Integrand size = 18, antiderivative size = 222 \[ \int (e+f x) (a+b \arctan (c+d x))^2 \, dx=-\frac {a b f x}{d}-\frac {b^2 f (c+d x) \arctan (c+d x)}{d^2}+\frac {i (d e-c f) (a+b \arctan (c+d x))^2}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) (a+b \arctan (c+d x))^2}{2 d^2 f}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^2}{2 f}+\frac {2 b (d e-c f) (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}+\frac {i b^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2} \] Output:
-a*b*f*x/d-b^2*f*(d*x+c)*arctan(d*x+c)/d^2+I*(-c*f+d*e)*(a+b*arctan(d*x+c) )^2/d^2-1/2*(-c*f+d*e+f)*(d*e-(1+c)*f)*(a+b*arctan(d*x+c))^2/d^2/f+1/2*(f* x+e)^2*(a+b*arctan(d*x+c))^2/f+2*b*(-c*f+d*e)*(a+b*arctan(d*x+c))*ln(2/(1+ I*(d*x+c)))/d^2+1/2*b^2*f*ln(1+(d*x+c)^2)/d^2+I*b^2*(-c*f+d*e)*polylog(2,1 -2/(1+I*(d*x+c)))/d^2
Time = 0.86 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.19 \[ \int (e+f x) (a+b \arctan (c+d x))^2 \, dx=\frac {2 a^2 c d e-2 a b c f-a^2 c^2 f+2 a^2 d^2 e x-2 a b d f x+a^2 d^2 f x^2+b^2 (-i+c+d x) (2 d e+i f-c f+d f x) \arctan (c+d x)^2-2 b \arctan (c+d x) \left (b f (c+d x)+a \left (-2 c d e+c^2 f-2 d^2 e x-f \left (1+d^2 x^2\right )\right )-2 b (d e-c f) \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )+4 a b d e \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )-2 b^2 f \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )-4 a b c f \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )-2 i b^2 (d e-c f) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )}{2 d^2} \] Input:
Integrate[(e + f*x)*(a + b*ArcTan[c + d*x])^2,x]
Output:
(2*a^2*c*d*e - 2*a*b*c*f - a^2*c^2*f + 2*a^2*d^2*e*x - 2*a*b*d*f*x + a^2*d ^2*f*x^2 + b^2*(-I + c + d*x)*(2*d*e + I*f - c*f + d*f*x)*ArcTan[c + d*x]^ 2 - 2*b*ArcTan[c + d*x]*(b*f*(c + d*x) + a*(-2*c*d*e + c^2*f - 2*d^2*e*x - f*(1 + d^2*x^2)) - 2*b*(d*e - c*f)*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + 4*a*b*d*e*Log[1/Sqrt[1 + (c + d*x)^2]] - 2*b^2*f*Log[1/Sqrt[1 + (c + d*x)^ 2]] - 4*a*b*c*f*Log[1/Sqrt[1 + (c + d*x)^2]] - (2*I)*b^2*(d*e - c*f)*PolyL og[2, -E^((2*I)*ArcTan[c + d*x])])/(2*d^2)
Time = 0.61 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, 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.
| \(\displaystyle \int (e+f x) (a+b \arctan (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 5570 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right ) (a+b \arctan (c+d x))^2}{d}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x)) (a+b \arctan (c+d x))^2d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^2 (a+b \arctan (c+d x))^2}{2 f}-\frac {b \int \left ((a+b \arctan (c+d x)) f^2+\frac {((d e-c f+f) (d e-(c+1) f)+2 f (d e-c f) (c+d x)) (a+b \arctan (c+d x))}{(c+d x)^2+1}\right )d(c+d x)}{f}}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^2 (a+b \arctan (c+d x))^2}{2 f}-\frac {b \left (-\frac {i f (d e-c f) (a+b \arctan (c+d x))^2}{b}+\frac {(-c f+d e+f) (d e-(c+1) f) (a+b \arctan (c+d x))^2}{2 b}-2 f (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))+a f^2 (c+d x)+b f^2 (c+d x) \arctan (c+d x)-i b f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )-\frac {1}{2} b f^2 \log \left ((c+d x)^2+1\right )\right )}{f}}{d^2}\) |
Input:
Int[(e + f*x)*(a + b*ArcTan[c + d*x])^2,x]
Output:
(((d*e - c*f + f*(c + d*x))^2*(a + b*ArcTan[c + d*x])^2)/(2*f) - (b*(a*f^2 *(c + d*x) + b*f^2*(c + d*x)*ArcTan[c + d*x] - (I*f*(d*e - c*f)*(a + b*Arc Tan[c + d*x])^2)/b + ((d*e + f - c*f)*(d*e - (1 + c)*f)*(a + b*ArcTan[c + d*x])^2)/(2*b) - 2*f*(d*e - c*f)*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))] - (b*f^2*Log[1 + (c + d*x)^2])/2 - I*b*f*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))]))/f)/d^2
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]
Time = 0.40 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.86
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{2} f \,x^{2}+e x \right )+\frac {b^{2} \left (\frac {\arctan \left (d x +c \right )^{2} \left (d x +c \right )^{2} f}{2 d}-\frac {\arctan \left (d x +c \right )^{2} c f \left (d x +c \right )}{d}+\arctan \left (d x +c \right )^{2} e \left (d x +c \right )-\frac {-\ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) c f +\ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) d e -\frac {\arctan \left (d x +c \right )^{2} f}{2}+\arctan \left (d x +c \right ) \left (d x +c \right ) f -\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\left (-2 c f +2 d e \right ) \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2}}{d}\right )}{d}+\frac {2 a b \left (\frac {\arctan \left (d x +c \right ) \left (d x +c \right )^{2} f}{2 d}-\frac {\arctan \left (d x +c \right ) c f \left (d x +c \right )}{d}+\arctan \left (d x +c \right ) e \left (d x +c \right )-\frac {f \left (d x +c \right )+\frac {\left (-2 c f +2 d e \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-f \arctan \left (d x +c \right )}{2 d}\right )}{d}\) | \(413\) |
