3.165 \(\int x^3 (c+a^2 c x^2)^3 \tan ^{-1}(a x) \, dx\)

Optimal. Leaf size=141 \[ -\frac{1}{90} a^5 c^3 x^9-\frac{11}{280} a^3 c^3 x^7+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{c^3 x}{40 a^3}-\frac{c^3 \tan ^{-1}(a x)}{40 a^4}-\frac{9}{200} a c^3 x^5-\frac{c^3 x^3}{120 a}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x) \]

[Out]

(c^3*x)/(40*a^3) - (c^3*x^3)/(120*a) - (9*a*c^3*x^5)/200 - (11*a^3*c^3*x^7)/280 - (a^5*c^3*x^9)/90 - (c^3*ArcT
an[a*x])/(40*a^4) + (c^3*x^4*ArcTan[a*x])/4 + (a^2*c^3*x^6*ArcTan[a*x])/2 + (3*a^4*c^3*x^8*ArcTan[a*x])/8 + (a
^6*c^3*x^10*ArcTan[a*x])/10

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Rubi [A]  time = 0.207256, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4948, 4852, 302, 203} \[ -\frac{1}{90} a^5 c^3 x^9-\frac{11}{280} a^3 c^3 x^7+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{c^3 x}{40 a^3}-\frac{c^3 \tan ^{-1}(a x)}{40 a^4}-\frac{9}{200} a c^3 x^5-\frac{c^3 x^3}{120 a}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*(c + a^2*c*x^2)^3*ArcTan[a*x],x]

[Out]

(c^3*x)/(40*a^3) - (c^3*x^3)/(120*a) - (9*a*c^3*x^5)/200 - (11*a^3*c^3*x^7)/280 - (a^5*c^3*x^9)/90 - (c^3*ArcT
an[a*x])/(40*a^4) + (c^3*x^4*ArcTan[a*x])/4 + (a^2*c^3*x^6*ArcTan[a*x])/2 + (3*a^4*c^3*x^8*ArcTan[a*x])/8 + (a
^6*c^3*x^10*ArcTan[a*x])/10

Rule 4948

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e,
c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x) \, dx &=\int \left (c^3 x^3 \tan ^{-1}(a x)+3 a^2 c^3 x^5 \tan ^{-1}(a x)+3 a^4 c^3 x^7 \tan ^{-1}(a x)+a^6 c^3 x^9 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int x^3 \tan ^{-1}(a x) \, dx+\left (3 a^2 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx+\left (3 a^4 c^3\right ) \int x^7 \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^9 \tan ^{-1}(a x) \, dx\\ &=\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)-\frac{1}{4} \left (a c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{2} \left (a^3 c^3\right ) \int \frac{x^6}{1+a^2 x^2} \, dx-\frac{1}{8} \left (3 a^5 c^3\right ) \int \frac{x^8}{1+a^2 x^2} \, dx-\frac{1}{10} \left (a^7 c^3\right ) \int \frac{x^{10}}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)-\frac{1}{4} \left (a c^3\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac{1}{2} \left (a^3 c^3\right ) \int \left (\frac{1}{a^6}-\frac{x^2}{a^4}+\frac{x^4}{a^2}-\frac{1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx-\frac{1}{8} \left (3 a^5 c^3\right ) \int \left (-\frac{1}{a^8}+\frac{x^2}{a^6}-\frac{x^4}{a^4}+\frac{x^6}{a^2}+\frac{1}{a^8 \left (1+a^2 x^2\right )}\right ) \, dx-\frac{1}{10} \left (a^7 c^3\right ) \int \left (\frac{1}{a^{10}}-\frac{x^2}{a^8}+\frac{x^4}{a^6}-\frac{x^6}{a^4}+\frac{x^8}{a^2}-\frac{1}{a^{10} \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{c^3 x}{40 a^3}-\frac{c^3 x^3}{120 a}-\frac{9}{200} a c^3 x^5-\frac{11}{280} a^3 c^3 x^7-\frac{1}{90} a^5 c^3 x^9+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)+\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{10 a^3}-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{4 a^3}-\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{8 a^3}+\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{2 a^3}\\ &=\frac{c^3 x}{40 a^3}-\frac{c^3 x^3}{120 a}-\frac{9}{200} a c^3 x^5-\frac{11}{280} a^3 c^3 x^7-\frac{1}{90} a^5 c^3 x^9-\frac{c^3 \tan ^{-1}(a x)}{40 a^4}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.147731, size = 141, normalized size = 1. \[ -\frac{1}{90} a^5 c^3 x^9-\frac{11}{280} a^3 c^3 x^7+\frac{1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)+\frac{3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac{1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac{c^3 x}{40 a^3}-\frac{c^3 \tan ^{-1}(a x)}{40 a^4}-\frac{9}{200} a c^3 x^5-\frac{c^3 x^3}{120 a}+\frac{1}{4} c^3 x^4 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(c + a^2*c*x^2)^3*ArcTan[a*x],x]

