Integrand size = 19, antiderivative size = 48 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=-\frac {1}{2} b e x+\frac {b e \arctan (c+d x)}{2 d}+\frac {e (c+d x)^2 (a+b \arctan (c+d x))}{2 d} \] Output:
-1/2*b*e*x+1/2*b*e*arctan(d*x+c)/d+1/2*e*(d*x+c)^2*(a+b*arctan(d*x+c))/d
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\frac {e \left (b (-d x+\arctan (c+d x))+(c+d x)^2 (a+b \arctan (c+d x))\right )}{2 d} \] Input:
Integrate[(c*e + d*e*x)*(a + b*ArcTan[c + d*x]),x]
Output:
(e*(b*(-(d*x) + ArcTan[c + d*x]) + (c + d*x)^2*(a + b*ArcTan[c + d*x])))/( 2*d)
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5566, 27, 5361, 262, 216}
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.
\(\displaystyle \int (c e+d e x) (a+b \arctan (c+d x)) \, dx\) |
\(\Big \downarrow \) 5566 |
\(\displaystyle \frac {\int e (c+d x) (a+b \arctan (c+d x))d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e \int (c+d x) (a+b \arctan (c+d x))d(c+d x)}{d}\) |
\(\Big \downarrow \) 5361 |
\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))-\frac {1}{2} b \int \frac {(c+d x)^2}{(c+d x)^2+1}d(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))-\frac {1}{2} b \left (-\int \frac {1}{(c+d x)^2+1}d(c+d x)+c+d x\right )\right )}{d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))-\frac {1}{2} b (-\arctan (c+d x)+c+d x)\right )}{d}\) |
Input:
Int[(c*e + d*e*x)*(a + b*ArcTan[c + d*x]),x]
Output:
(e*(-1/2*(b*(c + d*x - ArcTan[c + d*x])) + ((c + d*x)^2*(a + b*ArcTan[c + d*x]))/2))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[(f*(x/d))^m*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+e b \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\) | \(51\) |
default | \(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+e b \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\) | \(51\) |
parts | \(e a \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e b \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\) | \(52\) |
parallelrisch | \(\frac {d^{3} e b \,x^{2} \arctan \left (d x +c \right )+x^{2} a \,d^{3} e +2 c e b \arctan \left (d x +c \right ) x \,d^{2}+2 x a c \,d^{2} e +\arctan \left (d x +c \right ) b \,c^{2} d e -x b \,d^{2} e -5 a \,c^{2} d e +e b \arctan \left (d x +c \right ) d +2 b c d e -d e a}{2 d^{2}}\) | \(105\) |
risch | \(-\frac {i e b \left (d \,x^{2}+2 c x \right ) \ln \left (1+i \left (d x +c \right )\right )}{4}+\frac {i e d b \,x^{2} \ln \left (1-i \left (d x +c \right )\right )}{4}+\frac {i e b c x \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {a d e \,x^{2}}{2}+\frac {e \arctan \left (d x +c \right ) b \,c^{2}}{2 d}+a c e x -\frac {b e x}{2}+\frac {b e \arctan \left (d x +c \right )}{2 d}\) | \(113\) |
orering | \(\frac {\left (2 d^{3} x^{3}+5 c \,d^{2} x^{2}+4 c^{2} d x +c^{3}+2 d x +c \right ) \left (d e x +c e \right ) \left (a +b \arctan \left (d x +c \right )\right )}{2 \left (d x +c \right )^{2} d}-\frac {x \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) \left (d e \left (a +b \arctan \left (d x +c \right )\right )+\frac {\left (d e x +c e \right ) b d}{1+\left (d x +c \right )^{2}}\right )}{2 d \left (d x +c \right )}\) | \(131\) |
Input:
int((d*e*x+c*e)*(a+b*arctan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/2*e*a*(d*x+c)^2+e*b*(1/2*(d*x+c)^2*arctan(d*x+c)-1/2*d*x-1/2*c+1/2* arctan(d*x+c)))
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.25 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\frac {a d^{2} e x^{2} + {\left (2 \, a c - b\right )} d e x + {\left (b d^{2} e x^{2} + 2 \, b c d e x + {\left (b c^{2} + b\right )} e\right )} \arctan \left (d x + c\right )}{2 \, d} \] Input:
integrate((d*e*x+c*e)*(a+b*arctan(d*x+c)),x, algorithm="fricas")
Output:
1/2*(a*d^2*e*x^2 + (2*a*c - b)*d*e*x + (b*d^2*e*x^2 + 2*b*c*d*e*x + (b*c^2 + b)*e)*arctan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (41) = 82\).
