTan[c + d*x])/d^3 + a^2*b *x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcTan[c + d*x] + ((-3*a^2*b*d^2*e^2 + 6*a^ 2*b*c*d*e*f + a^2*b*f^2 - 3*a^2*b*c^2*f^2)*Log[1 + c^2 + 2*c*d*x + d^2*x^2 ])/(2*d^3) + (3*a*b^2*e^2*((-I)*ArcTan[c + d*x]^2 + (c + d*x)*ArcTan[c + d *x]^2 + 2*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] - I*PolyLog[2 , -E^((2*I)*ArcTan[c + d*x])]))/d + 6*a*b^2*e*f*(-(((c + d*x)*ArcTan[c + d *x])/d^2) + (I*c*ArcTan[c + d*x]^2)/d^2 - (c*(c + d*x)*ArcTan[c + d*x]^2)/ d^2 + ((1 + (c + d*x)^2)*ArcTan[c + d*x]^2)/(2*d^2) - (2*c*ArcTan[c + d*x] *Log[1 + E^((2*I)*ArcTan[c + d*x])])/d^2 - Log[1/Sqrt[1 + (c + d*x)^2]]/d^ 2 + (I*c*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/d^2) + (b^3*e^2*((-I)*Arc Tan[c + d*x]^3 + (c + d*x)*ArcTan[c + d*x]^3 + 3*ArcTan[c + d*x]^2*Log[1 + E^((2*I)*ArcTan[c + d*x])] - (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*A rcTan[c + d*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/2))/d + (b^3 *e*f*(ArcTan[c + d*x]*((3*I)*ArcTan[c + d*x] + (2*I)*c*ArcTan[c + d*x]^2 + (1 + (c + d*x)^2)*ArcTan[c + d*x]^2 - (c + d*x)*ArcTan[c + d*x]*(3 + 2*c* ArcTan[c + d*x]) - 6*Log[1 + E^((2*I)*ArcTan[c + d*x])] - 6*c*ArcTan[c + d *x]*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + (3*I)*(1 + 2*c*ArcTan[c + d*x])* PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] - 3*c*PolyLog[3, -E^((2*I)*ArcTa...
Time = 1.02 (sec) , antiderivative size = 554, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
| \(\displaystyle \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 5570 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2 (a+b \arctan (c+d x))^3}{d^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x))^2 (a+b \arctan (c+d x))^3d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {b \int \left ((c+d x) (a+b \arctan (c+d x))^2 f^3+3 (d e-c f) (a+b \arctan (c+d x))^2 f^2+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d f e-\left (3-c^2\right ) f^2\right )+f \left (3 d^2 e^2-6 c d f e-\left (1-3 c^2\right ) f^2\right ) (c+d x)\right ) (a+b \arctan (c+d x))^2}{(c+d x)^2+1}\right )d(c+d x)}{f}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 (a+b \arctan (c+d x))^3}{3 f}-\frac {b \left (-i b f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))-\frac {i f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) (a+b \arctan (c+d x))^3}{3 b}+\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) (a+b \arctan (c+d x))^3}{3 b}-f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^2+3 i f^2 (d e-c f) (a+b \arctan (c+d x))^2+3 f^2 (c+d x) (d e-c f) (a+b \arctan (c+d x))^2+6 b f^2 (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))+\frac {1}{2} f^3 (c+d x)^2 (a+b \arctan (c+d x))^2+\frac {1}{2} f^3 (a+b \arctan (c+d x))^2-a b f^3 (c+d x)-b^2 f^3 (c+d x) \arctan (c+d x)-\frac {1}{2} b^2 f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )+3 i b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )+\frac {1}{2} b^2 f^3 \log \left ((c+d x)^2+1\right )\right )}{f}}{d^3}\) |
(((d*e - c*f + f*(c + d*x))^3*(a + b*ArcTan[c + d*x])^3)/(3*f) - (b*(-(a*b *f^3*(c + d*x)) - b^2*f^3*(c + d*x)*ArcTan[c + d*x] + (f^3*(a + b*ArcTan[c + d*x])^2)/2 + (3*I)*f^2*(d*e - c*f)*(a + b*ArcTan[c + d*x])^2 + 3*f^2*(d *e - c*f)*(c + d*x)*(a + b*ArcTan[c + d*x])^2 + (f^3*(c + d*x)^2*(a + b*Ar cTan[c + d*x])^2)/2 - ((I/3)*f*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*( a + b*ArcTan[c + d*x])^3)/b + ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f - (3 - c^2 )*f^2)*(a + b*ArcTan[c + d*x])^3)/(3*b) + 6*b*f^2*(d*e - c*f)*(a + b*ArcTa n[c + d*x])*Log[2/(1 + I*(c + d*x))] - f*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c ^2)*f^2)*(a + b*ArcTan[c + d*x])^2*Log[2/(1 + I*(c + d*x))] + (b^2*f^3*Log [1 + (c + d*x)^2])/2 + (3*I)*b^2*f^2*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*( c + d*x))] - I*b*f*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcTan [c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))] - (b^2*f*(3*d^2*e^2 - 6*c*d *e*f - (1 - 3*c^2)*f^2)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/2))/f)/d^3
3.1.36.3.1 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 = 28.49 (sec) , antiderivative size = 5843, normalized size of antiderivative = 10.36
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(5843\) |
| default | \(\text {Expression too large to display}\) | \(5843\) |
| parts | \(\text {Expression too large to display}\) | \(6026\) |
\[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
integral(a^3*f^2*x^2 + 2*a^3*e*f*x + a^3*e^2 + (b^3*f^2*x^2 + 2*b^3*e*f*x + b^3*e^2)*arctan(d*x + c)^3 + 3*(a*b^2*f^2*x^2 + 2*a*b^2*e*f*x + a*b^2*e^ 2)*arctan(d*x + c)^2 + 3*(a^2*b*f^2*x^2 + 2*a^2*b*e*f*x + a^2*b*e^2)*arcta n(d*x + c), x)
Timed out. \[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\text {Timed out} \]
\[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
7/8*b^3*c^2*e^2*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 3*a*b^2*c^2* e^2*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - (3*arctan(d*x + c)*arcta n((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a*b^2*c^2*e^2 - 7/32 *(6*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)^2/d - 4*arctan(d*x + c)*arct an((d^2*x + c*d)/d)^3/d + arctan((d^2*x + c*d)/d)^4/d)*b^3*c^2*e^2 + 1/3*a ^3*f^2*x^3 + 7/8*b^3*e^2*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 28* b^3*d^2*f^2*integrate(1/32*x^4*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d^2*f^2*integrate(1/32*x^4*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) + 96*a*b^2*d^2*f^2* integrate(1/32*x^4*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 5 6*b^3*d^2*e*f*integrate(1/32*x^3*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^ 2 + 1), x) + 56*b^3*c*d*f^2*integrate(1/32*x^3*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 4*b^3*d^2*f^2*integrate(1/32*x^4*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*f*integrate(1/32*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) + 6*b^3*c*d*f^2*integrate(1/32*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*e*f*integrate(1/32*x^3*arctan(d*x + c)^2/(d^2 *x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*c*d*f^2*integrate(1/32*x^3*arcta n(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 28*b^3*d^2*e^2*integra...
\[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (e+f x)^2 (a+b \arctan (c+d x))^3 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x \]