Integrand size = 26, antiderivative size = 213 \[ \int x^2 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {2 a e x}{3 c^2}+\frac {5 b e x^2}{18 c}-\frac {2}{9} a e x^3-\frac {2 a e \arctan (c x)}{3 c^3}+\frac {2 b e x \arctan (c x)}{3 c^2}-\frac {2}{9} b e x^3 \arctan (c x)-\frac {b e \arctan (c x)^2}{3 c^3}-\frac {11 b e \log \left (1+c^2 x^2\right )}{18 c^3}-\frac {b e \log ^2\left (1+c^2 x^2\right )}{12 c^3}-\frac {b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 c}+\frac {1}{3} x^3 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )+\frac {b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 c^3} \]
2/3*a*e*x/c^2+5/18*b*e*x^2/c-2/9*a*e*x^3-2/3*a*e*arctan(c*x)/c^3+2/3*b*e*x *arctan(c*x)/c^2-2/9*b*e*x^3*arctan(c*x)-1/3*b*e*arctan(c*x)^2/c^3-11/18*b *e*ln(c^2*x^2+1)/c^3-1/12*b*e*ln(c^2*x^2+1)^2/c^3-1/6*b*x^2*(d+e*ln(c^2*x^ 2+1))/c+1/3*x^3*(a+b*arctan(c*x))*(d+e*ln(c^2*x^2+1))+1/6*b*ln(c^2*x^2+1)* (d+e*ln(c^2*x^2+1))/c^3
Time = 0.09 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.80 \[ \int x^2 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {2 c x \left (b c (-3 d+5 e) x+6 a c^2 d x^2-4 a e \left (-3+c^2 x^2\right )\right )-12 b e \arctan (c x)^2+2 \left (3 b d+6 a c^3 e x^3-b e \left (11+3 c^2 x^2\right )\right ) \log \left (1+c^2 x^2\right )+3 b e \log ^2\left (1+c^2 x^2\right )-4 \arctan (c x) \left (6 a e+b c x \left (-6 e-3 c^2 d x^2+2 c^2 e x^2\right )-3 b c^3 e x^3 \log \left (1+c^2 x^2\right )\right )}{36 c^3} \]
(2*c*x*(b*c*(-3*d + 5*e)*x + 6*a*c^2*d*x^2 - 4*a*e*(-3 + c^2*x^2)) - 12*b* e*ArcTan[c*x]^2 + 2*(3*b*d + 6*a*c^3*e*x^3 - b*e*(11 + 3*c^2*x^2))*Log[1 + c^2*x^2] + 3*b*e*Log[1 + c^2*x^2]^2 - 4*ArcTan[c*x]*(6*a*e + b*c*x*(-6*e - 3*c^2*d*x^2 + 2*c^2*e*x^2) - 3*b*c^3*e*x^3*Log[1 + c^2*x^2]))/(36*c^3)
Time = 0.72 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5556, 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.