Time = 0.61 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.98 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\begin {cases} a c e x + \frac {a d e x^{2}}{2} + \frac {b c^{2} e \operatorname {atan}{\left (c + d x \right )}}{2 d} + b c e x \operatorname {atan}{\left (c + d x \right )} + \frac {b d e x^{2} \operatorname {atan}{\left (c + d x \right )}}{2} - \frac {b e x}{2} + \frac {b e \operatorname {atan}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {atan}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*e*x+c*e)*(a+b*atan(d*x+c)),x)
Output:
Piecewise((a*c*e*x + a*d*e*x**2/2 + b*c**2*e*atan(c + d*x)/(2*d) + b*c*e*x *atan(c + d*x) + b*d*e*x**2*atan(c + d*x)/2 - b*e*x/2 + b*e*atan(c + d*x)/ (2*d), Ne(d, 0)), (c*e*x*(a + b*atan(c)), True))
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (42) = 84\).
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.50 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\frac {1}{2} \, a d e x^{2} + \frac {1}{2} \, {\left (x^{2} \arctan \left (d x + c\right ) - d {\left (\frac {x}{d^{2}} + \frac {{\left (c^{2} - 1\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{3}} - \frac {c \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{3}}\right )}\right )} b d e + a c e x + \frac {{\left (2 \, {\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b c e}{2 \, d} \] Input:
integrate((d*e*x+c*e)*(a+b*arctan(d*x+c)),x, algorithm="maxima")
Output:
1/2*a*d*e*x^2 + 1/2*(x^2*arctan(d*x + c) - d*(x/d^2 + (c^2 - 1)*arctan((d^ 2*x + c*d)/d)/d^3 - c*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^3))*b*d*e + a*c*e *x + 1/2*(2*(d*x + c)*arctan(d*x + c) - log((d*x + c)^2 + 1))*b*c*e/d
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (42) = 84\).
Time = 0.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.52 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\frac {2 \, b d^{2} e x^{2} \arctan \left (d x + c\right ) + 2 \, a d^{2} e x^{2} + 4 \, b c d e x \arctan \left (d x + c\right ) + \pi b c^{2} e \mathrm {sgn}\left (d x + c\right ) - \pi b c^{2} e + 4 \, a c d e x - 2 \, b c^{2} e \arctan \left (\frac {1}{d x + c}\right ) - 2 \, b d e x + \pi b e \mathrm {sgn}\left (d x + c\right ) - \pi b e - 2 \, b e \arctan \left (\frac {1}{d x + c}\right )}{4 \, d} \] Input:
integrate((d*e*x+c*e)*(a+b*arctan(d*x+c)),x, algorithm="giac")
Output:
1/4*(2*b*d^2*e*x^2*arctan(d*x + c) + 2*a*d^2*e*x^2 + 4*b*c*d*e*x*arctan(d* x + c) + pi*b*c^2*e*sgn(d*x + c) - pi*b*c^2*e + 4*a*c*d*e*x - 2*b*c^2*e*ar ctan(1/(d*x + c)) - 2*b*d*e*x + pi*b*e*sgn(d*x + c) - pi*b*e - 2*b*e*arcta n(1/(d*x + c)))/d
Time = 1.95 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.52 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=a\,c\,e\,x-\frac {b\,e\,x}{2}+\frac {b\,e\,\mathrm {atan}\left (c+d\,x\right )}{2\,d}+\frac {a\,d\,e\,x^2}{2}+\frac {b\,c^2\,e\,\mathrm {atan}\left (c+d\,x\right )}{2\,d}+b\,c\,e\,x\,\mathrm {atan}\left (c+d\,x\right )+\frac {b\,d\,e\,x^2\,\mathrm {atan}\left (c+d\,x\right )}{2} \] Input:
int((c*e + d*e*x)*(a + b*atan(c + d*x)),x)
Output:
a*c*e*x - (b*e*x)/2 + (b*e*atan(c + d*x))/(2*d) + (a*d*e*x^2)/2 + (b*c^2*e *atan(c + d*x))/(2*d) + b*c*e*x*atan(c + d*x) + (b*d*e*x^2*atan(c + d*x))/ 2
Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.48 \[ \int (c e+d e x) (a+b \arctan (c+d x)) \, dx=\frac {e \left (\mathit {atan} \left (d x +c \right ) b \,c^{2}+2 \mathit {atan} \left (d x +c \right ) b c d x +\mathit {atan} \left (d x +c \right ) b \,d^{2} x^{2}+\mathit {atan} \left (d x +c \right ) b +2 a c d x +a \,d^{2} x^{2}-b d x \right )}{2 d} \] Input:
int((d*e*x+c*e)*(a+b*atan(d*x+c)),x)
Output:
(e*(atan(c + d*x)*b*c**2 + 2*atan(c + d*x)*b*c*d*x + atan(c + d*x)*b*d**2* x**2 + atan(c + d*x)*b + 2*a*c*d*x + a*d**2*x**2 - b*d*x))/(2*d)