Integrand size = 18, antiderivative size = 231 \[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {b^2 e x}{3 c^2}-\frac {b^2 e \arctan (c x)}{3 c^3}-\frac {b e x^2 (a+b \arctan (c x))}{3 c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {i e (a+b \arctan (c x))^2}{3 c^3}+d x (a+b \arctan (c x))^2+\frac {1}{3} e x^3 (a+b \arctan (c x))^2+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \]
1/3*b^2*e*x/c^2-1/3*b^2*e*arctan(c*x)/c^3-1/3*b*e*x^2*(a+b*arctan(c*x))/c+ I*d*(a+b*arctan(c*x))^2/c-1/3*I*e*(a+b*arctan(c*x))^2/c^3+d*x*(a+b*arctan( c*x))^2+1/3*e*x^3*(a+b*arctan(c*x))^2+2*b*d*(a+b*arctan(c*x))*ln(2/(1+I*c* x))/c-2/3*b*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^3+I*b^2*d*polylog(2,1-2/ (1+I*c*x))/c-1/3*I*b^2*e*polylog(2,1-2/(1+I*c*x))/c^3
Time = 0.31 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.90 \[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {3 a^2 c^3 d x+b^2 c e x-a b c^2 e x^2+a^2 c^3 e x^3+b^2 \left (-3 i c^2 d+i e+c^3 \left (3 d x+e x^3\right )\right ) \arctan (c x)^2-b \arctan (c x) \left (-2 a c^3 x \left (3 d+e x^2\right )+b \left (e+c^2 e x^2\right )+2 b \left (-3 c^2 d+e\right ) \log \left (1+e^{2 i \arctan (c x)}\right )\right )-3 a b c^2 d \log \left (1+c^2 x^2\right )+a b e \log \left (1+c^2 x^2\right )-i b^2 \left (3 c^2 d-e\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{3 c^3} \]
(3*a^2*c^3*d*x + b^2*c*e*x - a*b*c^2*e*x^2 + a^2*c^3*e*x^3 + b^2*((-3*I)*c ^2*d + I*e + c^3*(3*d*x + e*x^3))*ArcTan[c*x]^2 - b*ArcTan[c*x]*(-2*a*c^3* x*(3*d + e*x^2) + b*(e + c^2*e*x^2) + 2*b*(-3*c^2*d + e)*Log[1 + E^((2*I)* ArcTan[c*x])]) - 3*a*b*c^2*d*Log[1 + c^2*x^2] + a*b*e*Log[1 + c^2*x^2] - I *b^2*(3*c^2*d - e)*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(3*c^3)
Time = 0.55 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5449, 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.
| \(\displaystyle \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5449 |
| \(\displaystyle \int \left (d (a+b \arctan (c x))^2+e x^2 (a+b \arctan (c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i e (a+b \arctan (c x))^2}{3 c^3}-\frac {2 b e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3}+d x (a+b \arctan (c x))^2+\frac {i d (a+b \arctan (c x))^2}{c}+\frac {2 b d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {1}{3} e x^3 (a+b \arctan (c x))^2-\frac {b e x^2 (a+b \arctan (c x))}{3 c}-\frac {b^2 e \arctan (c x)}{3 c^3}-\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 e x}{3 c^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c}\) |
(b^2*e*x)/(3*c^2) - (b^2*e*ArcTan[c*x])/(3*c^3) - (b*e*x^2*(a + b*ArcTan[c *x]))/(3*c) + (I*d*(a + b*ArcTan[c*x])^2)/c - ((I/3)*e*(a + b*ArcTan[c*x]) ^2)/c^3 + d*x*(a + b*ArcTan[c*x])^2 + (e*x^3*(a + b*ArcTan[c*x])^2)/3 + (2 *b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c - (2*b*e*(a + b*ArcTan[c*x] )*Log[2/(1 + I*c*x)])/(3*c^3) + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c - ((I/3)*b^2*e*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3
3.13.50.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcTan[c*x])^p, (d + e*x^2)^q, x], x ] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]
Time = 0.54 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.39
| method | result | size |
