Integrand size = 24, antiderivative size = 137 \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=-\frac {b (d-e) x}{2 c}+\frac {b e x}{c}+\frac {b (d-e) \arctan (c x)}{2 c^2}-\frac {b e \arctan (c x)}{c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-\frac {b e x \log \left (1+c^2 x^2\right )}{2 c}+\frac {e \left (1+c^2 x^2\right ) (a+b \arctan (c x)) \log \left (1+c^2 x^2\right )}{2 c^2} \]
-1/2*b*(d-e)*x/c+b*e*x/c+1/2*b*(d-e)*arctan(c*x)/c^2-b*e*arctan(c*x)/c^2+1 /2*d*x^2*(a+b*arctan(c*x))-1/2*e*x^2*(a+b*arctan(c*x))-1/2*b*e*x*ln(c^2*x^ 2+1)/c+1/2*e*(c^2*x^2+1)*(a+b*arctan(c*x))*ln(c^2*x^2+1)/c^2
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.77 \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {c x (-b (d-3 e)+a c (d-e) x)+e \left (a-b c x+a c^2 x^2\right ) \log \left (1+c^2 x^2\right )+b \arctan (c x) \left (d+c^2 d x^2-e \left (3+c^2 x^2\right )+\left (e+c^2 e x^2\right ) \log \left (1+c^2 x^2\right )\right )}{2 c^2} \]
(c*x*(-(b*(d - 3*e)) + a*c*(d - e)*x) + e*(a - b*c*x + a*c^2*x^2)*Log[1 + c^2*x^2] + b*ArcTan[c*x]*(d + c^2*d*x^2 - e*(3 + c^2*x^2) + (e + c^2*e*x^2 )*Log[1 + c^2*x^2]))/(2*c^2)
Time = 0.33 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5554, 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.
\(\displaystyle \int x (a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right ) \, dx\) |
\(\Big \downarrow \) 5554 |
\(\displaystyle -b c \int \left (\frac {(d-e) x^2}{2 \left (c^2 x^2+1\right )}+\frac {e \log \left (c^2 x^2+1\right )}{2 c^2}\right )dx+\frac {e \left (c^2 x^2+1\right ) \log \left (c^2 x^2+1\right ) (a+b \arctan (c x))}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e \left (c^2 x^2+1\right ) \log \left (c^2 x^2+1\right ) (a+b \arctan (c x))}{2 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))-\frac {1}{2} e x^2 (a+b \arctan (c x))-b c \left (-\frac {(d-e) \arctan (c x)}{2 c^3}+\frac {e \arctan (c x)}{c^3}+\frac {x (d-e)}{2 c^2}+\frac {e x \log \left (c^2 x^2+1\right )}{2 c^2}-\frac {e x}{c^2}\right )\) |
(d*x^2*(a + b*ArcTan[c*x]))/2 - (e*x^2*(a + b*ArcTan[c*x]))/2 + (e*(1 + c^ 2*x^2)*(a + b*ArcTan[c*x])*Log[1 + c^2*x^2])/(2*c^2) - b*c*(((d - e)*x)/(2 *c^2) - (e*x)/c^2 - ((d - e)*ArcTan[c*x])/(2*c^3) + (e*ArcTan[c*x])/c^3 + (e*x*Log[1 + c^2*x^2])/(2*c^2))
3.13.89.3.1 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*(d + e*Log[f + g*x^2]) , x]}, Simp[(a + b*ArcTan[c*x]) u, x] - Simp[b*c Int[ExpandIntegrand[u/ (1 + c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
Time = 1.24 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.23
method | result | size |
parallelrisch | \(\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) b \,c^{2} e \,x^{2}+\arctan \left (c x \right ) b \,c^{2} d \,x^{2}-\arctan \left (c x \right ) b \,c^{2} e \,x^{2}+\ln \left (c^{2} x^{2}+1\right ) a \,c^{2} e \,x^{2}+a \,c^{2} d \,x^{2}-a \,c^{2} e \,x^{2}-\ln \left (c^{2} x^{2}+1\right ) b c e x -b c d x +3 b c e x +\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) b e +\arctan \left (c x \right ) b d -3 e b \arctan \left (c x \right )+\ln \left (c^{2} x^{2}+1\right ) a e}{2 c^{2}}\) | \(168\) |
default | \(\text {Expression too large to display}\) | \(2929\) |
parts | \(\text {Expression too large to display}\) | \(2929\) |
risch | \(\text {Expression too large to display}\) | \(21445\) |
1/2*(arctan(c*x)*ln(c^2*x^2+1)*b*c^2*e*x^2+arctan(c*x)*b*c^2*d*x^2-arctan( c*x)*b*c^2*e*x^2+ln(c^2*x^2+1)*a*c^2*e*x^2+a*c^2*d*x^2-a*c^2*e*x^2-ln(c^2* x^2+1)*b*c*e*x-b*c*d*x+3*b*c*e*x+arctan(c*x)*ln(c^2*x^2+1)*b*e+arctan(c*x) *b*d-3*e*b*arctan(c*x)+ln(c^2*x^2+1)*a*e)/c^2
Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {{\left (a c^{2} d - a c^{2} e\right )} x^{2} - {\left (b c d - 3 \, b c e\right )} x + {\left ({\left (b c^{2} d - b c^{2} e\right )} x^{2} + b d - 3 \, b e\right )} \arctan \left (c x\right ) + {\left (a c^{2} e x^{2} - b c e x + a e + {\left (b c^{2} e x^{2} + b e\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )}{2 \, c^{2}} \]
1/2*((a*c^2*d - a*c^2*e)*x^2 - (b*c*d - 3*b*c*e)*x + ((b*c^2*d - b*c^2*e)* x^2 + b*d - 3*b*e)*arctan(c*x) + (a*c^2*e*x^2 - b*c*e*x + a*e + (b*c^2*e*x ^2 + b*e)*arctan(c*x))*log(c^2*x^2 + 1))/c^2
Time = 0.53 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.47 \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\begin {cases} \frac {a d x^{2}}{2} + \frac {a e x^{2} \log {\left (c^{2} x^{2} + 1 \right )}}{2} - \frac {a e x^{2}}{2} + \frac {a e \log {\left (c^{2} x^{2} + 1 \right )}}{2 c^{2}} + \frac {b d x^{2} \operatorname {atan}{\left (c x \right )}}{2} + \frac {b e x^{2} \log {\left (c^{2} x^{2} + 1 \right )} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b e x^{2} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b d x}{2 c} - \frac {b e x \log {\left (c^{2} x^{2} + 1 \right )}}{2 c} + \frac {3 b e x}{2 c} + \frac {b d \operatorname {atan}{\left (c x \right )}}{2 c^{2}} + \frac {b e \log {\left (c^{2} x^{2} + 1 \right )} \operatorname {atan}{\left (c x \right )}}{2 c^{2}} - \frac {3 b e \operatorname {atan}{\left (c x \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d x^{2}}{2} & \text {otherwise} \end {cases} \]
Piecewise((a*d*x**2/2 + a*e*x**2*log(c**2*x**2 + 1)/2 - a*e*x**2/2 + a*e*l og(c**2*x**2 + 1)/(2*c**2) + b*d*x**2*atan(c*x)/2 + b*e*x**2*log(c**2*x**2 + 1)*atan(c*x)/2 - b*e*x**2*atan(c*x)/2 - b*d*x/(2*c) - b*e*x*log(c**2*x* *2 + 1)/(2*c) + 3*b*e*x/(2*c) + b*d*atan(c*x)/(2*c**2) + b*e*log(c**2*x**2 + 1)*atan(c*x)/(2*c**2) - 3*b*e*atan(c*x)/(2*c**2), Ne(c, 0)), (a*d*x**2/ 2, True))
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.09 \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {1}{2} \, a d x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d - \frac {{\left (x \log \left (c^{2} x^{2} + 1\right ) - 3 \, x + \frac {2 \, \arctan \left (c x\right )}{c}\right )} b e}{2 \, c} - \frac {{\left (c^{2} x^{2} - {\left (c^{2} x^{2} + 1\right )} \log \left (c^{2} x^{2} + 1\right ) + 1\right )} b e \arctan \left (c x\right )}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - {\left (c^{2} x^{2} + 1\right )} \log \left (c^{2} x^{2} + 1\right ) + 1\right )} a e}{2 \, c^{2}} \]
1/2*a*d*x^2 + 1/2*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*b*d - 1/ 2*(x*log(c^2*x^2 + 1) - 3*x + 2*arctan(c*x)/c)*b*e/c - 1/2*(c^2*x^2 - (c^2 *x^2 + 1)*log(c^2*x^2 + 1) + 1)*b*e*arctan(c*x)/c^2 - 1/2*(c^2*x^2 - (c^2* x^2 + 1)*log(c^2*x^2 + 1) + 1)*a*e/c^2
\[ \int x (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )} x \,d x } \]
Time = 1.42 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.66 \[ \int x (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx=\frac {a\,d\,x^2}{2}-\frac {a\,e\,x^2}{2}-\frac {b\,d\,x}{2\,c}+\frac {3\,b\,e\,x}{2\,c}+\frac {b\,d\,x^2\,\mathrm {atan}\left (c\,x\right )}{2}-\frac {b\,e\,x^2\,\mathrm {atan}\left (c\,x\right )}{2}+\frac {a\,e\,\ln \left (c^2\,x^2+1\right )}{2\,c^2}+\frac {b\,d\,\mathrm {atan}\left (\frac {b\,c\,d\,x}{b\,d-3\,b\,e}-\frac {3\,b\,c\,e\,x}{b\,d-3\,b\,e}\right )}{2\,c^2}-\frac {3\,b\,e\,\mathrm {atan}\left (\frac {b\,c\,d\,x}{b\,d-3\,b\,e}-\frac {3\,b\,c\,e\,x}{b\,d-3\,b\,e}\right )}{2\,c^2}+\frac {a\,e\,x^2\,\ln \left (c^2\,x^2+1\right )}{2}-\frac {b\,e\,x\,\ln \left (c^2\,x^2+1\right )}{2\,c}+\frac {b\,e\,\mathrm {atan}\left (c\,x\right )\,\ln \left (c^2\,x^2+1\right )}{2\,c^2}+\frac {b\,e\,x^2\,\mathrm {atan}\left (c\,x\right )\,\ln \left (c^2\,x^2+1\right )}{2} \]
(a*d*x^2)/2 - (a*e*x^2)/2 - (b*d*x)/(2*c) + (3*b*e*x)/(2*c) + (b*d*x^2*ata n(c*x))/2 - (b*e*x^2*atan(c*x))/2 + (a*e*log(c^2*x^2 + 1))/(2*c^2) + (b*d* atan((b*c*d*x)/(b*d - 3*b*e) - (3*b*c*e*x)/(b*d - 3*b*e)))/(2*c^2) - (3*b* e*atan((b*c*d*x)/(b*d - 3*b*e) - (3*b*c*e*x)/(b*d - 3*b*e)))/(2*c^2) + (a* e*x^2*log(c^2*x^2 + 1))/2 - (b*e*x*log(c^2*x^2 + 1))/(2*c) + (b*e*atan(c*x )*log(c^2*x^2 + 1))/(2*c^2) + (b*e*x^2*atan(c*x)*log(c^2*x^2 + 1))/2