Integrand size = 20, antiderivative size = 382 \[ \int (e+f x)^2 (a+b \arctan (c+d x))^2 \, dx=\frac {b^2 f^2 x}{3 d^2}-\frac {2 a b f (d e-c f) x}{d^2}-\frac {b^2 f^2 \arctan (c+d x)}{3 d^3}-\frac {2 b^2 f (d e-c f) (c+d x) \arctan (c+d x)}{d^3}-\frac {b f^2 (c+d x)^2 (a+b \arctan (c+d x))}{3 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x))^2}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) (a+b \arctan (c+d x))^2}{3 d^3 f}+\frac {(e+f x)^3 (a+b \arctan (c+d x))^2}{3 f}+\frac {2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{3 d^3} \]
1/3*b^2*f^2*x/d^2-2*a*b*f*(-c*f+d*e)*x/d^2-1/3*b^2*f^2*arctan(d*x+c)/d^3-2 *b^2*f*(-c*f+d*e)*(d*x+c)*arctan(d*x+c)/d^3-1/3*b*f^2*(d*x+c)^2*(a+b*arcta n(d*x+c))/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 ))^2/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 ))^2/d^3/f+1/3*(f*x+e)^3*(a+b*arctan(d*x+c))^2/f+2/3*b*(3*d^2*e^2-6*c*d*e* f-(-3*c^2+1)*f^2)*(a+b*arctan(d*x+c))*ln(2/(1+I*(d*x+c)))/d^3+b^2*f*(-c*f+ d*e)*ln(1+(d*x+c)^2)/d^3+1/3*I*b^2*(3*d^2*e^2-6*c*d*e*f-(-3*c^2+1)*f^2)*po lylog(2,1-2/(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. \(801\) vs. \(2(382)=764\).
Time = 7.96 (sec) , antiderivative size = 801, normalized size of antiderivative = 2.10 \[ \int (e+f x)^2 (a+b \arctan (c+d x))^2 \, dx=a^2 e^2 x+a^2 e f x^2+\frac {1}{3} a^2 f^2 x^3+\frac {a b \left (-d f x (6 d e-4 c f+d f x)+2 \left (3 d e f-3 c^2 d e f+c^3 f^2+3 c \left (d^2 e^2-f^2\right )+d^3 x \left (3 e^2+3 e f x+f^2 x^2\right )\right ) \arctan (c+d x)+\left (-3 d^2 e^2+6 c d e f+\left (1-3 c^2\right ) f^2\right ) \log \left (1+(c+d x)^2\right )\right )}{3 d^3}+\frac {b^2 e^2 \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 )}{d}+\frac {b^2 e f \left (\left (1+2 i c-c^2+d^2 x^2\right ) \arctan (c+d x)^2-2 \arctan (c+d x) \left (c+d x+2 c \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )+\log \left (1+(c+d x)^2\right )+2 i c \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )}{d^2}+\frac {b^2 f^2 \left (1+(c+d x)^2\right )^{3/2} \left (\frac {c+d x}{\sqrt {1+(c+d x)^2}}+\frac {6 c (c+d x) \arctan (c+d x)}{\sqrt {1+(c+d x)^2}}+\frac {3 (c+d x) \arctan (c+d x)^2}{\sqrt {1+(c+d x)^2}}+\frac {3 c^2 (c+d x) \arctan (c+d x)^2}{\sqrt {1+(c+d x)^2}}+i \arctan (c+d x)^2 \cos (3 \arctan (c+d x))-3 i c^2 \arctan (c+d x)^2 \cos (3 \arctan (c+d x))-2 \arctan (c+d x) \cos (3 \arctan (c+d x)) \log \left (1+e^{2 i \arctan (c+d x)}\right )+6 c^2 \arctan (c+d x) \cos (3 \arctan (c+d x)) \log \left (1+e^{2 i \arctan (c+d x)}\right )+6 c \cos (3 \arctan (c+d x)) \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )+\frac {\left (3 i-12 c-9 i c^2\right ) \arctan (c+d x)^2+2 \arctan (c+d x) \left (-2+\left (-3+9 c^2\right ) \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )+18 c \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )}{\sqrt {1+(c+d x)^2}}-\frac {4 i \left (-1+3 c^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )}{\left (1+(c+d x)^2\right )^{3/2}}+\sin (3 \arctan (c+d x))+6 c \arctan (c+d x) \sin (3 \arctan (c+d x))-\arctan (c+d x)^2 \sin (3 \arctan (c+d x))+3 c^2 \arctan (c+d x)^2 \sin (3 \arctan (c+d x))\right )}{12 d^3} \]