| \(\displaystyle \int x^2 (a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right ) \, dx\) |
| \(\Big \downarrow \) 5556 |
| \(\displaystyle -2 c^2 e \int \left (\frac {b x \log \left (c^2 x^2+1\right )}{6 c^3 \left (c^2 x^2+1\right )}-\frac {x^3 (-2 c x \arctan (c x) b+b-2 a c x)}{6 c \left (c^2 x^2+1\right )}\right )dx+\frac {1}{3} x^3 (a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )-\frac {b x^2 \left (e \log \left (c^2 x^2+1\right )+d\right )}{6 c}+\frac {b \log \left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{6 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )-2 c^2 e \left (\frac {a \arctan (c x)}{3 c^5}-\frac {a x}{3 c^4}+\frac {a x^3}{9 c^2}+\frac {b \arctan (c x)^2}{6 c^5}-\frac {b x \arctan (c x)}{3 c^4}+\frac {b x^3 \arctan (c x)}{9 c^2}-\frac {5 b x^2}{36 c^3}+\frac {b \log ^2\left (c^2 x^2+1\right )}{24 c^5}+\frac {11 b \log \left (c^2 x^2+1\right )}{36 c^5}\right )-\frac {b x^2 \left (e \log \left (c^2 x^2+1\right )+d\right )}{6 c}+\frac {b \log \left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{6 c^3}\) |
-1/6*(b*x^2*(d + e*Log[1 + c^2*x^2]))/c + (x^3*(a + b*ArcTan[c*x])*(d + e* Log[1 + c^2*x^2]))/3 + (b*Log[1 + c^2*x^2]*(d + e*Log[1 + c^2*x^2]))/(6*c^ 3) - 2*c^2*e*(-1/3*(a*x)/c^4 - (5*b*x^2)/(36*c^3) + (a*x^3)/(9*c^2) + (a*A rcTan[c*x])/(3*c^5) - (b*x*ArcTan[c*x])/(3*c^4) + (b*x^3*ArcTan[c*x])/(9*c ^2) + (b*ArcTan[c*x]^2)/(6*c^5) + (11*b*Log[1 + c^2*x^2])/(36*c^5) + (b*Lo g[1 + c^2*x^2]^2)/(24*c^5))
3.13.88.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*( e_.))*(x_)^(m_.), x_Symbol] :> With[{u = IntHide[x^m*(a + b*ArcTan[c*x]), x ]}, Simp[(d + e*Log[f + g*x^2]) u, x] - Simp[2*e*g Int[ExpandIntegrand[ x*(u/(f + g*x^2)), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Intege rQ[m] && NeQ[m, -1]
Time = 1.82 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.02
| method | result | size |
| parallelrisch | \(\frac {12 e b \ln \left (c^{2} x^{2}+1\right ) \arctan \left (c x \right ) x^{3} c^{3}+12 b \arctan \left (c x \right ) x^{3} c^{3} d -8 x^{3} \arctan \left (c x \right ) b \,c^{3} e +12 e a \ln \left (c^{2} x^{2}+1\right ) x^{3} c^{3}+12 a \,c^{3} d \,x^{3}-8 a \,c^{3} e \,x^{3}-6 x^{2} \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} e -6 c^{2} x^{2} b d +10 b \,c^{2} e \,x^{2}+24 e b \arctan \left (c x \right ) x c +24 x a c e -12 e b \arctan \left (c x \right )^{2}+3 e b \ln \left (c^{2} x^{2}+1\right )^{2}-24 e a \arctan \left (c x \right )+6 \ln \left (c^{2} x^{2}+1\right ) b d -22 \ln \left (c^{2} x^{2}+1\right ) b e}{36 c^{3}}\) | \(217\) |
| default | \(\text {Expression too large to display}\) | \(4010\) |
| parts | \(\text {Expression too large to display}\) | \(4010\) |
| risch | \(\text {Expression too large to display}\) | \(22991\) |
