3.2.59 \(\int x (c+a^2 c x^2)^2 \text {ArcTan}(a x) \, dx\) [159]

Optimal. Leaf size=61 \[ -\frac {c^2 x}{6 a}-\frac {1}{9} a c^2 x^3-\frac {1}{30} a^3 c^2 x^5+\frac {c^2 \left (1+a^2 x^2\right )^3 \text {ArcTan}(a x)}{6 a^2} \]

[Out]

-1/6*c^2*x/a-1/9*a*c^2*x^3-1/30*a^3*c^2*x^5+1/6*c^2*(a^2*x^2+1)^3*arctan(a*x)/a^2

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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5050, 200} \begin {gather*} -\frac {1}{30} a^3 c^2 x^5+\frac {c^2 \left (a^2 x^2+1\right )^3 \text {ArcTan}(a x)}{6 a^2}-\frac {1}{9} a c^2 x^3-\frac {c^2 x}{6 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(c^2*x)/a - (a*c^2*x^3)/9 - (a^3*c^2*x^5)/30 + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x])/(6*a^2)

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 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*} \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x) \, dx &=\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac {\int \left (c+a^2 c x^2\right )^2 \, dx}{6 a}\\ &=\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{6 a^2}-\frac {\int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx}{6 a}\\ &=-\frac {c^2 x}{6 a}-\frac {1}{9} a c^2 x^3-\frac {1}{30} a^3 c^2 x^5+\frac {c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)}{6 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 50, normalized size = 0.82 \begin {gather*} \frac {c^2 \left (-a x \left (15+10 a^2 x^2+3 a^4 x^4\right )+15 \left (1+a^2 x^2\right )^3 \text {ArcTan}(a x)\right )}{90 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(-(a*x*(15 + 10*a^2*x^2 + 3*a^4*x^4)) + 15*(1 + a^2*x^2)^3*ArcTan[a*x]))/(90*a^2)

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Maple [A]
time = 0.10, size = 85, normalized size = 1.39

method	result	size
derivativedivides	\(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{6} x^{6}}{6}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )}{2}+\frac {a^{2} c^{2} x^{2} \arctan \left (a x \right )}{2}+\frac {c^{2} \arctan \left (a x \right )}{6}-\frac {c^{2} \left (\frac {1}{5} a^{5} x^{5}+\frac {2}{3} a^{3} x^{3}+a x \right )}{6}}{a^{2}}\)	\(85\)
default	\(\frac {\frac {c^{2} \arctan \left (a x \right ) a^{6} x^{6}}{6}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )}{2}+\frac {a^{2} c^{2} x^{2} \arctan \left (a x \right )}{2}+\frac {c^{2} \arctan \left (a x \right )}{6}-\frac {c^{2} \left (\frac {1}{5} a^{5} x^{5}+\frac {2}{3} a^{3} x^{3}+a x \right )}{6}}{a^{2}}\)	\(85\)
risch	\(-\frac {i c^{2} \left (a^{2} x^{2}+1\right )^{3} \ln \left (i a x +1\right )}{12 a^{2}}+\frac {i c^{2} a^{4} x^{6} \ln \left (-i a x +1\right )}{12}-\frac {a^{3} c^{2} x^{5}}{30}+\frac {i c^{2} a^{2} x^{4} \ln \left (-i a x +1\right )}{4}-\frac {a \,c^{2} x^{3}}{9}+\frac {i c^{2} x^{2} \ln \left (-i a x +1\right )}{4}-\frac {c^{2} x}{6 a}+\frac {i c^{2} \ln \left (a^{2} x^{2}+1\right )}{24 a^{2}}+\frac {c^{2} \arctan \left (a x \right )}{12 a^{2}}\)	\(147\)
meijerg	\(\frac {c^{2} \left (-\frac {2 x a \left (21 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{315}+\frac {2 x a \left (7 a^{6} x^{6}+7\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{4 a^{2}}+\frac {c^{2} \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{2 a^{2}}+\frac {c^{2} \left (-2 a x +\frac {2 \left (3 a^{2} x^{2}+3\right ) \arctan \left (a x \right )}{3}\right )}{4 a^{2}}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2*(1/6*c^2*arctan(a*x)*a^6*x^6+1/2*a^4*c^2*x^4*arctan(a*x)+1/2*a^2*c^2*x^2*arctan(a*x)+1/6*c^2*arctan(a*x)
-1/6*c^2*(1/5*a^5*x^5+2/3*a^3*x^3+a*x))

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Maxima [A]
time = 0.25, size = 62, normalized size = 1.02 \begin {gather*} \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )}{6 \, a^{2} c} - \frac {3 \, a^{4} c^{3} x^{5} + 10 \, a^{2} c^{3} x^{3} + 15 \, c^{3} x}{90 \, a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(a^2*c*x^2 + c)^3*arctan(a*x)/(a^2*c) - 1/90*(3*a^4*c^3*x^5 + 10*a^2*c^3*x^3 + 15*c^3*x)/(a*c)

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Fricas [A]
time = 2.44, size = 77, normalized size = 1.26 \begin {gather*} -\frac {3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x - 15 \, {\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 )}{90 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(3*a^5*c^2*x^5 + 10*a^3*c^2*x^3 + 15*a*c^2*x - 15*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 3*a^2*c^2*x^2 + c^2)*ar
ctan(a*x))/a^2

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Sympy [A]
time = 0.75, size = 92, normalized size = 1.51 \begin {gather*} \begin {cases} \frac {a^{4} c^{2} x^{6} \operatorname {atan}{\left (a x \right )}}{6} - \frac {a^{3} c^{2} x^{5}}{30} + \frac {a^{2} c^{2} x^{4} \operatorname {atan}{\left (a x \right )}}{2} - \frac {a c^{2} x^{3}}{9} + \frac {c^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{2} - \frac {c^{2} x}{6 a} + \frac {c^{2} \operatorname {atan}{\left (a x \right )}}{6 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*c**2*x**6*atan(a*x)/6 - a**3*c**2*x**5/30 + a**2*c**2*x**4*atan(a*x)/2 - a*c**2*x**3/9 + c**2*
x**2*atan(a*x)/2 - c**2*x/(6*a) + c**2*atan(a*x)/(6*a**2), Ne(a, 0)), (0, True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [B]
time = 0.55, size = 71, normalized size = 1.16 \begin {gather*} \frac {c^2\,\left (15\,\mathrm {atan}\left (a\,x\right )-15\,a\,x-10\,a^3\,x^3-3\,a^5\,x^5+45\,a^2\,x^2\,\mathrm {atan}\left (a\,x\right )+45\,a^4\,x^4\,\mathrm {atan}\left (a\,x\right )+15\,a^6\,x^6\,\mathrm {atan}\left (a\,x\right )\right )}{90\,a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*atan(a*x)*(c + a^2*c*x^2)^2,x)

[Out]

(c^2*(15*atan(a*x) - 15*a*x - 10*a^3*x^3 - 3*a^5*x^5 + 45*a^2*x^2*atan(a*x) + 45*a^4*x^4*atan(a*x) + 15*a^6*x^
6*atan(a*x)))/(90*a^2)

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