a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(-(d*f*x*(6*d*e - 4*c*f + d*f*x)) + 2*(3*d*e*f - 3*c^2*d*e*f + c^3*f^2 + 3*c*(d^2*e^2 - f^2) + d^3* x*(3*e^2 + 3*e*f*x + f^2*x^2))*ArcTan[c + d*x] + (-3*d^2*e^2 + 6*c*d*e*f + (1 - 3*c^2)*f^2)*Log[1 + (c + d*x)^2]))/(3*d^3) + (b^2*e^2*(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])]))/d + (b^2*e*f*((1 + (2*I)*c - c^2 + d^2*x^2)*ArcTan[c + d*x]^2 - 2*ArcTan[c + d*x]*(c + d*x + 2*c*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + Log[1 + (c + d*x)^2] + (2*I)*c*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]))/d^2 + (b^2*f^2*(1 + (c + d*x)^2)^(3/2)*((c + d*x)/Sqrt[1 + (c + d*x)^2] + (6*c*(c + d*x)*ArcTan[c + d*x])/Sqrt[1 + (c + d*x)^2] + (3*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] + (3*c^ 2*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] + I*ArcTan[c + d*x]^2 *Cos[3*ArcTan[c + d*x]] - (3*I)*c^2*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x ]] - 2*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 6*c^2*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*Arc Tan[c + d*x])] + 6*c*Cos[3*ArcTan[c + d*x]]*Log[1/Sqrt[1 + (c + d*x)^2]] + ((3*I - 12*c - (9*I)*c^2)*ArcTan[c + d*x]^2 + 2*ArcTan[c + d*x]*(-2 + (-3 + 9*c^2)*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + 18*c*Log[1/Sqrt[1 + (c + d *x)^2]])/Sqrt[1 + (c + d*x)^2] - ((4*I)*(-1 + 3*c^2)*PolyLog[2, -E^((2*I)* ArcTan[c + d*x])])/(1 + (c + d*x)^2)^(3/2) + Sin[3*ArcTan[c + d*x]] + 6...
Time = 0.70 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int (e+f x)^2 (a+b \arctan (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 5570 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2 (a+b \arctan (c+d x))^2}{d^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x))^2 (a+b \arctan (c+d x))^2d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 (a+b \arctan (c+d x))^2}{3 f}-\frac {2 b \int \left ((c+d x) (a+b \arctan (c+d x)) f^3+3 (d e-c f) (a+b \arctan (c+d x)) f^2+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d f e-\left (3-c^2\right ) f^2\right )+f \left (3 d^2 e^2-6 c d f e-\left (1-3 c^2\right ) f^2\right ) (c+d x)\right ) (a+b \arctan (c+d x))}{(c+d x)^2+1}\right )d(c+d x)}{3 f}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 (a+b \arctan (c+d x))^2}{3 f}-\frac {2 b \left (-\frac {i f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) (a+b \arctan (c+d x))^2}{2 b}+\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) (a+b \arctan (c+d x))^2}{2 b}-f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))+\frac {1}{2} f^3 (c+d x)^2 (a+b \arctan (c+d x))+3 a f^2 (c+d x) (d e-c f)+3 b f^2 (c+d x) \arctan (c+d x) (d e-c f)+\frac {1}{2} b f^3 \arctan (c+d x)-\frac {1}{2} i b f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )-\frac {3}{2} b f^2 (d e-c f) \log \left ((c+d x)^2+1\right )-\frac {1}{2} b f^3 (c+d x)\right )}{3 f}}{d^3}\) |
(((d*e - c*f + f*(c + d*x))^3*(a + b*ArcTan[c + d*x])^2)/(3*f) - (2*b*(-1/ 2*(b*f^3*(c + d*x)) + 3*a*f^2*(d*e - c*f)*(c + d*x) + (b*f^3*ArcTan[c + d* x])/2 + 3*b*f^2*(d*e - c*f)*(c + d*x)*ArcTan[c + d*x] + (f^3*(c + d*x)^2*( a + b*ArcTan[c + d*x]))/2 - ((I/2)*f*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)* f^2)*(a + b*ArcTan[c + d*x])^2)/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])^2)/(2*b) - f*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))] - (3*b* f^2*(d*e - c*f)*Log[1 + (c + d*x)^2])/2 - (I/2)*b*f*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))]))/(3*f))/d^3
3.1.31.3.1 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1017 vs. \(2 (362 ) = 724\).
