\(\int x (c+a^2 c x^2) \arctan (a x)^2 \, dx\) [260]

Optimal result

Integrand size = 18, antiderivative size = 96 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c \left (1+a^2 x^2\right )}{12 a^2}-\frac {c x \arctan (a x)}{3 a}-\frac {c x \left (1+a^2 x^2\right ) \arctan (a x)}{6 a}+\frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{4 a^2}+\frac {c \log \left (1+a^2 x^2\right )}{6 a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5050, 4998, 4930, 266} \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c \left (a^2 x^2+1\right )^2 \arctan (a x)^2}{4 a^2}-\frac {c x \left (a^2 x^2+1\right ) \arctan (a x)}{6 a}+\frac {c \left (a^2 x^2+1\right )}{12 a^2}+\frac {c \log \left (a^2 x^2+1\right )}{6 a^2}-\frac {c x \arctan (a x)}{3 a} \]

[In]

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

[Out]

(c*(1 + a^2*x^2))/(12*a^2) - (c*x*ArcTan[a*x])/(3*a) - (c*x*(1 + a^2*x^2)*ArcTan[a*x])/(6*a) + (c*(1 + a^2*x^2
)^2*ArcTan[a*x]^2)/(4*a^2) + (c*Log[1 + a^2*x^2])/(6*a^2)

Rule 266

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]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[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])

Rule 4998

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 5050

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]

Rubi steps \begin{align*} \text {integral}& = \frac {c \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{4 a^2}-\frac {\int \left (c+a^2 c x^2\right ) \arctan (a x) \, dx}{2 a} \\ & = \frac {c \left (1+a^2 x^2\right )}{12 a^2}-\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 \int \arctan (a x) \, dx}{3 a} \\ & = \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 {1}{3} c \int \frac {x}{1+a^2 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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c \left (a^2 x^2-2 a x \left (3+a^2 x^2\right ) \arctan (a x)+3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2+2 \log \left (1+a^2 x^2\right )\right )}{12 a^2} \]

[In]

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

[Out]

(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
)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91

method	result	size
parts	\(\frac {a^{2} c \,x^{4} \arctan \left (a x \right )^{2}}{4}+\frac {c \,x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c \arctan \left (a x \right )^{2}}{4 a^{2}}-\frac {c \left (\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}+x \arctan \left (a x \right ) a -\frac {a^{2} x^{2}}{6}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}\right )}{2 a^{2}}\)	\(87\)
derivativedivides	\(\frac {\frac {c \arctan \left (a x \right )^{2} a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c \arctan \left (a x \right )^{2}}{4}-\frac {c \left (\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}+x \arctan \left (a x \right ) a -\frac {a^{2} x^{2}}{6}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}\right )}{2}}{a^{2}}\)	\(88\)
default	\(\frac {\frac {c \arctan \left (a x \right )^{2} a^{4} x^{4}}{4}+\frac {a^{2} c \,x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c \arctan \left (a x \right )^{2}}{4}-\frac {c \left (\frac {\arctan \left (a x \right ) x^{3} a^{3}}{3}+x \arctan \left (a x \right ) a -\frac {a^{2} x^{2}}{6}-\frac {\ln \left (a^{2} x^{2}+1\right )}{3}\right )}{2}}{a^{2}}\)	\(88\)
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	\(-\frac {c \left (a^{2} x^{2}+1\right )^{2} \ln \left (i a x +1\right )^{2}}{16 a^{2}}+\frac {c \left (3 x^{4} \ln \left (-i a x +1\right ) a^{4}+2 i a^{3} x^{3}+6 a^{2} x^{2} \ln \left (-i a x +1\right )+6 i a x +3 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{24 a^{2}}-\frac {a^{2} c \,x^{4} \ln \left (-i a x +1\right )^{2}}{16}-\frac {i a c \,x^{3} \ln \left (-i a x +1\right )}{12}-\frac {c \,x^{2} \ln \left (-i a x +1\right )^{2}}{8}-\frac {i c x \ln \left (-i a x +1\right )}{4 a}+\frac {c \,x^{2}}{12}-\frac {c \ln \left (-i a x +1\right )^{2}}{16 a^{2}}+\frac {c \ln \left (-a^{2} x^{2}-1\right )}{6 a^{2}}\)	\(206\)

[In]

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

[Out]

1/4*a^2*c*x^4*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
+x*arctan(a*x)*a-1/6*a^2*x^2-1/3*ln(a^2*x^2+1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {a^{2} c x^{2} + 3 \, {\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2} - 2 \, {\left (a^{3} c x^{3} + 3 \, a c x\right )} \arctan \left (a x\right ) + 2 \, c \log \left (a^{2} x^{2} + 1\right )}{12 \, a^{2}} \]

[In]

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

[Out]

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*lo
g(a^2*x^2 + 1))/a^2

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.98 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\begin {cases} \frac {a^{2} c x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{4} - \frac {a c x^{3} \operatorname {atan}{\left (a x \right )}}{6} + \frac {c x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {c x^{2}}{12} - \frac {c x \operatorname {atan}{\left (a x \right )}}{2 a} + \frac {c \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{6 a^{2}} + \frac {c \operatorname {atan}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {{\left (a^{2} c x^{2} + c\right )}^{2} \arctan \left (a x\right )^{2}}{4 \, a^{2} c} + \frac {{\left (c^{2} x^{2} + \frac {2 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a - 2 \, {\left (a^{2} c^{2} x^{3} + 3 \, c^{2} x\right )} \arctan \left (a x\right )}{12 \, a c} \]

[In]

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

[Out]

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)

Giac [F]

\[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )} x \arctan \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int x \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx=\frac {c\,\left (6\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+x^2\right )}{12}+\frac {\frac {c\,\left (3\,{\mathrm {atan}\left (a\,x\right )}^2+2\,\ln \left (a^2\,x^2+1\right )\right )}{12}-\frac {a\,c\,x\,\mathrm {atan}\left (a\,x\right )}{2}}{a^2}+\frac {a^2\,c\,x^4\,{\mathrm {atan}\left (a\,x\right )}^2}{4}-\frac {a\,c\,x^3\,\mathrm {atan}\left (a\,x\right )}{6} \]

[In]

int(x*atan(a*x)^2*(c + a^2*c*x^2),x)

[Out]

(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*atan(a*x))/6