Integrand size = 20, antiderivative size = 382 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \arctan (c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{3 d^3} \]
1/3*b^2*f^2*x/d^2+2*a*b*f*(-c*f+d*e)*x/d^2+2*b^2*f*(-c*f+d*e)*(d*x+c)*arcc ot(d*x+c)/d^3+1/3*b*f^2*(d*x+c)^2*(a+b*arccot(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*arccot(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*arccot(d*x+c))^2/d^3/f+1/3*(f*x+e)^3*(a+b* arccot(d*x+c))^2/f-1/3*b^2*f^2*arctan(d*x+c)/d^3-2/3*b*(3*d^2*e^2-6*c*d*e* f-(-3*c^2+1)*f^2)*(a+b*arccot(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
Time = 6.67 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.53 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 c f^2+3 a^2 d^3 e^2 x+6 a b d^2 e f x+b^2 d f^2 x-4 a b c d f^2 x+3 a^2 d^3 e f x^2+a b d^2 f^2 x^2+a^2 d^3 f^2 x^3+b^2 (i+c+d x) \left ((i+c)^2 f^2-(i+c) d f (3 e+f x)+d^2 \left (3 e^2+3 e f x+f^2 x^2\right )\right ) \cot ^{-1}(c+d x)^2-6 a b c d^2 e^2 \arctan (c+d x)-6 a b d e f \arctan (c+d x)+6 a b c^2 d e f \arctan (c+d x)+6 a b c f^2 \arctan (c+d x)-2 a b c^3 f^2 \arctan (c+d x)-b \cot ^{-1}(c+d x) \left (-2 a d^3 x \left (3 e^2+3 e f x+f^2 x^2\right )+b f \left (5 c^2 f-6 d^2 e x+c d (-6 e+4 f x)-f \left (1+d^2 x^2\right )\right )+2 b \left (3 d^2 e^2-6 c d e f+\left (-1+3 c^2\right ) f^2\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+6 b^2 c f^2 \log \left (\frac {1}{c+d x}\right )+3 a b d^2 e^2 \log \left (1+c^2+2 c d x+d^2 x^2\right )-6 a b c d e f \log \left (1+c^2+2 c d x+d^2 x^2\right )-a b f^2 \log \left (1+c^2+2 c d x+d^2 x^2\right )+3 a b c^2 f^2 \log \left (1+c^2+2 c d x+d^2 x^2\right )+6 b^2 c f^2 \log \left (\frac {1}{\sqrt {1+\frac {1}{(c+d x)^2}}}\right )-6 b^2 d e f \log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )+i b^2 \left (3 d^2 e^2-6 c d e f+\left (-1+3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )}{3 d^3} \]
(b^2*c*f^2 + 3*a^2*d^3*e^2*x + 6*a*b*d^2*e*f*x + b^2*d*f^2*x - 4*a*b*c*d*f ^2*x + 3*a^2*d^3*e*f*x^2 + a*b*d^2*f^2*x^2 + a^2*d^3*f^2*x^3 + b^2*(I + c + d*x)*((I + c)^2*f^2 - (I + c)*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f ^2*x^2))*ArcCot[c + d*x]^2 - 6*a*b*c*d^2*e^2*ArcTan[c + d*x] - 6*a*b*d*e*f *ArcTan[c + d*x] + 6*a*b*c^2*d*e*f*ArcTan[c + d*x] + 6*a*b*c*f^2*ArcTan[c + d*x] - 2*a*b*c^3*f^2*ArcTan[c + d*x] - b*ArcCot[c + d*x]*(-2*a*d^3*x*(3* e^2 + 3*e*f*x + f^2*x^2) + b*f*(5*c^2*f - 6*d^2*e*x + c*d*(-6*e + 4*f*x) - f*(1 + d^2*x^2)) + 2*b*(3*d^2*e^2 - 6*c*d*e*f + (-1 + 3*c^2)*f^2)*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + 6*b^2*c*f^2*Log[(c + d*x)^(-1)] + 3*a*b*d^2 *e^2*Log[1 + c^2 + 2*c*d*x + d^2*x^2] - 6*a*b*c*d*e*f*Log[1 + c^2 + 2*c*d* x + d^2*x^2] - a*b*f^2*Log[1 + c^2 + 2*c*d*x + d^2*x^2] + 3*a*b*c^2*f^2*Lo g[1 + c^2 + 2*c*d*x + d^2*x^2] + 6*b^2*c*f^2*Log[1/Sqrt[1 + (c + d*x)^(-2) ]] - 6*b^2*d*e*f*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] + I*b^2*(3*d^ 2*e^2 - 6*c*d*e*f + (-1 + 3*c^2)*f^2)*PolyLog[2, E^((2*I)*ArcCot[c + d*x]) ])/(3*d^3)
Time = 0.77 (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 = {5571, 27, 5390, 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.