| parts | \(a^{2} \left (\frac {1}{3} e \,x^{3}+x d \right )+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c \,x^{3} e}{3}+\arctan \left (c x \right )^{2} c x d -\frac {2 \left (\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d}{2}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e}{2}-\frac {e \left (c x -\arctan \left (c x \right )\right )}{2}-\frac {\left (3 c^{2} d -e \right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2}\right )}{3 c^{2}}\right )}{c}+\frac {2 a b \left (\frac {c \arctan \left (c x \right ) x^{3} e}{3}+\arctan \left (c x \right ) c x d -\frac {\frac {e \,c^{2} x^{2}}{2}+\frac {\left (3 c^{2} d -e \right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{3 c^{2}}\right )}{c}\) | \(321\) |
| derivativedivides | \(\frac {\frac {a^{2} \left (c^{3} x d +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (\arctan \left (c x \right )^{2} c^{3} x d +\frac {\arctan \left (c x \right )^{2} e \,c^{3} x^{3}}{3}-\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{3}-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d +\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e}{3}+\frac {e \left (c x -\arctan \left (c x \right )\right )}{3}+\frac {\left (3 c^{2} d -e \right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{3}\right )}{c^{2}}+\frac {2 a b \left (\arctan \left (c x \right ) c^{3} x d +\frac {\arctan \left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \,c^{2} x^{2}}{6}-\frac {\left (3 c^{2} d -e \right ) \ln \left (c^{2} x^{2}+1\right )}{6}\right )}{c^{2}}}{c}\) | \(330\) |
| default | \(\frac {\frac {a^{2} \left (c^{3} x d +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (\arctan \left (c x \right )^{2} c^{3} x d +\frac {\arctan \left (c x \right )^{2} e \,c^{3} x^{3}}{3}-\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{3}-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d +\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e}{3}+\frac {e \left (c x -\arctan \left (c x \right )\right )}{3}+\frac {\left (3 c^{2} d -e \right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{3}\right )}{c^{2}}+\frac {2 a b \left (\arctan \left (c x \right ) c^{3} x d +\frac {\arctan \left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \,c^{2} x^{2}}{6}-\frac {\left (3 c^{2} d -e \right ) \ln \left (c^{2} x^{2}+1\right )}{6}\right )}{c^{2}}}{c}\) | \(330\) |
| risch | \(-\frac {a b e \,x^{2}}{3 c}+\frac {2 a b d}{c}-\frac {11 a b e}{9 c^{3}}+\frac {i b^{2} d \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {i b^{2} d \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{c}+\frac {b^{2} d \ln \left (i c x +1\right ) \ln \left (-i c x +1\right ) x}{2}+\frac {b^{2} e \ln \left (i c x +1\right ) \ln \left (-i c x +1\right ) x^{3}}{6}-\frac {b a d \ln \left (i c x +1\right )}{c}-\frac {i b^{2} d \ln \left (-i c x +1\right )}{2 c}-\frac {2 i b^{2} e \ln \left (c^{2} x^{2}+1\right )}{9 c^{3}}+\frac {5 i b^{2} e \ln \left (-i c x +1\right )}{36 c^{3}}-\frac {i b^{2} e \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}+\frac {i b^{2} \ln \left (i c x +1\right )^{2} d}{4 c}-\frac {i b^{2} \ln \left (i c x +1\right ) d}{2 c}+\frac {b^{2} e x}{3 c^{2}}-\frac {b^{2} e \arctan \left (c x \right )}{6 c^{3}}+\frac {b e a \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}+\frac {i a b d \arctan \left (c x \right )}{c}+\frac {i b^{2} e \ln \left (i c x +1\right ) x^{2}}{6 c}-\frac {i