[Out]

(c^3*x)/(40*a^3) - (c^3*x^3)/(120*a) - (9*a*c^3*x^5)/200 - (11*a^3*c^3*x^7)/280 - (a^5*c^3*x^9)/90 - (c^3*ArcT
an[a*x])/(40*a^4) + (c^3*x^4*ArcTan[a*x])/4 + (a^2*c^3*x^6*ArcTan[a*x])/2 + (3*a^4*c^3*x^8*ArcTan[a*x])/8 + (a
^6*c^3*x^10*ArcTan[a*x])/10

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Maple [A]  time = 0.023, size = 122, normalized size = 0.9 \begin{align*}{\frac{{c}^{3}x}{40\,{a}^{3}}}-{\frac{{c}^{3}{x}^{3}}{120\,a}}-{\frac{9\,a{c}^{3}{x}^{5}}{200}}-{\frac{11\,{a}^{3}{c}^{3}{x}^{7}}{280}}-{\frac{{a}^{5}{c}^{3}{x}^{9}}{90}}-{\frac{{c}^{3}\arctan \left ( ax \right ) }{40\,{a}^{4}}}+{\frac{{c}^{3}{x}^{4}\arctan \left ( ax \right ) }{4}}+{\frac{{a}^{2}{c}^{3}{x}^{6}\arctan \left ( ax \right ) }{2}}+{\frac{3\,{a}^{4}{c}^{3}{x}^{8}\arctan \left ( ax \right ) }{8}}+{\frac{{a}^{6}{c}^{3}{x}^{10}\arctan \left ( ax \right ) }{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a^2*c*x^2+c)^3*arctan(a*x),x)

[Out]

1/40*c^3*x/a^3-1/120*c^3*x^3/a-9/200*a*c^3*x^5-11/280*a^3*c^3*x^7-1/90*a^5*c^3*x^9-1/40*c^3*arctan(a*x)/a^4+1/
4*c^3*x^4*arctan(a*x)+1/2*a^2*c^3*x^6*arctan(a*x)+3/8*a^4*c^3*x^8*arctan(a*x)+1/10*a^6*c^3*x^10*arctan(a*x)

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Maxima [A]  time = 1.47741, size = 162, normalized size = 1.15 \begin{align*} -\frac{1}{12600} \, a{\left (\frac{315 \, c^{3} \arctan \left (a x\right )}{a^{5}} + \frac{140 \, a^{8} c^{3} x^{9} + 495 \, a^{6} c^{3} x^{7} + 567 \, a^{4} c^{3} x^{5} + 105 \, a^{2} c^{3} x^{3} - 315 \, c^{3} x}{a^{4}}\right )} + \frac{1}{40} \,{\left (4 \, a^{6} c^{3} x^{10} + 15 \, a^{4} c^{3} x^{8} + 20 \, a^{2} c^{3} x^{6} + 10 \, c^{3} x^{4}\right )} \arctan \left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="maxima")

[Out]