Time = 0.67 (sec) , antiderivative size = 1018, normalized size of antiderivative = 2.66
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1018\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1042\) |
| default | \(\text {Expression too large to display}\) | \(1042\) |
| risch | \(\text {Expression too large to display}\) | \(2416\) |
1/3*a^2*(f*x+e)^3/f+b^2/d*(1/3/d^2*f^2*arctan(d*x+c)^2*(d*x+c)^3-1/d^2*f^2 *arctan(d*x+c)^2*(d*x+c)^2*c+1/d*f*arctan(d*x+c)^2*(d*x+c)^2*e+1/d^2*f^2*a rctan(d*x+c)^2*(d*x+c)*c^2-2/d*f*arctan(d*x+c)^2*(d*x+c)*c*e+arctan(d*x+c) ^2*(d*x+c)*e^2-1/3/d^2*f^2*arctan(d*x+c)^2*c^3+1/d*f*arctan(d*x+c)^2*c^2*e -arctan(d*x+c)^2*c*e^2+1/3*d/f*arctan(d*x+c)^2*e^3-2/3/d^2/f*(1/2*arctan(d *x+c)*f^3*(d*x+c)^2-3*arctan(d*x+c)*c*f^3*(d*x+c)+3*arctan(d*x+c)*d*e*f^2* (d*x+c)+3/2*arctan(d*x+c)*ln(1+(d*x+c)^2)*c^2*f^3-3*arctan(d*x+c)*ln(1+(d* x+c)^2)*c*d*e*f^2+3/2*arctan(d*x+c)*ln(1+(d*x+c)^2)*d^2*e^2*f-1/2*arctan(d *x+c)*ln(1+(d*x+c)^2)*f^3-arctan(d*x+c)^2*c^3*f^3+3*arctan(d*x+c)^2*c^2*d* e*f^2-3*arctan(d*x+c)^2*c*d^2*e^2*f+arctan(d*x+c)^2*d^3*e^3+3*arctan(d*x+c )^2*c*f^3-3*arctan(d*x+c)^2*d*e*f^2-1/2*f^2*(f*(d*x+c)+1/2*(-6*c*f+6*d*e)* ln(1+(d*x+c)^2)-f*arctan(d*x+c))-1/2*f*(3*c^2*f^2-6*c*d*e*f+3*d^2*e^2-f^2) *(-1/2*I*(ln(d*x+c-I)*ln(1+(d*x+c)^2)-1/2*ln(d*x+c-I)^2-dilog(-1/2*I*(d*x+ c+I))-ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I)))+1/2*I*(ln(d*x+c+I)*ln(1+(d*x+c)^2) -1/2*ln(d*x+c+I)^2-dilog(1/2*I*(d*x+c-I))-ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))) )-1/4*(-2*c^3*f^3+6*c^2*d*e*f^2-6*c*d^2*e^2*f+2*d^3*e^3+6*c*f^3-6*d*e*f^2) *arctan(d*x+c)^2))+1/3*a*b/d^3*f^2*ln(1+(d*x+c)^2)+2/d*a*b*c*e^2*arctan(d* x+c)+2/3/d^3*f^2*b*a*c^3*arctan(d*x+c)+4/3/d^2*c*f^2*x*b*a-2/d*e*x*f*b*a-a *b/d^3*f^2*ln(1+(d*x+c)^2)*c^2-a*b/d*ln(1+(d*x+c)^2)*e^2-2/d^3*c*f^2*b*a*a rctan(d*x+c)-1/3/d*f^2*b*a*x^2-2*b/d^2*arctan(d*x+c)*a*c^2*e*f+2*a*b/d^...
\[ \int (e+f x)^2 (a+b \arctan (c+d x))^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \]
integral(a^2*f^2*x^2 + 2*a^2*e*f*x + a^2*e^2 + (b^2*f^2*x^2 + 2*b^2*e*f*x + b^2*e^2)*arctan(d*x + c)^2 + 2*(a*b*f^2*x^2 + 2*a*b*e*f*x + a*b*e^2)*arc tan(d*x + c), x)
Timed out. \[ \int (e+f x)^2 (a+b \arctan (c+d x))^2 \, dx=\text {Timed out} \]
\[ \int (e+f x)^2 (a+b \arctan (c+d x))^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \]
3/4*b^2*c^2*e^2*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 1/4*(3*arcta n(d*x + c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*b^2* c^2*e^2 + 1/3*a^2*f^2*x^3 + 36*b^2*d^2*f^2*integrate(1/48*x^4*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^2*d^2*f^2*integrate(1/48*x^4 *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) + 72 *b^2*d^2*e*f*integrate(1/48*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 72*b^2*c*d*f^2*integrate(1/48*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 4*b^2*d^2*f^2*integrate(1/48*x^4*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^2*d^2*e*f*integr ate(1/48*x^3*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^2*c*d*f^2*integrate(1/48*x^3*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) + 36*b^2*d^2*e^2*integrate(1/48*x^2* arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 144*b^2*c*d*e*f*inte grate(1/48*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 36*b^ 2*c^2*f^2*integrate(1/48*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^2*d^2*e*f*integrate(1/48*x^3*log(d^2*x^2 + 2*c*d*x + c^2 + 1 )/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 4*b^2*c*d*f^2*integrate(1/48*x^3*log (d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^2*d^ 2*e^2*integrate(1/48*x^2*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^2*c*d*e*f*integrate(1/48*x^2*log(d^2*x^2 + 2...
\[ \int (e+f x)^2 (a+b \arctan (c+d x))^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e+f x)^2 (a+b \arctan (c+d x))^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2 \,d x \]