| derivativedivides | \(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\arctan \left (d x +c \right )^{2} f c \left (d x +c \right )-\arctan \left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\arctan \left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}-\ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) c f +\ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) d e -\frac {\arctan \left (d x +c \right )^{2} f}{2}+\arctan \left (d x +c \right ) \left (d x +c \right ) f -\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\frac {\left (2 c f -2 d e \right ) \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2}\right )}{d}-\frac {2 a b \left (\arctan \left (d x +c \right ) f c \left (d x +c \right )-\arctan \left (d x +c \right ) e d \left (d x +c \right )-\frac {\arctan \left (d x +c \right ) f \left (d x +c \right )^{2}}{2}+\frac {f \left (d x +c \right )}{2}-\frac {\left (2 c f -2 d e \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4}-\frac {f \arctan \left (d x +c \right )}{2}\right )}{d}}{d}\) | \(419\) |
| default | \(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\arctan \left (d x +c \right )^{2} f c \left (d x +c \right )-\arctan \left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\arctan \left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}-\ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) c f +\ln \left (1+\left (d x +c \right )^{2}\right ) \arctan \left (d x +c \right ) d e -\frac {\arctan \left (d x +c \right )^{2} f}{2}+\arctan \left (d x +c \right ) \left (d x +c \right ) f -\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\frac {\left (2 c f -2 d e \right ) \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2}\right )}{d}-\frac {2 a b \left (\arctan \left (d x +c \right ) f c \left (d x +c \right )-\arctan \left (d x +c \right ) e d \left (d x +c \right )-\frac {\arctan \left (d x +c \right ) f \left (d x +c \right )^{2}}{2}+\frac {f \left (d x +c \right )}{2}-\frac {\left (2 c f -2 d e \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4}-\frac {f \arctan \left (d x +c \right )}{2}\right )}{d}}{d}\) | \(419\) |
| risch | \(\text {Expression too large to display}\) | \(1175\) |
Input:
int((f*x+e)*(a+b*arctan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
a^2*(1/2*f*x^2+e*x)+b^2/d*(1/2/d*arctan(d*x+c)^2*(d*x+c)^2*f-1/d*arctan(d* x+c)^2*c*f*(d*x+c)+arctan(d*x+c)^2*e*(d*x+c)-1/d*(-ln(1+(d*x+c)^2)*arctan( d*x+c)*c*f+ln(1+(d*x+c)^2)*arctan(d*x+c)*d*e-1/2*arctan(d*x+c)^2*f+arctan( d*x+c)*(d*x+c)*f-1/2*f*ln(1+(d*x+c)^2)-1/2*(-2*c*f+2*d*e)*(-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))))))+2*a*b/d*(1/2/ d*arctan(d*x+c)*(d*x+c)^2*f-1/d*arctan(d*x+c)*c*f*(d*x+c)+arctan(d*x+c)*e* (d*x+c)-1/2/d*(f*(d*x+c)+1/2*(-2*c*f+2*d*e)*ln(1+(d*x+c)^2)-f*arctan(d*x+c )))
\[ \int (e+f x) (a+b \arctan (c+d x))^2 \, dx=\int { {\left (f x + e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((f*x+e)*(a+b*arctan(d*x+c))^2,x, algorithm="fricas")
Output:
integral(a^2*f*x + a^2*e + (b^2*f*x + b^2*e)*arctan(d*x + c)^2 + 2*(a*b*f* x + a*b*e)*arctan(d*x + c), x)
\[ \int (e+f x) (a+b \arctan (c+d x))^2 \, dx=\int \left (a + b \operatorname {atan}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \] Input:
integrate((f*x+e)*(a+b*atan(d*x+c))**2,x)
Output:
Integral((a + b*atan(c + d*x))**2*(e + f*x), x)
\[ \int (e+f x) (a+b \arctan (c+d x))^2 \, dx=\int { {\left (f x + e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((f*x+e)*(a+b*arctan(d*x+c))^2,x, algorithm="maxima")
Output:
3/4*b^2*c^2*e*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 1/4*(3*arctan( 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 + 12*b^2*d^2*f*integrate(1/16*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + b^2*d^2*f*integrate(1/16*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) + 12*b^2*d^2*e*integrate(1/16*x^ 2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 24*b^2*c*d*f*integ rate(1/16*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 2*b^2* d^2*f*integrate(1/16*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) + b^2*d^2*e*integrate(1/16*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) + 2*b^2*c*d*f*integrate(1/16*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) + 24*b^2*c*d*e*integrate(1/16*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^2*c^2*f*integrate(1/16*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d *x + c^2 + 1), x) + 4*b^2*d^2*e*integrate(1/16*x^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 2*b^2*c*d*f*integrate(1/16*x ^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 2* b^2*c*d*e*integrate(1/16*x*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) + b^2*c^2*f*integrate(1/16*x*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) + 4*b^2*c*d*e*integrate(1/16 *x*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) +...
