Integrand size = 14, antiderivative size = 76 \[ \int (d+e x) (a+b \arctan (c x)) \, dx=-\frac {b e x}{2 c}-\frac {b \left (d^2-\frac {e^2}{c^2}\right ) \arctan (c x)}{2 e}+\frac {(d+e x)^2 (a+b \arctan (c x))}{2 e}-\frac {b d \log \left (1+c^2 x^2\right )}{2 c} \] Output:
-1/2*b*e*x/c-1/2*b*(d^2-e^2/c^2)*arctan(c*x)/e+1/2*(e*x+d)^2*(a+b*arctan(c *x))/e-1/2*b*d*ln(c^2*x^2+1)/c
Time = 0.00 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int (d+e x) (a+b \arctan (c x)) \, dx=a d x-\frac {b e x}{2 c}+\frac {1}{2} a e x^2+\frac {b e \arctan (c x)}{2 c^2}+b d x \arctan (c x)+\frac {1}{2} b e x^2 \arctan (c x)-\frac {b d \log \left (1+c^2 x^2\right )}{2 c} \] Input:
Integrate[(d + e*x)*(a + b*ArcTan[c*x]),x]
Output:
a*d*x - (b*e*x)/(2*c) + (a*e*x^2)/2 + (b*e*ArcTan[c*x])/(2*c^2) + b*d*x*Ar cTan[c*x] + (b*e*x^2*ArcTan[c*x])/2 - (b*d*Log[1 + c^2*x^2])/(2*c)
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5387, 478, 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.
\(\displaystyle \int (d+e x) (a+b \arctan (c x)) \, dx\) |
\(\Big \downarrow \) 5387 |
\(\displaystyle \frac {(d+e x)^2 (a+b \arctan (c x))}{2 e}-\frac {b c \int \frac {(d+e x)^2}{c^2 x^2+1}dx}{2 e}\) |
\(\Big \downarrow \) 478 |
\(\displaystyle \frac {(d+e x)^2 (a+b \arctan (c x))}{2 e}-\frac {b c \int \left (\frac {e^2}{c^2}+\frac {d^2 c^2+2 d e x c^2-e^2}{c^2 \left (c^2 x^2+1\right )}\right )dx}{2 e}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^2 (a+b \arctan (c x))}{2 e}-\frac {b c \left (\frac {\arctan (c x) (c d-e) (c d+e)}{c^3}+\frac {d e \log \left (c^2 x^2+1\right )}{c^2}+\frac {e^2 x}{c^2}\right )}{2 e}\) |
Input:
Int[(d + e*x)*(a + b*ArcTan[c*x]),x]
Output:
((d + e*x)^2*(a + b*ArcTan[c*x]))/(2*e) - (b*c*((e^2*x)/c^2 + ((c*d - e)*( c*d + e)*ArcTan[c*x])/c^3 + (d*e*Log[1 + c^2*x^2])/c^2))/(2*e)
Int[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[Expand Integrand[(c + d*x)^n/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ [n, 1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])/(e*(q + 1))), x] - Simp[b*( c/(e*(q + 1))) Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{a, b , c, d, e, q}, x] && NeQ[q, -1]
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91
method | result | size |
parts | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \arctan \left (c x \right ) x^{2} e}{2}+b \arctan \left (c x \right ) d x -\frac {b d \ln \left (c^{2} x^{2}+1\right )}{2 c}-\frac {b e x}{2 c}+\frac {\arctan \left (c x \right ) b e}{2 c^{2}}\) | \(69\) |
parallelrisch | \(-\frac {-\arctan \left (c x \right ) b \,c^{2} e \,x^{2}-a \,c^{2} e \,x^{2}-2 d b \arctan \left (c x \right ) x \,c^{2}-2 a \,c^{2} d x +b c d \ln \left (c^{2} x^{2}+1\right )+b c e x -\arctan \left (c x \right ) b e}{2 c^{2}}\) | \(78\) |
derivativedivides | \(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\arctan \left (c x \right ) d \,c^{2} x +\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {c e x}{2}-\frac {d c \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {e \arctan \left (c x \right )}{2}\right )}{c}}{c}\) | \(82\) |
default | \(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\arctan \left (c x \right ) d \,c^{2} x +\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {c e x}{2}-\frac {d c \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {e \arctan \left (c x \right )}{2}\right )}{c}}{c}\) | \(82\) |
risch | \(-\frac {i b \left (e \,x^{2}+2 d x \right ) \ln \left (i c x +1\right )}{4}+\frac {i b e \,x^{2} \ln \left (-i c x +1\right )}{4}+\frac {i b d x \ln \left (-i c x +1\right )}{2}+\frac {a e \,x^{2}}{2}+a d x -\frac {b d \ln \left (c^{2} x^{2}+1\right )}{2 c}-\frac {b e x}{2 c}+\frac {\arctan \left (c x \right ) b e}{2 c^{2}}\) | \(101\) |