-1/12600*a*(315*c^3*arctan(a*x)/a^5 + (140*a^8*c^3*x^9 + 495*a^6*c^3*x^7 + 567*a^4*c^3*x^5 + 105*a^2*c^3*x^3 -
315*c^3*x)/a^4) + 1/40*(4*a^6*c^3*x^10 + 15*a^4*c^3*x^8 + 20*a^2*c^3*x^6 + 10*c^3*x^4)*arctan(a*x)

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Fricas [A]  time = 1.6332, size = 261, normalized size = 1.85 \begin{align*} -\frac{140 \, a^{9} c^{3} x^{9} + 495 \, a^{7} c^{3} x^{7} + 567 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3} - 315 \, a c^{3} x - 315 \,{\left (4 \, a^{10} c^{3} x^{10} + 15 \, a^{8} c^{3} x^{8} + 20 \, a^{6} c^{3} x^{6} + 10 \, a^{4} c^{3} x^{4} - c^{3}\right )} \arctan \left (a x\right )}{12600 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="fricas")

[Out]

-1/12600*(140*a^9*c^3*x^9 + 495*a^7*c^3*x^7 + 567*a^5*c^3*x^5 + 105*a^3*c^3*x^3 - 315*a*c^3*x - 315*(4*a^10*c^
3*x^10 + 15*a^8*c^3*x^8 + 20*a^6*c^3*x^6 + 10*a^4*c^3*x^4 - c^3)*arctan(a*x))/a^4

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Sympy [A]  time = 5.6534, size = 138, normalized size = 0.98 \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{10} \operatorname{atan}{\left (a x \right )}}{10} - \frac{a^{5} c^{3} x^{9}}{90} + \frac{3 a^{4} c^{3} x^{8} \operatorname{atan}{\left (a x \right )}}{8} - \frac{11 a^{3} c^{3} x^{7}}{280} + \frac{a^{2} c^{3} x^{6} \operatorname{atan}{\left (a x \right )}}{2} - \frac{9 a c^{3} x^{5}}{200} + \frac{c^{3} x^{4} \operatorname{atan}{\left (a x \right )}}{4} - \frac{c^{3} x^{3}}{120 a} + \frac{c^{3} x}{40 a^{3}} - \frac{c^{3} \operatorname{atan}{\left (a x \right )}}{40 a^{4}} & \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**3*(a**2*c*x**2+c)**3*atan(a*x),x)

[Out]

Piecewise((a**6*c**3*x**10*atan(a*x)/10 - a**5*c**3*x**9/90 + 3*a**4*c**3*x**8*atan(a*x)/8 - 11*a**3*c**3*x**7
/280 + a**2*c**3*x**6*atan(a*x)/2 - 9*a*c**3*x**5/200 + c**3*x**4*atan(a*x)/4 - c**3*x**3/(120*a) + c**3*x/(40
*a**3) - c**3*atan(a*x)/(40*a**4), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.13402, size = 162, normalized size = 1.15 \begin{align*} \frac{1}{40} \,{\left (4 \, a^{6} c^{3} x^{10} + 15 \, a^{4} c^{3} x^{8} + 20 \, a^{2} c^{3} x^{6} + 10 \, c^{3} x^{4}\right )} \arctan \left (a x\right ) - \frac{c^{3} \arctan \left (a x\right )}{40 \, a^{4}} - \frac{140 \, a^{23} c^{3} x^{9} + 495 \, a^{21} c^{3} x^{7} + 567 \, a^{19} c^{3} x^{5} + 105 \, a^{17} c^{3} x^{3} - 315 \, a^{15} c^{3} x}{12600 \, a^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a^2*c*x^2+c)^3*arctan(a*x),x, algorithm="giac")

[Out]

1/40*(4*a^6*c^3*x^10 + 15*a^4*c^3*x^8 + 20*a^2*c^3*x^6 + 10*c^3*x^4)*arctan(a*x) - 1/40*c^3*arctan(a*x)/a^4 -
1/12600*(140*a^23*c^3*x^9 + 495*a^21*c^3*x^7 + 567*a^19*c^3*x^5 + 105*a^17*c^3*x^3 - 315*a^15*c^3*x)/a^18