Integrand size = 18, antiderivative size = 96 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c \left (1+a^2 x^2\right )}{12 a^2}-\frac {c x \arctan (a x)}{3 a}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)}{6 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{4 a^2}+\frac {c \log \left (1+a^2 x^2\right )}{6 a^2} \] Output:
1/12*c*(a^2*x^2+1)/a^2-1/3*c*x*arctan(a*x)/a-1/6*c*x*(a^2*x^2+1)*arctan(a* x)/a+1/4*c*(a^2*x^2+1)^2*arctan(a*x)^2/a^2+1/6*c*ln(a^2*x^2+1)/a^2
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c \left (a^2 x^2-2 a x \left (3+a^2 x^2\right ) \arctan (a x)+3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2+2 \log \left (1+a^2 x^2\right )\right )}{12 a^2} \] Input:
Integrate[x*(c + a^2*c*x^2)*ArcTan[a*x]^2,x]
Output:
(c*(a^2*x^2 - 2*a*x*(3 + a^2*x^2)*ArcTan[a*x] + 3*(1 + a^2*x^2)^2*ArcTan[a *x]^2 + 2*Log[1 + a^2*x^2]))/(12*a^2)
Time = 0.35 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5465, 27, 5413, 5345, 240}
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.
\(\displaystyle \int x \arctan (a x)^2 \left (a^2 c x^2+c\right ) \, dx\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle \frac {c \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{4 a^2}-\frac {\int c \left (a^2 x^2+1\right ) \arctan (a x)dx}{2 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {c \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{4 a^2}-\frac {c \int \left (a^2 x^2+1\right ) \arctan (a x)dx}{2 a}\) |
\(\Big \downarrow \) 5413 |
\(\displaystyle \frac {c \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{4 a^2}-\frac {c \left (\frac {2}{3} \int \arctan (a x)dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )}{2 a}\) |
\(\Big \downarrow \) 5345 |
\(\displaystyle \frac {c \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{4 a^2}-\frac {c \left (\frac {2}{3} \left (x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )}{2 a}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {c \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{4 a^2}-\frac {c \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )}{2 a}\) |
Input:
Int[x*(c + a^2*c*x^2)*ArcTan[a*x]^2,x]
Output:
(c*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(4*a^2) - (c*(-1/6*(1 + a^2*x^2)/a + (x* (1 + a^2*x^2)*ArcTan[a*x])/3 + (2*(x*ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a)) )/3))/(2*a)
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), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2) ^q*((a + b*ArcTan[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 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 = 0.76 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91
method | result | size |
parts | \(\frac {x^{4} c \,a^{2} \arctan \left (a x \right )^{2}}{4}+\frac {c \,x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c \arctan \left (a x \right )^{2}}{4 a^{2}}-\frac {c \left (\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}+\arctan \left (a x \right ) a x -\frac {a^{2} x^{2}}{6}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}\right )}{2 a^{2}}\) | \(87\) |
derivativedivides | \(\frac {\frac {c \arctan \left (a x \right )^{2} a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c \arctan \left (a x \right )^{2}}{4}-\frac {c \left (\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}+\arctan \left (a x \right ) a x -\frac {a^{2} x^{2}}{6}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}\right )}{2}}{a^{2}}\) | \(88\) |
default | \(\frac {\frac {c \arctan \left (a x \right )^{2} a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c \arctan \left (a x \right )^{2}}{4}-\frac {c \left (\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}+\arctan \left (a x \right ) a x -\frac {a^{2} x^{2}}{6}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}\right )}{2}}{a^{2}}\) | \(88\) |
parallelrisch | \(\frac {3 c \arctan \left (a x \right )^{2} a^{4} x^{4}-2 c \arctan \left (a x \right ) a^{3} x^{3}+6 a^{2} c \,x^{2} \arctan \left (a x \right )^{2}+a^{2} c \,x^{2}-6 a c x \arctan \left (a x \right )+3 c \arctan \left (a x \right )^{2}+2 c \ln \left (a^{2} x^{2}+1\right )}{12 a^{2}}\) | \(89\) |
