Integrand size = 20, antiderivative size = 91 \[ \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {1}{4 a^2 c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)}{2 a c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^2}{4 a^2 c^2}-\frac {\arctan (a x)^2}{2 a^2 c^2 \left (1+a^2 x^2\right )} \] Output:
1/4/a^2/c^2/(a^2*x^2+1)+1/2*x*arctan(a*x)/a/c^2/(a^2*x^2+1)+1/4*arctan(a*x )^2/a^2/c^2-1/2*arctan(a*x)^2/a^2/c^2/(a^2*x^2+1)
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.52 \[ \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {1+2 a x \arctan (a x)+\left (-1+a^2 x^2\right ) \arctan (a x)^2}{4 a^2 c^2 \left (1+a^2 x^2\right )} \] Input:
Integrate[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2,x]
Output:
(1 + 2*a*x*ArcTan[a*x] + (-1 + a^2*x^2)*ArcTan[a*x]^2)/(4*a^2*c^2*(1 + a^2 *x^2))
Time = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5465, 27, 5427, 241}
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.
\(\displaystyle \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^2} \, dx\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle \frac {\int \frac {\arctan (a x)}{c^2 \left (a^2 x^2+1\right )^2}dx}{a}-\frac {\arctan (a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a c^2}-\frac {\arctan (a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}\) |
\(\Big \downarrow \) 5427 |
\(\displaystyle \frac {-\frac {1}{2} a \int \frac {x}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a c^2}-\frac {\arctan (a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}\) |
\(\Big \downarrow \) 241 |
\(\displaystyle \frac {\frac {x \arctan (a x)}{2 \left (a^2 x^2+1\right )}+\frac {1}{4 a \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^2}{4 a}}{a c^2}-\frac {\arctan (a x)^2}{2 a^2 c^2 \left (a^2 x^2+1\right )}\) |
Input:
Int[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^2,x]
Output:
-1/2*ArcTan[a*x]^2/(a^2*c^2*(1 + a^2*x^2)) + (1/(4*a*(1 + a^2*x^2)) + (x*A rcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a))/(a*c^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2)^2, x_Sym bol] :> Simp[x*((a + b*ArcTan[c*x])^p/(2*d*(d + e*x^2))), x] + (Simp[(a + b *ArcTan[c*x])^(p + 1)/(2*b*c*d^2*(p + 1)), x] - Simp[b*c*(p/2) Int[x*((a + b*ArcTan[c*x])^(p - 1)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Time = 0.60 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(\frac {\arctan \left (a x \right )^{2} x^{2} a^{2}-a^{2} x^{2}+2 \arctan \left (a x \right ) a x -\arctan \left (a x \right )^{2}}{4 c^{2} \left (a^{2} x^{2}+1\right ) a^{2}}\) | \(58\) |
derivativedivides | \(\frac {-\frac {\arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\frac {\arctan \left (a x \right ) a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}}{c^{2}}}{a^{2}}\) | \(73\) |
default | \(\frac {-\frac {\arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\frac {\arctan \left (a x \right ) a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}}{c^{2}}}{a^{2}}\) | \(73\) |
parts | \(-\frac {\arctan \left (a x \right )^{2}}{2 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\frac {\arctan \left (a x \right ) a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}}{a^{2} c^{2}}\) | \(75\) |
risch | \(-\frac {\left (a^{2} x^{2}-1\right ) \ln \left (i a x +1\right )^{2}}{16 a^{2} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\left (-\ln \left (-i a x +1\right )+a^{2} x^{2} \ln \left (-i a x +1\right )-2 i a x \right ) \ln \left (i a x +1\right )}{8 \left (a x +i\right ) a^{2} c^{2} \left (a x -i\right )}-\frac {-4+a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-\ln \left (-i a x +1\right )^{2}-4 i a x \ln \left (-i a x +1\right )}{16 \left (a x +i\right ) a^{2} c^{2} \left (a x -i\right )}\) | \(171\) |