\[ \int (e+f x) (a+b \arctan (c+d x))^2 \, dx=\int { {\left (f x + e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((f*x+e)*(a+b*arctan(d*x+c))^2,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int (e+f x) (a+b \arctan (c+d x))^2 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2 \,d x \] Input:
int((e + f*x)*(a + b*atan(c + d*x))^2,x)
Output:
int((e + f*x)*(a + b*atan(c + d*x))^2, x)
\[ \int (e+f x) (a+b \arctan (c+d x))^2 \, dx=\frac {\mathit {atan} \left (d x +c \right )^{2} b^{2} c^{2} f +2 \mathit {atan} \left (d x +c \right )^{2} b^{2} d^{2} e x +\mathit {atan} \left (d x +c \right )^{2} b^{2} d^{2} f \,x^{2}+\mathit {atan} \left (d x +c \right )^{2} b^{2} f -2 \mathit {atan} \left (d x +c \right ) a b \,c^{2} f +4 \mathit {atan} \left (d x +c \right ) a b c d e +4 \mathit {atan} \left (d x +c \right ) a b \,d^{2} e x +2 \mathit {atan} \left (d x +c \right ) a b \,d^{2} f \,x^{2}+2 \mathit {atan} \left (d x +c \right ) a b f -2 \mathit {atan} \left (d x +c \right ) b^{2} c f -2 \mathit {atan} \left (d x +c \right ) b^{2} d f x +4 \left (\int \frac {\mathit {atan} \left (d x +c \right ) x}{d^{2} x^{2}+2 c d x +c^{2}+1}d x \right ) b^{2} c \,d^{2} f -4 \left (\int \frac {\mathit {atan} \left (d x +c \right ) x}{d^{2} x^{2}+2 c d x +c^{2}+1}d x \right ) b^{2} d^{3} e +2 \,\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) a b c f -2 \,\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) a b d e +\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} f +2 a^{2} d^{2} e x +a^{2} d^{2} f \,x^{2}-2 a b d f x}{2 d^{2}} \] Input:
int((f*x+e)*(a+b*atan(d*x+c))^2,x)
Output:
(atan(c + d*x)**2*b**2*c**2*f + 2*atan(c + d*x)**2*b**2*d**2*e*x + atan(c + d*x)**2*b**2*d**2*f*x**2 + atan(c + d*x)**2*b**2*f - 2*atan(c + d*x)*a*b *c**2*f + 4*atan(c + d*x)*a*b*c*d*e + 4*atan(c + d*x)*a*b*d**2*e*x + 2*ata n(c + d*x)*a*b*d**2*f*x**2 + 2*atan(c + d*x)*a*b*f - 2*atan(c + d*x)*b**2* c*f - 2*atan(c + d*x)*b**2*d*f*x + 4*int((atan(c + d*x)*x)/(c**2 + 2*c*d*x + d**2*x**2 + 1),x)*b**2*c*d**2*f - 4*int((atan(c + d*x)*x)/(c**2 + 2*c*d *x + d**2*x**2 + 1),x)*b**2*d**3*e + 2*log(c**2 + 2*c*d*x + d**2*x**2 + 1) *a*b*c*f - 2*log(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b*d*e + log(c**2 + 2*c* d*x + d**2*x**2 + 1)*b**2*f + 2*a**2*d**2*e*x + a**2*d**2*f*x**2 - 2*a*b*d *f*x)/(2*d**2)