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

Optimal. Leaf size=136 \[ -\frac{1}{72} a^5 c^3 x^8-\frac{10}{189} a^3 c^3 x^6+\frac{8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)-\frac{89 a c^3 x^4}{1260}-\frac{8 c^3 x^2}{315 a}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x) \]

[Out]

(-8*c^3*x^2)/(315*a) - (89*a*c^3*x^4)/1260 - (10*a^3*c^3*x^6)/189 - (a^5*c^3*x^8)/72 + (c^3*x^3*ArcTan[a*x])/3
+ (3*a^2*c^3*x^5*ArcTan[a*x])/5 + (3*a^4*c^3*x^7*ArcTan[a*x])/7 + (a^6*c^3*x^9*ArcTan[a*x])/9 + (8*c^3*Log[1
+ a^2*x^2])/(315*a^3)

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Rubi [A]  time = 0.233357, antiderivative size = 136, 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, 266, 43} \[ -\frac{1}{72} a^5 c^3 x^8-\frac{10}{189} a^3 c^3 x^6+\frac{8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)-\frac{89 a c^3 x^4}{1260}-\frac{8 c^3 x^2}{315 a}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*c^3*x^2)/(315*a) - (89*a*c^3*x^4)/1260 - (10*a^3*c^3*x^6)/189 - (a^5*c^3*x^8)/72 + (c^3*x^3*ArcTan[a*x])/3
+ (3*a^2*c^3*x^5*ArcTan[a*x])/5 + (3*a^4*c^3*x^7*ArcTan[a*x])/7 + (a^6*c^3*x^9*ArcTan[a*x])/9 + (8*c^3*Log[1
+ a^2*x^2])/(315*a^3)

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x) \, dx &=\int \left (c^3 x^2 \tan ^{-1}(a x)+3 a^2 c^3 x^4 \tan ^{-1}(a x)+3 a^4 c^3 x^6 \tan ^{-1}(a x)+a^6 c^3 x^8 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int x^2 \tan ^{-1}(a x) \, dx+\left (3 a^2 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\left (3 a^4 c^3\right ) \int x^6 \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^8 \tan ^{-1}(a x) \, dx\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac{1}{3} \left (a c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^3 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx-\frac{1}{7} \left (3 a^5 c^3\right ) \int \frac{x^7}{1+a^2 x^2} \, dx-\frac{1}{9} \left (a^7 c^3\right ) \int \frac{x^9}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac{1}{6} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{10} \left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{14} \left (3 a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{18} \left (a^7 c^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac{1}{6} \left (a c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{10} \left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{14} \left (3 a^5 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^6}-\frac{x}{a^4}+\frac{x^2}{a^2}-\frac{1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{18} \left (a^7 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^8}+\frac{x}{a^6}-\frac{x^2}{a^4}+\frac{x^3}{a^2}+\frac{1}{a^8 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{8 c^3 x^2}{315 a}-\frac{89 a c^3 x^4}{1260}-\frac{10}{189} a^3 c^3 x^6-\frac{1}{72} a^5 c^3 x^8+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{8 c^3 \log \left (1+a^2 x^2\right )}{315 a^3}\\ \end{align*}

Mathematica [A]  time = 0.084463, size = 136, normalized size = 1. \[ -\frac{1}{72} a^5 c^3 x^8-\frac{10}{189} a^3 c^3 x^6+\frac{8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)-\frac{89 a c^3 x^4}{1260}-\frac{8 c^3 x^2}{315 a}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*c^3*x^2)/(315*a) - (89*a*c^3*x^4)/1260 - (10*a^3*c^3*x^6)/189 - (a^5*c^3*x^8)/72 + (c^3*x^3*ArcTan[a*x])/3
+ (3*a^2*c^3*x^5*ArcTan[a*x])/5 + (3*a^4*c^3*x^7*ArcTan[a*x])/7 + (a^6*c^3*x^9*ArcTan[a*x])/9 + (8*c^3*Log[1
+ a^2*x^2])/(315*a^3)

