Integrand size = 20, antiderivative size = 567 \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\frac {i b^2 d \cot ^{-1}(c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (d e-c f) \cot ^{-1}(c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 a b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}-\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}+\frac {2 b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {2 b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {2 b^2 d \cot ^{-1}(c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {a b d \log \left (1+(c+d x)^2\right )}{f^2+(d e-c f)^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2} \]
I*b^2*d*arccot(d*x+c)^2/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)+b^2*d*(-c*f+d*e)*a rccot(d*x+c)^2/f/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)-(a+b*arccot(d*x+c))^2/f/( f*x+e)-2*a*b*d*(-c*f+d*e)*arctan(d*x+c)/f/(f^2+(-c*f+d*e)^2)-2*a*b*d*ln(f* x+e)/(f^2+(-c*f+d*e)^2)+2*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)-2*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)-2*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)+a*b*d*ln(1+(d*x+c)^2)/(f^2 +(-c*f+d*e)^2)+I*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)-I*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)+I*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)
Time = 6.12 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=-\frac {a^2+\frac {2 a b f \left (\left (-c d e+f+c^2 f-d^2 e x+c d f x\right ) \cot ^{-1}(c+d x)+d (e+f x) \log \left (-\frac {d (e+f x)}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (e+f x) \left (1+(c+d x)^2\right ) \left (\frac {e^{i \arctan \left (\frac {f}{d e-c f}\right )} \cot ^{-1}(c+d x)^2}{(-d e+c f) \sqrt {1+\frac {f^2}{(d e-c f)^2}}}+\frac {\cot ^{-1}(c+d x)^2}{d e+d f x}+\frac {f \left (i \pi \cot ^{-1}(c+d x)+\pi \log \left (1+e^{-2 i \cot ^{-1}(c+d x)}\right )+2 \cot ^{-1}(c+d x) \log \left (1-e^{2 i \left (\cot ^{-1}(c+d x)+\arctan \left (\frac {f}{d e-c f}\right )\right )}\right )-\pi \log \left (\frac {1}{\sqrt {1+\frac {1}{(c+d x)^2}}}\right )+2 \arctan \left (\frac {f}{-d e+c f}\right ) \left (i \cot ^{-1}(c+d x)-\log \left (1-e^{2 i \left (\cot ^{-1}(c+d x)+\arctan \left (\frac {f}{d e-c f}\right )\right )}\right )+\log \left (\sin \left (\cot ^{-1}(c+d x)+\arctan \left (\frac {f}{d e-c f}\right )\right )\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \left (\cot ^{-1}(c+d x)+\arctan \left (\frac {f}{d e-c f}\right )\right )}\right )\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}\right )}{(c+d x)^2 \left (1+\frac {1}{(c+d x)^2}\right )}}{f (e+f x)} \]
-((a^2 + (2*a*b*f*((-(c*d*e) + f + c^2*f - d^2*e*x + c*d*f*x)*ArcCot[c + d *x] + d*(e + f*x)*Log[-((d*(e + f*x))/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) )]))/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^2*d*(e + f*x)*(1 + (c + d* x)^2)*((E^(I*ArcTan[f/(d*e - c*f)])*ArcCot[c + d*x]^2)/((-(d*e) + c*f)*Sqr t[1 + f^2/(d*e - c*f)^2]) + ArcCot[c + d*x]^2/(d*e + d*f*x) + (f*(I*Pi*Arc Cot[c + d*x] + Pi*Log[1 + E^((-2*I)*ArcCot[c + d*x])] + 2*ArcCot[c + d*x]* Log[1 - E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))] - Pi*Log[1/Sq rt[1 + (c + d*x)^(-2)]] + 2*ArcTan[f/(-(d*e) + c*f)]*(I*ArcCot[c + d*x] - Log[1 - E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))] + Log[Sin[Arc Cot[c + d*x] + ArcTan[f/(d*e - c*f)]]]) - I*PolyLog[2, E^((2*I)*(ArcCot[c + d*x] + ArcTan[f/(d*e - c*f)]))]))/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) )/((c + d*x)^2*(1 + (c + d*x)^(-2))))/(f*(e + f*x)))
Time = 1.55 (sec) , antiderivative size = 586, 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.