b^{2} e \ln \left (-i c x +1\right ) x^{2}}{6 c}+\frac {e \,a^{2} x^{3}}{3}+d x \,a^{2}-\frac {i b e a \ln \left (i c x +1\right ) x^{3}}{3}-i b a d \ln \left (i c x +1\right ) x +i \ln \left (-i c x +1\right ) x a b d +\frac {i e b a \ln \left (-i c x +1\right ) x^{3}}{3}+\frac {i b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{c}-\frac {i b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{c}-\frac {i b^{2} e \ln \left (i c x +1\right ) \ln \left (-i c x +1\right )}{6 c^{3}}+\frac {i b^{2} e \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{3}}-\frac {i b^{2} e \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}+\frac {i b^{2} d \ln \left (i c x +1\right ) \ln \left (-i c x +1\right )}{2 c}+\frac {i b^{2} d}{c}-\frac {17 i b^{2} e}{54 c^{3}}+\frac {i d \,a^{2}}{c}-\frac {i e \,a^{2}}{3 c^{3}}-\frac {i b^{2} e \ln \left (i c x +1\right )^{2}}{12 c^{3}}+\frac {11 i b^{2} e \ln \left (i c x +1\right )}{36 c^{3}}+\frac {i b^{2} e \ln \left (-i c x +1\right )^{2}}{12 c^{3}}-\frac {i \ln \left (-i c x +1\right )^{2} b^{2} d}{4 c}-\frac {a b d \ln \left (c^{2} x^{2}+1\right )}{2 c}-\frac {b^{2} e \ln \left (i c x +1\right )^{2} x^{3}}{12}-\frac {b^{2} \ln \left (i c x +1\right )^{2} x d}{4}-\frac {\ln \left (-i c x +1\right )^{2} x \,b^{2} d}{4}-\frac {b^{2} e \ln \left (-i c x +1\right )^{2} x^{3}}{12}\) | \(782\) |
a^2*(1/3*e*x^3+x*d)+b^2/c*(1/3*arctan(c*x)^2*c*x^3*e+arctan(c*x)^2*c*x*d-2 /3/c^2*(1/2*arctan(c*x)*e*c^2*x^2+3/2*arctan(c*x)*ln(c^2*x^2+1)*c^2*d-1/2* arctan(c*x)*ln(c^2*x^2+1)*e-1/2*e*(c*x-arctan(c*x))-1/2*(3*c^2*d-e)*(-1/2* I*(ln(c*x-I)*ln(c^2*x^2+1)-1/2*ln(c*x-I)^2-dilog(-1/2*I*(I+c*x))-ln(c*x-I) *ln(-1/2*I*(I+c*x)))+1/2*I*(ln(I+c*x)*ln(c^2*x^2+1)-1/2*ln(I+c*x)^2-dilog( 1/2*I*(c*x-I))-ln(I+c*x)*ln(1/2*I*(c*x-I))))))+2*a*b/c*(1/3*c*arctan(c*x)* x^3*e+arctan(c*x)*c*x*d-1/3/c^2*(1/2*e*c^2*x^2+1/2*(3*c^2*d-e)*ln(c^2*x^2+ 1)))
\[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
integral(a^2*e*x^2 + a^2*d + (b^2*e*x^2 + b^2*d)*arctan(c*x)^2 + 2*(a*b*e* x^2 + a*b*d)*arctan(c*x), x)
\[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )\, dx \]
\[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
1/3*a^2*e*x^3 + 36*b^2*c^2*e*integrate(1/48*x^4*arctan(c*x)^2/(c^2*x^2 + 1 ), x) + 3*b^2*c^2*e*integrate(1/48*x^4*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x ) + 4*b^2*c^2*e*integrate(1/48*x^4*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 36 *b^2*c^2*d*integrate(1/48*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^2*c^2* d*integrate(1/48*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 12*b^2*c^2*d*i ntegrate(1/48*x^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 1/4*b^2*d*arctan(c* x)^3/c - 8*b^2*c*e*integrate(1/48*x^3*arctan(c*x)/(c^2*x^2 + 1), x) - 24*b ^2*c*d*integrate(1/48*x*arctan(c*x)/(c^2*x^2 + 1), x) + 1/3*(2*x^3*arctan( c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*e + a^2*d*x + 36*b^2*e*inte grate(1/48*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^2*e*integrate(1/48*x^ 2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3*b^2*d*integrate(1/48*log(c^2*x^ 2 + 1)^2/(c^2*x^2 + 1), x) + (2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*d/ c + 1/12*(b^2*e*x^3 + 3*b^2*d*x)*arctan(c*x)^2 - 1/48*(b^2*e*x^3 + 3*b^2*d *x)*log(c^2*x^2 + 1)^2
\[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \]