Integrand size = 20, antiderivative size = 1233 \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\frac {3 i a b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}+\frac {i b^3 d \cot ^{-1}(c+d x)^3}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^3 d (d e-c f) \cot ^{-1}(c+d x)^3}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 a^2 b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {3 a^2 b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {6 a b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 b^3 d \cot ^{-1}(c+d x)^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 a^2 b d \log \left (1+(c+d x)^2\right )}{2 \left (f^2+(d e-c f)^2\right )}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {3 i b^3 d \cot ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i a b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 i b^3 d \cot ^{-1}(c+d x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {3 b^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \]
I*b^3*d*arccot(d*x+c)^3/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*a*b^2*d*(-c*f+d* e)*arccot(d*x+c)^2/f/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*a*b^2*d*polylog(2 ,1-2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+b^3*d*(-c*f+d*e)*arcco t(d*x+c)^3/f/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-(a+b*arccot(d*x+c))^3/f/(f*x+ e)-3*a^2*b*d*(-c*f+d*e)*arctan(d*x+c)/f/(f^2+(-c*f+d*e)^2)-3*a^2*b*d*ln(f* x+e)/(f^2+(-c*f+d*e)^2)+6*a*b^2*d*arccot(d*x+c)*ln(2/(1-I*(d*x+c)))/(d^2*e ^2-2*c*d*e*f+(c^2+1)*f^2)+3*b^3*d*arccot(d*x+c)^2*ln(2/(1-I*(d*x+c)))/(d^2 *e^2-2*c*d*e*f+(c^2+1)*f^2)-6*a*b^2*d*arccot(d*x+c)*ln(2*d*(f*x+e)/(d*e+I* f-c*f)/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3*b^3*d*arccot(d*x+c )^2*ln(2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1) *f^2)-6*a*b^2*d*arccot(d*x+c)*ln(2/(1+I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+ 1)*f^2)-3*b^3*d*arccot(d*x+c)^2*ln(2/(1+I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^ 2+1)*f^2)+3/2*a^2*b*d*ln(1+(d*x+c)^2)/(f^2+(-c*f+d*e)^2)+3*I*a*b^2*d*arcco t(d*x+c)^2/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*b^3*d*arccot(d*x+c)*polylog (2,1-2/(1+I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*b^3*d*arccot(d*x +c)*polylog(2,1-2/(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-3*I*b^3*d *arccot(d*x+c)*polylog(2,1-2*d*(f*x+e)/(d*e+I*f-c*f)/(1-I*(d*x+c)))/(d^2*e ^2-2*c*d*e*f+(c^2+1)*f^2)+3*I*a*b^2*d*polylog(2,1-2/(1+I*(d*x+c)))/(d^2*e^ 2-2*c*d*e*f+(c^2+1)*f^2)-3*I*a*b^2*d*polylog(2,1-2*d*(f*x+e)/(d*e+I*f-c*f) /(1-I*(d*x+c)))/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+3/2*b^3*d*polylog(3,1-2...
\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx \]
Time = 2.42 (sec) , antiderivative size = 1266, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5569, 7292, 5581, 27, 7276, 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.
| \(\displaystyle \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx\) |
| \(\Big \downarrow \) 5569 |
| \(\displaystyle -\frac {3 b d \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x) \left ((c+d x)^2+1\right )}dx}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {3 b d \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x) \left (c^2+2 d x c+d^2 x^2+1\right )}dx}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}\) |
| \(\Big \downarrow \) 5581 |
| \(\displaystyle -\frac {3 b \int \frac {d \left (a+b \cot ^{-1}(c+d x)\right )^2}{\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right ) \left ((c+d x)^2+1\right )}d(c+d x)}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 b d \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(d e-c f+f (c+d x)) \left ((c+d x)^2+1\right )}d(c+d x)}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 b d \int \left (\frac {a^2}{(d e-c f+f (c+d x)) \left ((c+d x)^2+1\right )}+\frac {2 b \cot ^{-1}(c+d x) a}{(d e-c f+f (c+d x)) \left ((c+d x)^2+1\right )}+\frac {b^2 \cot ^{-1}(c+d x)^2}{(d e-c f+f (c+d x)) \left ((c+d x)^2+1\right )}\right )d(c+d x)}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{f (e+f x)}-\frac {3 b d \left (-\frac {i b^2 f \cot ^{-1}(c+d x)^3}{3 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac {b^2 (d e-c f) \cot ^{-1}(c+d x)^3}{3 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}-\frac {b^2 f \log \left (\frac {2}{1-i (c+d x)}\right ) \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {b^2 f \log \left (\frac {2}{i (c+d x)+1}\right ) \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {b^2 f \log \left (\frac {2 (d e-c f+f (c+d x))}{(d e-c f+i f) (1-i (c+d x))}\right ) \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {i a b f \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {a b (d e-c f) \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {2 a b f \log \left (\frac {2}{1-i (c+d x)}\right ) \cot ^{-1}(c+d x)}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {2 a b f \log \left (\frac {2}{i (c+d x)+1}\right ) \cot ^{-1}(c+d x)}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {2 a b f \log \left (\frac {2 (d e-c f+f (c+d x))}{(d e-c f+i f) (1-i (c+d x))}\right ) \cot ^{-1}(c+d x)}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {i b^2 f \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right ) \cot ^{-1}(c+d x)}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {i b^2 f \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) \cot ^{-1}(c+d x)}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {i b^2 f \operatorname {PolyLog}\left (2,1-\frac {2 (d e-c f+f (c+d x))}{(d e-c f+i f) (1-i (c+d x))}\right ) \cot ^{-1}(c+d x)}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {a^2 (d e-c f) \arctan (c+d x)}{f^2+(d e-c f)^2}+\frac {a^2 f \log (d e-c f+f (c+d x))}{f^2+(d e-c f)^2}-\frac {a^2 f \log \left ((c+d x)^2+1\right )}{2 \left (f^2+(d e-c f)^2\right )}-\frac {i a b f \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {i a b f \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}+\frac {i a b f \operatorname {PolyLog}\left (2,1-\frac {2 (d e-c f+f (c+d x))}{(d e-c f+i f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}-\frac {b^2 f \operatorname {PolyLog}\left (3,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac {b^2 f \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}+\frac {b^2 f \operatorname {PolyLog}\left (3,1-\frac {2 (d e-c f+f (c+d x))}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 \left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2\right )}\right )}{f}\) |
-((a + b*ArcCot[c + d*x])^3/(f*(e + f*x))) - (3*b*d*(((-I)*a*b*f*ArcCot[c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (a*b*(d*e - c*f)*ArcCot[ c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - ((I/3)*b^2*f*ArcCot[c + d*x]^3)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (b^2*(d*e - c*f)*ArcCot[ c + d*x]^3)/(3*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (a^2*(d*e - c*f)*A rcTan[c + d*x])/(f^2 + (d*e - c*f)^2) - (2*a*b*f*ArcCot[c + d*x]*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (b^2*f*ArcCot[c + d*x]^2*Log[2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (2*a*b*f*ArcCot[c + d*x]*Log[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^2*f*ArcCot[c + d*x]^2*Log[2/(1 + I*(c + d*x))])/(d^2*e ^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (a^2*f*Log[d*e - c*f + f*(c + d*x)])/(f^ 2 + (d*e - c*f)^2) + (2*a*b*f*ArcCot[c + d*x]*Log[(2*(d*e - c*f + f*(c + d *x)))/((d*e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^2*f*ArcCot[c + d*x]^2*Log[(2*(d*e - c*f + f*(c + d*x)))/((d *e + I*f - c*f)*(1 - I*(c + d*x)))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (a^2*f*Log[1 + (c + d*x)^2])/(2*(f^2 + (d*e - c*f)^2)) - (I*a*b*f*PolyL og[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (I *b^2*f*ArcCot[c + d*x]*PolyLog[2, 1 - 2/(1 - I*(c + d*x))])/(d^2*e^2 - 2*c *d*e*f + (1 + c^2)*f^2) - (I*a*b*f*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/(d ^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (I*b^2*f*ArcCot[c + d*x]*PolyLog[...