Input:
int((e*x+d)*(a+b*arctan(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/2*e*x^2+d*x)+1/2*b*arctan(c*x)*x^2*e+b*arctan(c*x)*d*x-1/2*b*d*ln(c^2 *x^2+1)/c-1/2*b*e*x/c+1/2/c^2*arctan(c*x)*b*e
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int (d+e x) (a+b \arctan (c x)) \, dx=\frac {a c^{2} e x^{2} - b c d \log \left (c^{2} x^{2} + 1\right ) + {\left (2 \, a c^{2} d - b c e\right )} x + {\left (b c^{2} e x^{2} + 2 \, b c^{2} d x + b e\right )} \arctan \left (c x\right )}{2 \, c^{2}} \] Input:
integrate((e*x+d)*(a+b*arctan(c*x)),x, algorithm="fricas")
Output:
1/2*(a*c^2*e*x^2 - b*c*d*log(c^2*x^2 + 1) + (2*a*c^2*d - b*c*e)*x + (b*c^2 *e*x^2 + 2*b*c^2*d*x + b*e)*arctan(c*x))/c^2
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.14 \[ \int (d+e x) (a+b \arctan (c x)) \, dx=\begin {cases} a d x + \frac {a e x^{2}}{2} + b d x \operatorname {atan}{\left (c x \right )} + \frac {b e x^{2} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b d \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c} - \frac {b e x}{2 c} + \frac {b e \operatorname {atan}{\left (c x \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)*(a+b*atan(c*x)),x)
Output:
Piecewise((a*d*x + a*e*x**2/2 + b*d*x*atan(c*x) + b*e*x**2*atan(c*x)/2 - b *d*log(x**2 + c**(-2))/(2*c) - b*e*x/(2*c) + b*e*atan(c*x)/(2*c**2), Ne(c, 0)), (a*(d*x + e*x**2/2), True))
Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.93 \[ \int (d+e x) (a+b \arctan (c x)) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (c x\right ) - c {\left (\frac {x}{c^{2}} - \frac {\arctan \left (c x\right )}{c^{3}}\right )}\right )} b e + a d x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d}{2 \, c} \] Input:
integrate((e*x+d)*(a+b*arctan(c*x)),x, algorithm="maxima")
Output:
1/2*a*e*x^2 + 1/2*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*b*e + a* d*x + 1/2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*b*d/c
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.12 \[ \int (d+e x) (a+b \arctan (c x)) \, dx=\frac {b c^{2} e x^{2} \arctan \left (c x\right ) + a c^{2} e x^{2} + 2 \, b c^{2} d x \arctan \left (c x\right ) + 2 \, a c^{2} d x - \pi b e \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (x\right ) - b c e x - b c d \log \left (c^{2} x^{2} + 1\right ) + b e \arctan \left (c x\right )}{2 \, c^{2}} \] Input:
integrate((e*x+d)*(a+b*arctan(c*x)),x, algorithm="giac")
Output:
1/2*(b*c^2*e*x^2*arctan(c*x) + a*c^2*e*x^2 + 2*b*c^2*d*x*arctan(c*x) + 2*a *c^2*d*x - pi*b*e*sgn(c)*sgn(x) - b*c*e*x - b*c*d*log(c^2*x^2 + 1) + b*e*a rctan(c*x))/c^2
Time = 0.68 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.88 \[ \int (d+e x) (a+b \arctan (c x)) \, dx=a\,d\,x+\frac {a\,e\,x^2}{2}+b\,d\,x\,\mathrm {atan}\left (c\,x\right )-\frac {b\,e\,x}{2\,c}+\frac {b\,e\,\mathrm {atan}\left (c\,x\right )}{2\,c^2}+\frac {b\,e\,x^2\,\mathrm {atan}\left (c\,x\right )}{2}-\frac {b\,d\,\ln \left (c^2\,x^2+1\right )}{2\,c} \] Input:
int((a + b*atan(c*x))*(d + e*x),x)
Output:
a*d*x + (a*e*x^2)/2 + b*d*x*atan(c*x) - (b*e*x)/(2*c) + (b*e*atan(c*x))/(2 *c^2) + (b*e*x^2*atan(c*x))/2 - (b*d*log(c^2*x^2 + 1))/(2*c)
Time = 0.19 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int (d+e x) (a+b \arctan (c x)) \, dx=\frac {2 \mathit {atan} \left (c x \right ) b \,c^{2} d x +\mathit {atan} \left (c x \right ) b \,c^{2} e \,x^{2}+\mathit {atan} \left (c x \right ) b e -\mathrm {log}\left (c^{2} x^{2}+1\right ) b c d +2 a \,c^{2} d x +a \,c^{2} e \,x^{2}-b c e x}{2 c^{2}} \] Input:
int((e*x+d)*(a+b*atan(c*x)),x)
Output:
(2*atan(c*x)*b*c**2*d*x + atan(c*x)*b*c**2*e*x**2 + atan(c*x)*b*e - log(c* *2*x**2 + 1)*b*c*d + 2*a*c**2*d*x + a*c**2*e*x**2 - b*c*e*x)/(2*c**2)