3.268 \(\int x (c+a^2 c x^2)^2 \tan ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=153 \[ \frac{c^2 \left (a^2 x^2+1\right )^2}{60 a^2}+\frac{2 c^2 \left (a^2 x^2+1\right )}{45 a^2}+\frac{4 c^2 \log \left (a^2 x^2+1\right )}{45 a^2}+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac{c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{15 a}-\frac{4 c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{45 a}-\frac{8 c^2 x \tan ^{-1}(a x)}{45 a} \]

[Out]

(2*c^2*(1 + a^2*x^2))/(45*a^2) + (c^2*(1 + a^2*x^2)^2)/(60*a^2) - (8*c^2*x*ArcTan[a*x])/(45*a) - (4*c^2*x*(1 +
a^2*x^2)*ArcTan[a*x])/(45*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x])/(15*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^
2)/(6*a^2) + (4*c^2*Log[1 + a^2*x^2])/(45*a^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0930782, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4930, 4878, 4846, 260} \[ \frac{c^2 \left (a^2 x^2+1\right )^2}{60 a^2}+\frac{2 c^2 \left (a^2 x^2+1\right )}{45 a^2}+\frac{4 c^2 \log \left (a^2 x^2+1\right )}{45 a^2}+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac{c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{15 a}-\frac{4 c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{45 a}-\frac{8 c^2 x \tan ^{-1}(a x)}{45 a} \]

Antiderivative was successfully verified.

[In]

Int[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]

[Out]

(2*c^2*(1 + a^2*x^2))/(45*a^2) + (c^2*(1 + a^2*x^2)^2)/(60*a^2) - (8*c^2*x*ArcTan[a*x])/(45*a) - (4*c^2*x*(1 +
a^2*x^2)*ArcTan[a*x])/(45*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x])/(15*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^
2)/(6*a^2) + (4*c^2*Log[1 + a^2*x^2])/(45*a^2)

Rule 4930

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] - Dist[(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]

Rule 4878

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> -Simp[(b*(d + e*x^2)^q)/(2*c*
q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x] + Simp[(x*(d +
e*x^2)^q*(a + b*ArcTan[c*x]))/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx &=\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac{\int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx}{3 a}\\ &=\frac{c^2 \left (1+a^2 x^2\right )^2}{60 a^2}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{15 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac{(4 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx}{15 a}\\ &=\frac{2 c^2 \left (1+a^2 x^2\right )}{45 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2}{60 a^2}-\frac{4 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{45 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{15 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}-\frac{\left (8 c^2\right ) \int \tan ^{-1}(a x) \, dx}{45 a}\\ &=\frac{2 c^2 \left (1+a^2 x^2\right )}{45 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2}{60 a^2}-\frac{8 c^2 x \tan ^{-1}(a x)}{45 a}-\frac{4 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{45 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{15 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}+\frac{1}{45} \left (8 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx\\ &=\frac{2 c^2 \left (1+a^2 x^2\right )}{45 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2}{60 a^2}-\frac{8 c^2 x \tan ^{-1}(a x)}{45 a}-\frac{4 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{45 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{15 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^2}{6 a^2}+\frac{4 c^2 \log \left (1+a^2 x^2\right )}{45 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0649818, size = 84, normalized size = 0.55 \[ \frac{c^2 \left (3 a^4 x^4+14 a^2 x^2+16 \log \left (a^2 x^2+1\right )-4 a x \left (3 a^4 x^4+10 a^2 x^2+15\right ) \tan ^{-1}(a x)+30 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2\right )}{180 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]

[Out]

(c^2*(14*a^2*x^2 + 3*a^4*x^4 - 4*a*x*(15 + 10*a^2*x^2 + 3*a^4*x^4)*ArcTan[a*x] + 30*(1 + a^2*x^2)^3*ArcTan[a*x
]^2 + 16*Log[1 + a^2*x^2]))/(180*a^2)

________________________________________________________________________________________

Maple [A]  time = 0.031, size = 142, normalized size = 0.9 \begin{align*}{\frac{{a}^{4}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{6}}{6}}+{\frac{{a}^{2}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{4}}{2}}+{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}}{2}}-{\frac{{a}^{3}{c}^{2}\arctan \left ( ax \right ){x}^{5}}{15}}-{\frac{2\,a{c}^{2}\arctan \left ( ax \right ){x}^{3}}{9}}-{\frac{{c}^{2}x\arctan \left ( ax \right ) }{3\,a}}+{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{6\,{a}^{2}}}+{\frac{{a}^{2}{c}^{2}{x}^{4}}{60}}+{\frac{7\,{c}^{2}{x}^{2}}{90}}+{\frac{4\,{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{45\,{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a^2*c*x^2+c)^2*arctan(a*x)^2,x)

[Out]