| \(\displaystyle \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 5571 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x))^2 \left (a+b \cot ^{-1}(c+d x)\right )^2d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 5390 |
| \(\displaystyle \frac {\frac {2 b \int \left ((c+d x) \left (a+b \cot ^{-1}(c+d x)\right ) f^3+3 (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) f^2+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d f e-\left (3-c^2\right ) f^2\right )+f \left (3 d^2 e^2-6 c d f e-\left (1-3 c^2\right ) f^2\right ) (c+d x)\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{(c+d x)^2+1}\right )d(c+d x)}{3 f}+\frac {(f (c+d x)-c f+d e)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac {2 b \left (\frac {i f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 b}-\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 b}-f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )+\frac {1}{2} f^3 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )+3 a f^2 (c+d x) (d e-c f)-\frac {1}{2} b f^3 \arctan (c+d x)+\frac {1}{2} i b f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )+\frac {3}{2} b f^2 (d e-c f) \log \left ((c+d x)^2+1\right )+3 b f^2 (c+d x) (d e-c f) \cot ^{-1}(c+d x)+\frac {1}{2} b f^3 (c+d x)\right )}{3 f}}{d^3}\) |
(((d*e - c*f + f*(c + d*x))^3*(a + b*ArcCot[c + d*x])^2)/(3*f) + (2*b*((b* f^3*(c + d*x))/2 + 3*a*f^2*(d*e - c*f)*(c + d*x) + 3*b*f^2*(d*e - c*f)*(c + d*x)*ArcCot[c + d*x] + (f^3*(c + d*x)^2*(a + b*ArcCot[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*ArcCot[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*ArcCot[c + d*x])^2)/(2*b) - (b*f^3*ArcTan[c + d*x])/2 - f*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[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
3.2.36.3.1 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_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcCot[c*x])^p/(e*(q + 1))), x] + S imp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcCot[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_.) + ArcCot[(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 rcCot[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. 1071 vs. \(2 (362 ) = 724\).
Time = 1.34 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.81
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1072\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1087\) |
| default | \(\text {Expression too large to display}\) | \(1087\) |
| risch | \(\text {Expression too large to display}\) | \(3165\) |
