Integrand size = 20, antiderivative size = 565 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}+\frac {3 i b^3 f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^3} \]
a*b^2*f^2*x/d^2+b^3*f^2*(d*x+c)*arccot(d*x+c)/d^3+1/2*b*f^2*(a+b*arccot(d* x+c))^2/d^3+3*I*b*f*(-c*f+d*e)*(a+b*arccot(d*x+c))^2/d^3+3*b*f*(-c*f+d*e)* (d*x+c)*(a+b*arccot(d*x+c))^2/d^3+1/2*b*f^2*(d*x+c)^2*(a+b*arccot(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*arccot(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*arccot(d*x+c))^3/d^3 /f+1/3*(f*x+e)^3*(a+b*arccot(d*x+c))^3/f-6*b^2*f*(-c*f+d*e)*(a+b*arccot(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* arccot(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*arccot(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. \(2309\) vs. \(2(565)=1130\).
Time = 14.56 (sec) , antiderivative size = 2309, normalized size of antiderivative = 4.09 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\text {Result too large to show} \]
(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 + a^2*b*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCot[c + d*x] + ((-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 + ((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*(1 + (c + d*x)^2)*(-((c + d*x)*ArcCot[c + d*x]^2) + 2*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] - I*(ArcCot[c + d* x]^2 + PolyLog[2, E^((2*I)*ArcCot[c + d*x])])))/(d*(c + d*x)^2*(1 + (c + d *x)^(-2))) + (6*a*b^2*e*f*(1 + (c + d*x)^2)*(((c + d*x)*ArcCot[c + d*x])/d ^2 - (c*(c + d*x)*ArcCot[c + d*x]^2)/d^2 + ((c + d*x)^2*(1 + (c + d*x)^(-2 ))*ArcCot[c + d*x]^2)/(2*d^2) - Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) ]/d^2 + (2*c*(ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] - (I/2)*( ArcCot[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcCot[c + d*x])])))/d^2))/((c + d *x)^2*(1 + (c + d*x)^(-2))) - (a*b^2*f^2*x^2*(1 + (c + d*x)^2)*(-((c + d*x )*ArcCot[c + d*x]^2) + ArcCot[c + d*x]*(-1 + 3*c*ArcCot[c + d*x]) - (1 - 6 *c*ArcCot[c + d*x] - ArcCot[c + d*x]^2 + 3*c^2*ArcCot[c + d*x]^2)/((c + d* x)*(1 + (c + d*x)^(-2))) - (6*c*(Log[(c + d*x)^(-1)] + Log[1/Sqrt[1 + (c + d*x)^(-2)]]))/((c + d*x)^2*(1 + (c + d*x)^(-2))) + (I*(ArcCot[c + d*x]*(A rcCot[c + d*x] + (2*I)*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + PolyLog[2, E^ ((2*I)*ArcCot[c + d*x])]))/((c + d*x)^2*(1 + (c + d*x)^(-2))) + (6*c^2*...
Time = 1.13 (sec) , antiderivative size = 551, 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 = {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.
| \(\displaystyle \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 5571 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x))^2 \left (a+b \cot ^{-1}(c+d x)\right )^3d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 5390 |
| \(\displaystyle \frac {\frac {b \int \left ((c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2 f^3+3 (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2 f^2+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d f e-\left (3-c^2\right ) f^2\right )+f \left (3 d^2 e^2-6 c d f e-\left (1-3 c^2\right ) f^2\right ) (c+d x)\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{(c+d x)^2+1}\right )d(c+d x)}{f}+\frac {(f (c+d x)-c f+d e)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac {b \left (i b f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )+\frac {i f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 b}-\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 b}-f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2+3 i f^2 (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2+3 f^2 (c+d x) (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2-6 b f^2 (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )+a b f^3 (c+d x)+\frac {1}{2} f^3 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2+\frac {1}{2} f^3 \left (a+b \cot ^{-1}(c+d x)\right )^2-\frac {1}{2} b^2 f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )+3 i b^2 f^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )+\frac {1}{2} b^2 f^3 \log \left ((c+d x)^2+1\right )+b^2 f^3 (c+d x) \cot ^{-1}(c+d x)\right )}{f}}{d^3}\) |
(((d*e - c*f + f*(c + d*x))^3*(a + b*ArcCot[c + d*x])^3)/(3*f) + (b*(a*b*f ^3*(c + d*x) + b^2*f^3*(c + d*x)*ArcCot[c + d*x] + (f^3*(a + b*ArcCot[c + d*x])^2)/2 + (3*I)*f^2*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2 + 3*f^2*(d*e - c*f)*(c + d*x)*(a + b*ArcCot[c + d*x])^2 + (f^3*(c + d*x)^2*(a + b*ArcCo t[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*ArcCot[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*ArcCot[c + d*x])^3)/(3*b) - 6*b*f^2*(d*e - c*f)*(a + b*ArcCot[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*ArcCot[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*ArcCot[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
3.2.41.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 101.64 (sec) , antiderivative size = 6248, normalized size of antiderivative = 11.06
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(6248\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(10834\) |
| default | \(\text {Expression too large to display}\) | \(10834\) |
\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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)*arccot(d*x + c)^3 + 3*(a*b^2*f^2*x^2 + 2*a*b^2*e*f*x + a*b^2*e^ 2)*arccot(d*x + c)^2 + 3*(a^2*b*f^2*x^2 + 2*a^2*b*e*f*x + a^2*b*e^2)*arcco t(d*x + c), x)
Timed out. \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\text {Timed out} \]
\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
1/24*b^3*f^2*x^3*arctan2(1, d*x + c)^3 + 1/8*b^3*e*f*x^2*arctan2(1, d*x + c)^3 + 1/8*b^3*e^2*x*arctan2(1, d*x + c)^3 + 1/3*a^3*f^2*x^3 + a^3*e*f*x^2 + 3*(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^2*b*e*f + 1/2*(2*x^3*arcco t(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^2*b*f^2 + a^3 *e^2*x + 3/2*(2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a^2*b*e^ 2/d - 1/32*(b^3*f^2*x^3*arctan2(1, d*x + c) + 3*b^3*e*f*x^2*arctan2(1, d*x + c) + 3*b^3*e^2*x*arctan2(1, d*x + c))*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^ 2 + integrate(1/32*(4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d *x + c)^2)*d^2*f^2*x^4 + 4*(2*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arct an2(1, d*x + c)^2)*d^2*e*f + (b^3*arctan2(1, d*x + c)^2 + 2*(7*b^3*arctan2 (1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*f^2)*x^3 + 4*(7*b^3* arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2 + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2)*e^2 + 4*((7*b^3*arctan2 (1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*e^2 + (3*b^3*arctan2( 1, d*x + c)^2 + 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*e*f + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2 + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2 )*f^2)*x^2 + (3*b^3*d^2*f^2*x^4*arctan2(1, d*x + c) + (6*b^3*d^2*e*f*ar...
\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3 \,d x \]