1/36*(12*e*b*ln(c^2*x^2+1)*arctan(c*x)*x^3*c^3+12*b*arctan(c*x)*x^3*c^3*d- 8*x^3*arctan(c*x)*b*c^3*e+12*e*a*ln(c^2*x^2+1)*x^3*c^3+12*a*c^3*d*x^3-8*a* c^3*e*x^3-6*x^2*ln(c^2*x^2+1)*b*c^2*e-6*c^2*x^2*b*d+10*b*c^2*e*x^2+24*e*b* arctan(c*x)*x*c+24*x*a*c*e-12*e*b*arctan(c*x)^2+3*e*b*ln(c^2*x^2+1)^2-24*e *a*arctan(c*x)+6*ln(c^2*x^2+1)*b*d-22*ln(c^2*x^2+1)*b*e)/c^3
Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.79 \[ \int x^2 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {24 \, a c e x + 4 \, {\left (3 \, a c^{3} d - 2 \, a c^{3} e\right )} x^{3} - 12 \, b e \arctan \left (c x\right )^{2} + 3 \, b e \log \left (c^{2} x^{2} + 1\right )^{2} - 2 \, {\left (3 \, b c^{2} d - 5 \, b c^{2} e\right )} x^{2} + 4 \, {\left (6 \, b c e x + {\left (3 \, b c^{3} d - 2 \, b c^{3} e\right )} x^{3} - 6 \, a e\right )} \arctan \left (c x\right ) + 2 \, {\left (6 \, b c^{3} e x^{3} \arctan \left (c x\right ) + 6 \, a c^{3} e x^{3} - 3 \, b c^{2} e x^{2} + 3 \, b d - 11 \, b e\right )} \log \left (c^{2} x^{2} + 1\right )}{36 \, c^{3}} \]
1/36*(24*a*c*e*x + 4*(3*a*c^3*d - 2*a*c^3*e)*x^3 - 12*b*e*arctan(c*x)^2 + 3*b*e*log(c^2*x^2 + 1)^2 - 2*(3*b*c^2*d - 5*b*c^2*e)*x^2 + 4*(6*b*c*e*x + (3*b*c^3*d - 2*b*c^3*e)*x^3 - 6*a*e)*arctan(c*x) + 2*(6*b*c^3*e*x^3*arctan (c*x) + 6*a*c^3*e*x^3 - 3*b*c^2*e*x^2 + 3*b*d - 11*b*e)*log(c^2*x^2 + 1))/ c^3
Time = 0.76 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.21 \[ \int x^2 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\begin {cases} \frac {a d x^{3}}{3} + \frac {a e x^{3} \log {\left (c^{2} x^{2} + 1 \right )}}{3} - \frac {2 a e x^{3}}{9} + \frac {2 a e x}{3 c^{2}} - \frac {2 a e \operatorname {atan}{\left (c x \right )}}{3 c^{3}} + \frac {b d x^{3} \operatorname {atan}{\left (c x \right )}}{3} + \frac {b e x^{3} \log {\left (c^{2} x^{2} + 1 \right )} \operatorname {atan}{\left (c x \right )}}{3} - \frac {2 b e x^{3} \operatorname {atan}{\left (c x \right )}}{9} - \frac {b d x^{2}}{6 c} - \frac {b e x^{2} \log {\left (c^{2} x^{2} + 1 \right )}}{6 c} + \frac {5 b e x^{2}}{18 c} + \frac {2 b e x \operatorname {atan}{\left (c x \right )}}{3 c^{2}} + \frac {b d \log {\left (c^{2} x^{2} + 1 \right )}}{6 c^{3}} + \frac {b e \log {\left (c^{2} x^{2} + 1 \right )}^{2}}{12 c^{3}} - \frac {11 b e \log {\left (c^{2} x^{2} + 1 \right )}}{18 c^{3}} - \frac {b e \operatorname {atan}^{2}{\left (c x \right )}}{3 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d x^{3}}{3} & \text {otherwise} \end {cases} \]
Piecewise((a*d*x**3/3 + a*e*x**3*log(c**2*x**2 + 1)/3 - 2*a*e*x**3/9 + 2*a *e*x/(3*c**2) - 2*a*e*atan(c*x)/(3*c**3) + b*d*x**3*atan(c*x)/3 + b*e*x**3 *log(c**2*x**2 + 1)*atan(c*x)/3 - 2*b*e*x**3*atan(c*x)/9 - b*d*x**2/(6*c) - b*e*x**2*log(c**2*x**2 + 1)/(6*c) + 5*b*e*x**2/(18*c) + 2*b*e*x*atan(c*x )/(3*c**2) + b*d*log(c**2*x**2 + 1)/(6*c**3) + b*e*log(c**2*x**2 + 1)**2/( 12*c**3) - 11*b*e*log(c**2*x**2 + 1)/(18*c**3) - b*e*atan(c*x)**2/(3*c**3) , Ne(c, 0)), (a*d*x**3/3, True))