1/3*a^2*(f*x+e)^3/f+b^2/d*(1/3/d^2*f^2*arccot(d*x+c)^2*(d*x+c)^3-1/d^2*f^2 *arccot(d*x+c)^2*(d*x+c)^2*c+1/d*f*arccot(d*x+c)^2*(d*x+c)^2*e+1/d^2*f^2*a rccot(d*x+c)^2*(d*x+c)*c^2-2/d*f*arccot(d*x+c)^2*(d*x+c)*c*e+arccot(d*x+c) ^2*(d*x+c)*e^2-1/3/d^2*f^2*arccot(d*x+c)^2*c^3+1/d*f*arccot(d*x+c)^2*c^2*e -arccot(d*x+c)^2*c*e^2+1/3*d/f*arccot(d*x+c)^2*e^3+2/3/d^2/f*(1/2*arccot(d *x+c)*f^3*(d*x+c)^2-3*arccot(d*x+c)*c*f^3*(d*x+c)+3*arccot(d*x+c)*d*e*f^2* (d*x+c)+3/2*arccot(d*x+c)*ln(1+(d*x+c)^2)*c^2*f^3-3*arccot(d*x+c)*ln(1+(d* x+c)^2)*c*d*e*f^2+3/2*arccot(d*x+c)*ln(1+(d*x+c)^2)*d^2*e^2*f-1/2*arccot(d *x+c)*ln(1+(d*x+c)^2)*f^3-arccot(d*x+c)*arctan(d*x+c)*c^3*f^3+3*arccot(d*x +c)*arctan(d*x+c)*c^2*d*e*f^2-3*arccot(d*x+c)*arctan(d*x+c)*c*d^2*e^2*f+ar ccot(d*x+c)*arctan(d*x+c)*d^3*e^3+3*arccot(d*x+c)*arctan(d*x+c)*c*f^3-3*ar ccot(d*x+c)*arctan(d*x+c)*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)*a rctan(d*x+c)^2))+2/3*a*b/f*arccot(d*x+c)*e^3-5/3/d^3*c^2*f^2*b*a+1/3/d*f^2 *b*a*x^2-1/3*a*b/d^3*f^2*ln(1+(d*x+c)^2)-2*a*b/d^2*f*ln(1+(d*x+c)^2)*c*e+2 *b/d^2*arctan(d*x+c)*a*c^2*e*f-2*b/d*arctan(d*x+c)*a*c*e^2+2*a*b*f*arcc...
\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
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)*arccot(d*x + c)^2 + 2*(a*b*f^2*x^2 + 2*a*b*e*f*x + a*b*e^2)*arc cot(d*x + c), x)
Timed out. \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\text {Timed out} \]
\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
1/12*b^2*f^2*x^3*arctan2(1, d*x + c)^2 + 1/4*b^2*e*f*x^2*arctan2(1, d*x + c)^2 + 1/3*a^2*f^2*x^3 + 1/4*b^2*e^2*x*arctan2(1, d*x + c)^2 + a^2*e*f*x^2 + 2*(x^2*arccot(d*x + c) + d*(x/d^2 + (c^2 - 1)*arctan((d^2*x + c*d)/d)/d ^3 - c*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^3))*a*b*e*f + 1/3*(2*x^3*arccot( d*x + c) + d*((d*x^2 - 4*c*x)/d^3 - 2*(c^3 - 3*c)*arctan((d^2*x + c*d)/d)/ d^4 + (3*c^2 - 1)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^4))*a*b*f^2 + a^2*e^2 *x + (2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a*b*e^2/d - 1/48 *(b^2*f^2*x^3 + 3*b^2*e*f*x^2 + 3*b^2*e^2*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + integrate(1/48*(36*b^2*d^2*f^2*x^4*arctan2(1, d*x + c)^2 + 8*(9*b^ 2*d^2*e*f*arctan2(1, d*x + c)^2 + (9*b^2*c*arctan2(1, d*x + c)^2 + b^2*arc tan2(1, d*x + c))*d*f^2)*x^3 + 36*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*arc tan2(1, d*x + c)^2)*e^2 + 12*(3*b^2*d^2*e^2*arctan2(1, d*x + c)^2 + 2*(6*b ^2*c*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*e*f + 3*(b^2*c^2*a rctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c)^2)*f^2)*x^2 + 3*(b^2*d^2*f^ 2*x^4 + 2*(b^2*d^2*e*f + b^2*c*d*f^2)*x^3 + (b^2*c^2 + b^2)*e^2 + (b^2*d^2 *e^2 + 4*b^2*c*d*e*f + (b^2*c^2 + b^2)*f^2)*x^2 + 2*(b^2*c*d*e^2 + (b^2*c^ 2 + b^2)*e*f)*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 24*((3*b^2*c*arctan2 (1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*e^2 + 3*(b^2*c^2*arctan2(1, d* x + c)^2 + b^2*arctan2(1, d*x + c)^2)*e*f)*x + 4*(b^2*d^2*f^2*x^4 + 3*b^2* c*d*e^2*x + (3*b^2*d^2*e*f + b^2*c*d*f^2)*x^3 + 3*(b^2*d^2*e^2 + b^2*c*...
\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^2 \,d x \]