risch | \(-\frac {c \left (a^{2} x^{2}+1\right )^{2} \ln \left (i a x +1\right )^{2}}{16 a^{2}}+\frac {c \left (3 x^{4} \ln \left (-i a x +1\right ) a^{4}+2 i a^{3} x^{3}+6 a^{2} x^{2} \ln \left (-i a x +1\right )+6 i a x +3 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{24 a^{2}}-\frac {c \,a^{2} x^{4} \ln \left (-i a x +1\right )^{2}}{16}-\frac {i c a \,x^{3} \ln \left (-i a x +1\right )}{12}-\frac {c \,x^{2} \ln \left (-i a x +1\right )^{2}}{8}-\frac {i c x \ln \left (-i a x +1\right )}{4 a}+\frac {c \,x^{2}}{12}-\frac {c \ln \left (-i a x +1\right )^{2}}{16 a^{2}}+\frac {c \ln \left (-a^{2} x^{2}-1\right )}{6 a^{2}}\) | \(206\) |
Input:
int(x*(a^2*c*x^2+c)*arctan(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/4*x^4*c*a^2*arctan(a*x)^2+1/2*c*x^2*arctan(a*x)^2+1/4*c*arctan(a*x)^2/a^ 2-1/2*c/a^2*(1/3*arctan(a*x)*x^3*a^3+arctan(a*x)*a*x-1/6*a^2*x^2-1/3*ln(a^ 2*x^2+1))
Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {a^{2} c x^{2} + 3 \, {\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2} - 2 \, {\left (a^{3} c x^{3} + 3 \, a c x\right )} \arctan \left (a x\right ) + 2 \, c \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{2}} \] Input:
integrate(x*(a^2*c*x^2+c)*arctan(a*x)^2,x, algorithm="fricas")
Output:
1/12*(a^2*c*x^2 + 3*(a^4*c*x^4 + 2*a^2*c*x^2 + c)*arctan(a*x)^2 - 2*(a^3*c *x^3 + 3*a*c*x)*arctan(a*x) + 2*c*log(a^2*x^2 + 1))/a^2
Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.98 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\begin {cases} \frac {a^{2} c x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{4} - \frac {a c x^{3} \operatorname {atan}{\left (a x \right )}}{6} + \frac {c x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {c x^{2}}{12} - \frac {c x \operatorname {atan}{\left (a x \right )}}{2 a} + \frac {c \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{6 a^{2}} + \frac {c \operatorname {atan}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x*(a**2*c*x**2+c)*atan(a*x)**2,x)
Output:
Piecewise((a**2*c*x**4*atan(a*x)**2/4 - a*c*x**3*atan(a*x)/6 + c*x**2*atan (a*x)**2/2 + c*x**2/12 - c*x*atan(a*x)/(2*a) + c*log(x**2 + a**(-2))/(6*a* *2) + c*atan(a*x)**2/(4*a**2), Ne(a, 0)), (0, True))
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{4 \, a^{2} c} + \frac {{\left (c^{2} x^{2} + \frac {2 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a - 2 \, {\left (a^{2} c^{2} x^{3} + 3 \, c^{2} x\right )} \arctan \left (a x\right )}{12 \, a c} \] Input:
integrate(x*(a^2*c*x^2+c)*arctan(a*x)^2,x, algorithm="maxima")
Output:
1/4*(a^2*c*x^2 + c)^2*arctan(a*x)^2/(a^2*c) + 1/12*((c^2*x^2 + 2*c^2*log(a ^2*x^2 + 1)/a^2)*a - 2*(a^2*c^2*x^3 + 3*c^2*x)*arctan(a*x))/(a*c)
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {1}{4} \, {\left (a^{2} c x^{4} + 2 \, c x^{2}\right )} \arctan \left (a x\right )^{2} - \frac {2 \, a^{3} c x^{3} \arctan \left (a x\right ) - a^{2} c x^{2} + 6 \, a c x \arctan \left (a x\right ) - 3 \, c \arctan \left (a x\right )^{2} - 2 \, c \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{2}} \] Input:
integrate(x*(a^2*c*x^2+c)*arctan(a*x)^2,x, algorithm="giac")
Output:
1/4*(a^2*c*x^4 + 2*c*x^2)*arctan(a*x)^2 - 1/12*(2*a^3*c*x^3*arctan(a*x) - a^2*c*x^2 + 6*a*c*x*arctan(a*x) - 3*c*arctan(a*x)^2 - 2*c*log(a^2*x^2 + 1) )/a^2
Time = 0.80 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c\,\left (6\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+x^2\right )}{12}+\frac {\frac {c\,\left (3\,{\mathrm {atan}\left (a\,x\right )}^2+2\,\ln \left (a^2\,x^2+1\right )\right )}{12}-\frac {a\,c\,x\,\mathrm {atan}\left (a\,x\right )}{2}}{a^2}+\frac {a^2\,c\,x^4\,{\mathrm {atan}\left (a\,x\right )}^2}{4}-\frac {a\,c\,x^3\,\mathrm {atan}\left (a\,x\right )}{6} \] Input:
int(x*atan(a*x)^2*(c + a^2*c*x^2),x)
Output:
(c*(6*x^2*atan(a*x)^2 + x^2))/12 + ((c*(2*log(a^2*x^2 + 1) + 3*atan(a*x)^2 ))/12 - (a*c*x*atan(a*x))/2)/a^2 + (a^2*c*x^4*atan(a*x)^2)/4 - (a*c*x^3*at an(a*x))/6
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c \left (3 \mathit {atan} \left (a x \right )^{2} a^{4} x^{4}+6 \mathit {atan} \left (a x \right )^{2} a^{2} x^{2}+3 \mathit {atan} \left (a x \right )^{2}-2 \mathit {atan} \left (a x \right ) a^{3} x^{3}-6 \mathit {atan} \left (a x \right ) a x +2 \,\mathrm {log}\left (a^{2} x^{2}+1\right )+a^{2} x^{2}\right )}{12 a^{2}} \] Input:
int(x*(a^2*c*x^2+c)*atan(a*x)^2,x)
Output:
(c*(3*atan(a*x)**2*a**4*x**4 + 6*atan(a*x)**2*a**2*x**2 + 3*atan(a*x)**2 - 2*atan(a*x)*a**3*x**3 - 6*atan(a*x)*a*x + 2*log(a**2*x**2 + 1) + a**2*x** 2))/(12*a**2)