Time = 0.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00 \[ \int x^2 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {1}{3} \, a d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{2} {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b e \arctan \left (c x\right ) + \frac {1}{6} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d + \frac {1}{9} \, {\left (3 \, x^{3} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{2} {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{4}} + \frac {3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a e + \frac {{\left (10 \, c^{2} x^{2} + 12 \, \arctan \left (c x\right )^{2} - 2 \, {\left (3 \, c^{2} x^{2} + 11\right )} \log \left (c^{2} x^{2} + 1\right ) + 3 \, \log \left (c^{2} x^{2} + 1\right )^{2}\right )} b e}{36 \, c^{3}} \]
1/3*a*d*x^3 + 1/9*(3*x^3*log(c^2*x^2 + 1) - 2*c^2*((c^2*x^3 - 3*x)/c^4 + 3 *arctan(c*x)/c^5))*b*e*arctan(c*x) + 1/6*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*b*d + 1/9*(3*x^3*log(c^2*x^2 + 1) - 2*c^2*((c^2*x^ 3 - 3*x)/c^4 + 3*arctan(c*x)/c^5))*a*e + 1/36*(10*c^2*x^2 + 12*arctan(c*x) ^2 - 2*(3*c^2*x^2 + 11)*log(c^2*x^2 + 1) + 3*log(c^2*x^2 + 1)^2)*b*e/c^3
\[ \int x^2 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )} x^{2} \,d x } \]
Time = 2.79 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00 \[ \int x^2 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {a\,d\,x^3}{3}-\frac {2\,a\,e\,x^3}{9}+\frac {b\,e\,{\ln \left (c^2\,x^2+1\right )}^2}{12\,c^3}+\frac {2\,a\,e\,x}{3\,c^2}-\frac {2\,a\,e\,\mathrm {atan}\left (c\,x\right )}{3\,c^3}+\frac {b\,d\,x^3\,\mathrm {atan}\left (c\,x\right )}{3}-\frac {2\,b\,e\,x^3\,\mathrm {atan}\left (c\,x\right )}{9}+\frac {b\,d\,\ln \left (c^2\,x^2+1\right )}{6\,c^3}-\frac {11\,b\,e\,\ln \left (c^2\,x^2+1\right )}{18\,c^3}-\frac {b\,d\,x^2}{6\,c}+\frac {5\,b\,e\,x^2}{18\,c}+\frac {a\,e\,x^3\,\ln \left (c^2\,x^2+1\right )}{3}-\frac {b\,e\,{\mathrm {atan}\left (c\,x\right )}^2}{3\,c^3}+\frac {b\,e\,x^3\,\mathrm {atan}\left (c\,x\right )\,\ln \left (c^2\,x^2+1\right )}{3}-\frac {b\,e\,x^2\,\ln \left (c^2\,x^2+1\right )}{6\,c}+\frac {2\,b\,e\,x\,\mathrm {atan}\left (c\,x\right )}{3\,c^2} \]
(a*d*x^3)/3 - (2*a*e*x^3)/9 + (b*e*log(c^2*x^2 + 1)^2)/(12*c^3) + (2*a*e*x )/(3*c^2) - (2*a*e*atan(c*x))/(3*c^3) + (b*d*x^3*atan(c*x))/3 - (2*b*e*x^3 *atan(c*x))/9 + (b*d*log(c^2*x^2 + 1))/(6*c^3) - (11*b*e*log(c^2*x^2 + 1)) /(18*c^3) - (b*d*x^2)/(6*c) + (5*b*e*x^2)/(18*c) + (a*e*x^3*log(c^2*x^2 + 1))/3 - (b*e*atan(c*x)^2)/(3*c^3) + (b*e*x^3*atan(c*x)*log(c^2*x^2 + 1))/3 - (b*e*x^2*log(c^2*x^2 + 1))/(6*c) + (2*b*e*x*atan(c*x))/(3*c^2)