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Maple [A]  time = 0.026, size = 119, normalized size = 0.9 \begin{align*} -{\frac{8\,{c}^{3}{x}^{2}}{315\,a}}-{\frac{89\,a{c}^{3}{x}^{4}}{1260}}-{\frac{10\,{a}^{3}{c}^{3}{x}^{6}}{189}}-{\frac{{a}^{5}{c}^{3}{x}^{8}}{72}}+{\frac{{c}^{3}{x}^{3}\arctan \left ( ax \right ) }{3}}+{\frac{3\,{a}^{2}{c}^{3}{x}^{5}\arctan \left ( ax \right ) }{5}}+{\frac{3\,{a}^{4}{c}^{3}{x}^{7}\arctan \left ( ax \right ) }{7}}+{\frac{{a}^{6}{c}^{3}{x}^{9}\arctan \left ( ax \right ) }{9}}+{\frac{8\,{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{315\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/315*c^3*x^2/a-89/1260*a*c^3*x^4-10/189*a^3*c^3*x^6-1/72*a^5*c^3*x^8+1/3*c^3*x^3*arctan(a*x)+3/5*a^2*c^3*x^5
*arctan(a*x)+3/7*a^4*c^3*x^7*arctan(a*x)+1/9*a^6*c^3*x^9*arctan(a*x)+8/315*c^3*ln(a^2*x^2+1)/a^3

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Maxima [A]  time = 0.979308, size = 159, normalized size = 1.17 \begin{align*} \frac{1}{7560} \, a{\left (\frac{192 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{4}} - \frac{105 \, a^{6} c^{3} x^{8} + 400 \, a^{4} c^{3} x^{6} + 534 \, a^{2} c^{3} x^{4} + 192 \, c^{3} x^{2}}{a^{2}}\right )} + \frac{1}{315} \,{\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7560*a*(192*c^3*log(a^2*x^2 + 1)/a^4 - (105*a^6*c^3*x^8 + 400*a^4*c^3*x^6 + 534*a^2*c^3*x^4 + 192*c^3*x^2)/a
^2) + 1/315*(35*a^6*c^3*x^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)*arctan(a*x)

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Fricas [A]  time = 1.72335, size = 270, normalized size = 1.99 \begin{align*} -\frac{105 \, a^{8} c^{3} x^{8} + 400 \, a^{6} c^{3} x^{6} + 534 \, a^{4} c^{3} x^{4} + 192 \, a^{2} c^{3} x^{2} - 192 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 24 \,{\left (35 \, a^{9} c^{3} x^{9} + 135 \, a^{7} c^{3} x^{7} + 189 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3}\right )} \arctan \left (a x\right )}{7560 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7560*(105*a^8*c^3*x^8 + 400*a^6*c^3*x^6 + 534*a^4*c^3*x^4 + 192*a^2*c^3*x^2 - 192*c^3*log(a^2*x^2 + 1) - 24
*(35*a^9*c^3*x^9 + 135*a^7*c^3*x^7 + 189*a^5*c^3*x^5 + 105*a^3*c^3*x^3)*arctan(a*x))/a^3

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Sympy [A]  time = 4.32547, size = 138, normalized size = 1.01 \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{9} \operatorname{atan}{\left (a x \right )}}{9} - \frac{a^{5} c^{3} x^{8}}{72} + \frac{3 a^{4} c^{3} x^{7} \operatorname{atan}{\left (a x \right )}}{7} - \frac{10 a^{3} c^{3} x^{6}}{189} + \frac{3 a^{2} c^{3} x^{5} \operatorname{atan}{\left (a x \right )}}{5} - \frac{89 a c^{3} x^{4}}{1260} + \frac{c^{3} x^{3} \operatorname{atan}{\left (a x \right )}}{3} - \frac{8 c^{3} x^{2}}{315 a} + \frac{8 c^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{315 a^{3}} & \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**2*(a**2*c*x**2+c)**3*atan(a*x),x)

[Out]

Piecewise((a**6*c**3*x**9*atan(a*x)/9 - a**5*c**3*x**8/72 + 3*a**4*c**3*x**7*atan(a*x)/7 - 10*a**3*c**3*x**6/1
89 + 3*a**2*c**3*x**5*atan(a*x)/5 - 89*a*c**3*x**4/1260 + c**3*x**3*atan(a*x)/3 - 8*c**3*x**2/(315*a) + 8*c**3
*log(x**2 + a**(-2))/(315*a**3), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.17765, size = 158, normalized size = 1.16 \begin{align*} \frac{1}{315} \,{\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right ) + \frac{8 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{315 \, a^{3}} - \frac{105 \, a^{13} c^{3} x^{8} + 400 \, a^{11} c^{3} x^{6} + 534 \, a^{9} c^{3} x^{4} + 192 \, a^{7} c^{3} x^{2}}{7560 \, a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(35*a^6*c^3*x^9 + 135*a^4*c^3*x^7 + 189*a^2*c^3*x^5 + 105*c^3*x^3)*arctan(a*x) + 8/315*c^3*log(a^2*x^2 +
1)/a^3 - 1/7560*(105*a^13*c^3*x^8 + 400*a^11*c^3*x^6 + 534*a^9*c^3*x^4 + 192*a^7*c^3*x^2)/a^8