| \(\displaystyle \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx\) |
| \(\Big \downarrow \) 5569 |
| \(\displaystyle -\frac {2 b d \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x) \left ((c+d x)^2+1\right )}dx}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {2 b d \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x) \left (c^2+2 d x c+d^2 x^2+1\right )}dx}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 5581 |
| \(\displaystyle -\frac {2 b \int \frac {d \left (a+b \cot ^{-1}(c+d x)\right )}{\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right ) \left ((c+d x)^2+1\right )}d(c+d x)}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b d \int \frac {a+b \cot ^{-1}(c+d x)}{(d e-c f+f (c+d x)) \left ((c+d x)^2+1\right )}d(c+d x)}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 b d \int \left (\frac {a}{(d e-c f+f (c+d x)) \left ((c+d x)^2+1\right )}+\frac {b \cot ^{-1}(c+d x)}{(d e-c f+f (c+d x)) \left ((c+d x)^2+1\right )}\right )d(c+d x)}{f}-\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 b d \left (\frac {a \arctan (c+d x) (d e-c f)}{(d e-c f)^2+f^2}+\frac {a f \log (f (c+d x)-c f+d e)}{(d e-c f)^2+f^2}-\frac {a f \log \left ((c+d x)^2+1\right )}{2 \left ((d e-c f)^2+f^2\right )}-\frac {i b f \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}-\frac {i b f \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{2 \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}+\frac {i b f \operatorname {PolyLog}\left (2,1-\frac {2 (d e-c f+f (c+d x))}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}-\frac {i b f \cot ^{-1}(c+d x)^2}{2 \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}-\frac {b (d e-c f) \cot ^{-1}(c+d x)^2}{2 \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}-\frac {b f \log \left (\frac {2}{1-i (c+d x)}\right ) \cot ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {b f \log \left (\frac {2}{1+i (c+d x)}\right ) \cot ^{-1}(c+d x)}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {b f \cot ^{-1}(c+d x) \log \left (\frac {2 (f (c+d x)-c f+d e)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}\right )}{f}\) |
-((a + b*ArcCot[c + d*x])^2/(f*(e + f*x))) - (2*b*d*(((-1/2*I)*b*f*ArcCot[ c + d*x]^2)/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) - (b*(d*e - c*f)*ArcCot[ c + d*x]^2)/(2*(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2)) + (a*(d*e - c*f)*Arc Tan[c + d*x])/(f^2 + (d*e - c*f)^2) - (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*f*ArcCot[c + d*x]*Lo g[2/(1 + I*(c + d*x))])/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (a*f*Log[d *e - c*f + f*(c + d*x)])/(f^2 + (d*e - c*f)^2) + (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) - (a*f*Log[1 + (c + d*x)^2])/(2*(f^2 + ( d*e - c*f)^2)) - ((I/2)*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/2)*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/2)*b*f*PolyLog[2, 1 - (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)))/f
3.2.40.3.1 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]
Time = 4.18 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.38
| method | result | size |
| parts | \(-\frac {a^{2}}{\left (f x +e \right ) f}+\frac {b^{2} \left (-\frac {d^{2} \operatorname {arccot}\left (d x +c \right )^{2}}{\left (f \left (d x +c \right )-c f +d e \right ) f}-\frac {2 d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right ) f \ln \left (f \left (d x +c \right )-c f +d e \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {\operatorname {arccot}\left (d x +c \right ) f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) c f}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) d e}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {f^{2} \left (-\frac {i \ln \left (f \left (d x +c \right )-c f +d e \right ) \left (\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {f \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {\left (c f -d e \right ) \arctan \left (d x +c \right )^{2}}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}\right )}{f}\right )}{d}+\frac {2 a b \left (-\frac {d^{2} \operatorname {arccot}\left (d x +c \right )}{\left (f \left (d x +c \right )-c f +d e \right ) f}-\frac {d^{2} \left (\frac {f \ln \left (f \left (d x +c \right )-c f +d e \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {-\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (-c f +d e \right ) \arctan \left (d x +c \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}\right )}{f}\right )}{d}\) | \(784\) |
| derivativedivides | \(\frac {\frac {a^{2} d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b^{2} d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right )^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {-\frac {2 \,\operatorname {arccot}\left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {2 \,\operatorname {arccot}\left (d x +c \right ) f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 \,\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) c f}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {2 \,\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) d e}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {2 \left (c f -d e \right ) \arctan \left (d x +c \right )^{2}}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 f \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 f^{2} \left (-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )+2 a b \,d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {\frac {\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c f -d e \right ) \arctan \left (d x +c \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )}{d}\) | \(793\) |
| default | \(\frac {\frac {a^{2} d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b^{2} d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right )^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {-\frac {2 \,\operatorname {arccot}\left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {2 \,\operatorname {arccot}\left (d x +c \right ) f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 \,\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) c f}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {2 \,\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) d e}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {2 \left (c f -d e \right ) \arctan \left (d x +c \right )^{2}}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 f \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {2 f^{2} \left (-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )+2 a b \,d^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {\frac {\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c f -d e \right ) \arctan \left (d x +c \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )}{d}\) | \(793\) |
-a^2/(f*x+e)/f+b^2/d*(-d^2/(f*(d*x+c)-c*f+d*e)/f*arccot(d*x+c)^2-2*d^2/f*( arccot(d*x+c)/(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)/(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)/(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)/(c^2*f ^2-2*c*d*e*f+d^2*e^2+f^2)*arctan(d*x+c)*d*e+f^2/(c^2*f^2-2*c*d*e*f+d^2*e^2 +f^2)*(-1/2*I*ln(f*(d*x+c)-c*f+d*e)*(ln((I*f-f*(d*x+c))/(d*e+I*f-c*f))-ln( (I*f+f*(d*x+c))/(c*f-d*e+I*f)))/f-1/2*I*(dilog((I*f-f*(d*x+c))/(d*e+I*f-c* f))-dilog((I*f+f*(d*x+c))/(c*f-d*e+I*f)))/f)-1/2*f/(c^2*f^2-2*c*d*e*f+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))))-1/2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*(c*f-d*e)*arctan(d*x+c)^2))+ 2*a*b/d*(-d^2/(f*(d*x+c)-c*f+d*e)/f*arccot(d*x+c)-d^2/f*(1/(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/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2) *(-1/2*f*ln(1+(d*x+c)^2)+(-c*f+d*e)*arctan(d*x+c))))
\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
-(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*b - 1/16*(4*arctan2(1, d*x + c)^2 - 1 6*(f^2*x + e*f)*integrate(1/16*(12*d^2*f*x^2*arctan2(1, d*x + c)^2 + 8*(3* c*arctan2(1, d*x + c)^2 - arctan2(1, d*x + c))*d*f*x - 8*d*e*arctan2(1, d* x + c) + (d^2*f*x^2 + 2*c*d*f*x + (c^2 + 1)*f)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 12*(c^2*arctan2(1, d*x + c)^2 + arctan2(1, d*x + c)^2)*f - 4*(d^ 2*f*x^2 + c*d*e + (d^2*e + c*d*f)*x)*log(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) - log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2)*b^2/(f^2*x + e*f) - a^2/(f^2*x + e*f)
Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\left (a+b \cot ^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^2}{{\left (e+f\,x\right )}^2} \,d x \]