3.2.45.3.1 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_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _), x_Symbol] :> Simp[(e + f*x)^(m + 1)*((a + b*ArcCot[c + d*x])^p/(f*(m + 1))), x] + Simp[b*d*(p/(f*(m + 1))) Int[(e + f*x)^(m + 1)*((a + b*ArcCot[ c + d*x])^(p - 1)/(1 + (c + d*x)^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x ] && IGtQ[p, 0] && ILtQ[m, -1]
Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/d Subs t[Int[((d*e - c*f)/d + f*(x/d))^m*(C/d^2 + (C/d^2)*x^2)^q*(a + b*ArcCot[x]) ^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] & & EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 28.03 (sec) , antiderivative size = 4229, normalized size of antiderivative = 3.43
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(4229\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(4722\) |
| default | \(\text {Expression too large to display}\) | \(4722\) |
-a^3/(f*x+e)/f+b^3/d*(-d^2/(f*(d*x+c)-c*f+d*e)/f*arccot(d*x+c)^3-3*d^2/f*( arccot(d*x+c)^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*f*ln(f*(d*x+c)-c*f+d*e)-1/ 2*arccot(d*x+c)^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*f*ln(1+(d*x+c)^2)-arccot (d*x+c)^2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*arctan(d*x+c)*c*f+arccot(d*x+c)^ 2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*arctan(d*x+c)*d*e+1/(c^2*f^2-2*c*d*e*f+d ^2*e^2+f^2)*f*arctan(d*x+c)^2*ln((1+I*(d*x+c))/(1+(d*x+c)^2)^(1/2))-1/3*I/ (c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*f*arctan(d*x+c)^3+1/4/(c^2*f^2-2*c*d*e*f+d ^2*e^2+f^2)*(-2*Pi*c*f+2*Pi*d*e+4*f*ln(2)+I*f*Pi*csgn(I*(1+I*(d*x+c))^2/(1 +(d*x+c)^2))*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(1+(1+I*(d*x+c))^2/(1+(d *x+c)^2))^2)^2-2*I*f*Pi*csgn(I*(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))*csgn(I*( 1+(1+I*(d*x+c))^2/(1+(d*x+c)^2))^2)^2+4*I*f*Pi*csgn(I*(I*f*(1+I*(d*x+c))^2 /(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d *x+c)^2)-I*f+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))^2+2*I*f*Pi*csgn(I *(I*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e* (1+I*(d*x+c))^2/(1+(d*x+c)^2)-I*f+c*f-d*e))*csgn(I*(I*f*(1+I*(d*x+c))^2/(1 +(d*x+c)^2)+c*f*(1+I*(d*x+c))^2/(1+(d*x+c)^2)-d*e*(1+I*(d*x+c))^2/(1+(d*x+ c)^2)-I*f+c*f-d*e)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))*csgn(I/(1+(1+I*(d*x+ c))^2/(1+(d*x+c)^2)))-I*f*Pi*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(1+(1+I* (d*x+c))^2/(1+(d*x+c)^2))^2)^3+2*I*f*Pi*csgn(I*(1+I*(d*x+c))/(1+(d*x+c)^2) ^(1/2))*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2))^2+I*f*Pi*csgn(I*(1+(1+I*(...
\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \]
integral((b^3*arccot(d*x + c)^3 + 3*a*b^2*arccot(d*x + c)^2 + 3*a^2*b*arcc ot(d*x + c) + a^3)/(f^2*x^2 + 2*e*f*x + e^2), x)
Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3}}{{\left (f x + e\right )}^{2}} \,d x } \]
-3/2*(d*(2*(d^2*e - c*d*f)*arctan((d^2*x + c*d)/d)/((d^2*e^2*f - 2*c*d*e*f ^2 + (c^2 + 1)*f^3)*d) - log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*e^2 - 2*c*d *e*f + (c^2 + 1)*f^2) + 2*log(f*x + e)/(d^2*e^2 - 2*c*d*e*f + (c^2 + 1)*f^ 2)) + 2*arccot(d*x + c)/(f^2*x + e*f))*a^2*b - a^3/(f^2*x + e*f) - 1/32*(4 *b^3*arctan2(1, d*x + c)^3 - 3*b^3*arctan2(1, d*x + c)*log(d^2*x^2 + 2*c*d *x + c^2 + 1)^2 - 32*(f^2*x + e*f)*integrate(-1/32*(12*b^3*d*e*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)*d^2*f*x^2 + 4*(3*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*x - 3*(b^3*d^2*f*x^2*arctan 2(1, d*x + c) + b^3*d*e + (2*b^3*c*arctan2(1, d*x + c) + b^3)*d*f*x + (b^3 *c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*f)*log(d^2*x^2 + 2*c*d *x + c^2 + 1)^2 - 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)*f + 12*(b^3*d^2*f*x^2*arctan2(1, d*x + c) + b^3*c*d*e*arctan2(1, d*x + c) + (b^3*d^2*e*arctan2(1, d*x + c) + b^3*c*d*f*arctan2(1, d*x + c))*x)*l og(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^2*f^3*x^4 + (c^2 + 1)*e^2*f + 2*(d^2*e *f^2 + c*d*f^3)*x^3 + (d^2*e^2*f + 4*c*d*e*f^2 + (c^2 + 1)*f^3)*x^2 + 2*(c *d*e^2*f + (c^2 + 1)*e*f^2)*x), x))/(f^2*x + e*f)
Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^3}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3}{{\left (e+f\,x\right )}^2} \,d x \]