orering | \(\frac {\left (a^{2} x^{2}+1\right ) \left (10 a^{4} x^{4}-3 a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{2}}{4 a^{4} x^{2} \left (a^{2} c \,x^{2}+c \right )^{2}}+\frac {\left (a^{2} x^{2}+1\right )^{2} \left (5 a^{2} x^{2}-1\right ) \left (\frac {\arctan \left (a x \right )^{2}}{\left (a^{2} c \,x^{2}+c \right )^{2}}+\frac {2 x \arctan \left (a x \right ) a}{\left (a^{2} c \,x^{2}+c \right )^{2} \left (a^{2} x^{2}+1\right )}-\frac {4 x^{2} \arctan \left (a x \right )^{2} c \,a^{2}}{\left (a^{2} c \,x^{2}+c \right )^{3}}\right )}{4 a^{4} x^{2}}+\frac {\left (a^{2} x^{2}+1\right )^{3} \left (\frac {4 \arctan \left (a x \right ) a}{\left (a^{2} c \,x^{2}+c \right )^{2} \left (a^{2} x^{2}+1\right )}-\frac {12 \arctan \left (a x \right )^{2} c x \,a^{2}}{\left (a^{2} c \,x^{2}+c \right )^{3}}+\frac {2 x \,a^{2}}{\left (a^{2} x^{2}+1\right )^{2} \left (a^{2} c \,x^{2}+c \right )^{2}}-\frac {16 x^{2} \arctan \left (a x \right ) a^{3} c}{\left (a^{2} c \,x^{2}+c \right )^{3} \left (a^{2} x^{2}+1\right )}-\frac {4 x^{2} \arctan \left (a x \right ) a^{3}}{\left (a^{2} c \,x^{2}+c \right )^{2} \left (a^{2} x^{2}+1\right )^{2}}+\frac {24 x^{3} \arctan \left (a x \right )^{2} c^{2} a^{4}}{\left (a^{2} c \,x^{2}+c \right )^{4}}\right )}{8 a^{4} x}\) | \(366\) |
Input:
int(x*arctan(a*x)^2/(a^2*c*x^2+c)^2,x,method=_RETURNVERBOSE)
Output:
1/4*(arctan(a*x)^2*x^2*a^2-a^2*x^2+2*arctan(a*x)*a*x-arctan(a*x)^2)/c^2/(a ^2*x^2+1)/a^2
Time = 0.13 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.53 \[ \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {2 \, a x \arctan \left (a x\right ) + {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )^{2} + 1}{4 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} \] Input:
integrate(x*arctan(a*x)^2/(a^2*c*x^2+c)^2,x, algorithm="fricas")
Output:
1/4*(2*a*x*arctan(a*x) + (a^2*x^2 - 1)*arctan(a*x)^2 + 1)/(a^4*c^2*x^2 + a ^2*c^2)
\[ \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x \operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \] Input:
integrate(x*atan(a*x)**2/(a**2*c*x**2+c)**2,x)
Output:
Integral(x*atan(a*x)**2/(a**4*x**4 + 2*a**2*x**2 + 1), x)/c**2
Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14 \[ \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {{\left (\frac {x}{a^{2} c x^{2} + c} + \frac {\arctan \left (a x\right )}{a c}\right )} \arctan \left (a x\right )}{2 \, a c} - \frac {{\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1}{4 \, {\left (a^{4} c x^{2} + a^{2} c\right )} c} - \frac {\arctan \left (a x\right )^{2}}{2 \, {\left (a^{2} c x^{2} + c\right )} a^{2} c} \] Input:
integrate(x*arctan(a*x)^2/(a^2*c*x^2+c)^2,x, algorithm="maxima")
Output:
1/2*(x/(a^2*c*x^2 + c) + arctan(a*x)/(a*c))*arctan(a*x)/(a*c) - 1/4*((a^2* x^2 + 1)*arctan(a*x)^2 - 1)/((a^4*c*x^2 + a^2*c)*c) - 1/2*arctan(a*x)^2/(( a^2*c*x^2 + c)*a^2*c)
\[ \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \] Input:
integrate(x*arctan(a*x)^2/(a^2*c*x^2+c)^2,x, algorithm="giac")
Output:
integrate(x*arctan(a*x)^2/(a^2*c*x^2 + c)^2, x)
Time = 0.68 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.55 \[ \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+2\,a\,x\,\mathrm {atan}\left (a\,x\right )-{\mathrm {atan}\left (a\,x\right )}^2+1}{4\,a^2\,c^2\,\left (a^2\,x^2+1\right )} \] Input:
int((x*atan(a*x)^2)/(c + a^2*c*x^2)^2,x)
Output:
(2*a*x*atan(a*x) - atan(a*x)^2 + a^2*x^2*atan(a*x)^2 + 1)/(4*a^2*c^2*(a^2* x^2 + 1))
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.63 \[ \int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\mathit {atan} \left (a x \right )^{2} a^{2} x^{2}-\mathit {atan} \left (a x \right )^{2}+2 \mathit {atan} \left (a x \right ) a x -a^{2} x^{2}}{4 a^{2} c^{2} \left (a^{2} x^{2}+1\right )} \] Input:
int(x*atan(a*x)^2/(a^2*c*x^2+c)^2,x)
Output:
(atan(a*x)**2*a**2*x**2 - atan(a*x)**2 + 2*atan(a*x)*a*x - a**2*x**2)/(4*a **2*c**2*(a**2*x**2 + 1))