1/6*a^4*c^2*arctan(a*x)^2*x^6+1/2*a^2*c^2*arctan(a*x)^2*x^4+1/2*c^2*arctan(a*x)^2*x^2-1/15*a^3*c^2*arctan(a*x)
*x^5-2/9*a*c^2*arctan(a*x)*x^3-1/3*c^2*x*arctan(a*x)/a+1/6/a^2*c^2*arctan(a*x)^2+1/60*a^2*c^2*x^4+7/90*c^2*x^2
+4/45*c^2*ln(a^2*x^2+1)/a^2

________________________________________________________________________________________

Maxima [A]  time = 1.00444, size = 150, normalized size = 0.98 \begin{align*} \frac{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{6 \, a^{2} c} + \frac{{\left (3 \, a^{2} c^{3} x^{4} + 14 \, c^{3} x^{2} + \frac{16 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a - 4 \,{\left (3 \, a^{4} c^{3} x^{5} + 10 \, a^{2} c^{3} x^{3} + 15 \, c^{3} x\right )} \arctan \left (a x\right )}{180 \, a c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="maxima")

[Out]

1/6*(a^2*c*x^2 + c)^3*arctan(a*x)^2/(a^2*c) + 1/180*((3*a^2*c^3*x^4 + 14*c^3*x^2 + 16*c^3*log(a^2*x^2 + 1)/a^2
)*a - 4*(3*a^4*c^3*x^5 + 10*a^2*c^3*x^3 + 15*c^3*x)*arctan(a*x))/(a*c)

________________________________________________________________________________________

Fricas [A]  time = 2.21317, size = 274, normalized size = 1.79 \begin{align*} \frac{3 \, a^{4} c^{2} x^{4} + 14 \, a^{2} c^{2} x^{2} + 30 \,{\left (a^{6} c^{2} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2} + 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \,{\left (3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \arctan \left (a x\right )}{180 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="fricas")

[Out]

1/180*(3*a^4*c^2*x^4 + 14*a^2*c^2*x^2 + 30*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*arctan(a*x)^2 +
16*c^2*log(a^2*x^2 + 1) - 4*(3*a^5*c^2*x^5 + 10*a^3*c^2*x^3 + 15*a*c^2*x)*arctan(a*x))/a^2

________________________________________________________________________________________

Sympy [A]  time = 2.78782, size = 158, normalized size = 1.03 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{6} \operatorname{atan}^{2}{\left (a x \right )}}{6} - \frac{a^{3} c^{2} x^{5} \operatorname{atan}{\left (a x \right )}}{15} + \frac{a^{2} c^{2} x^{4} \operatorname{atan}^{2}{\left (a x \right )}}{2} + \frac{a^{2} c^{2} x^{4}}{60} - \frac{2 a c^{2} x^{3} \operatorname{atan}{\left (a x \right )}}{9} + \frac{c^{2} x^{2} \operatorname{atan}^{2}{\left (a x \right )}}{2} + \frac{7 c^{2} x^{2}}{90} - \frac{c^{2} x \operatorname{atan}{\left (a x \right )}}{3 a} + \frac{4 c^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{45 a^{2}} + \frac{c^{2} \operatorname{atan}^{2}{\left (a x \right )}}{6 a^{2}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a**2*c*x**2+c)**2*atan(a*x)**2,x)

[Out]

Piecewise((a**4*c**2*x**6*atan(a*x)**2/6 - a**3*c**2*x**5*atan(a*x)/15 + a**2*c**2*x**4*atan(a*x)**2/2 + a**2*
c**2*x**4/60 - 2*a*c**2*x**3*atan(a*x)/9 + c**2*x**2*atan(a*x)**2/2 + 7*c**2*x**2/90 - c**2*x*atan(a*x)/(3*a)
+ 4*c**2*log(x**2 + a**(-2))/(45*a**2) + c**2*atan(a*x)**2/(6*a**2), Ne(a, 0)), (0, True))

________________________________________________________________________________________

Giac [A]  time = 1.16454, size = 216, normalized size = 1.41 \begin{align*} \frac{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{6 \, a^{2} c} - \frac{3 \,{\left (4 \, x^{5} \arctan \left (a x\right ) - a{\left (\frac{a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac{2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )}\right )} a^{4} c^{2} + 20 \,{\left (2 \, x^{3} \arctan \left (a x\right ) - a{\left (\frac{x^{2}}{a^{2}} - \frac{\log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )}\right )} a^{2} c^{2} + \frac{30 \,{\left (2 \, a x \arctan \left (a x\right ) - \log \left (a^{2} x^{2} + 1\right )\right )} c^{2}}{a}}{180 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="giac")

[Out]

1/6*(a^2*c*x^2 + c)^3*arctan(a*x)^2/(a^2*c) - 1/180*(3*(4*x^5*arctan(a*x) - a*((a^2*x^4 - 2*x^2)/a^4 + 2*log(a
^2*x^2 + 1)/a^6))*a^4*c^2 + 20*(2*x^3*arctan(a*x) - a*(x^2/a^2 - log(a^2*x^2 + 1)/a^4))*a^2*c^2 + 30*(2*a*x*ar
ctan(a*x) - log(a^2*x^2 + 1))*c^2/a)/a