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

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

[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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Rubi [A]  time = 0.23, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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])

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 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 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])

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*}

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Mathematica [A]  time = 0.09, size = 136, normalized size = 1.00 \[ \frac {1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac {1}{72} a^5 c^3 x^8+\frac {3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)-\frac {10}{189} a^3 c^3 x^6+\frac {3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac {8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}-\frac {89 a c^3 x^4}{1260}+\frac {1}{3} c^3 x^3 \tan ^{-1}(a x)-\frac {8 c^3 x^2}{315 a} \]

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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fricas [A]  time = 0.50, size = 116, normalized size = 0.85 \[ -\frac {105 \, a^{8} c^{3} x^{8} + 400 \, a^{6} c^{3} x^{6} + 534 \, a^{4} c^{3} x^{4} + 192 \, a^{2} c^{3} x^{2} - 192 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 24 \, {\left (35 \, a^{9} c^{3} x^{9} + 135 \, a^{7} c^{3} x^{7} + 189 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3}\right )} \arctan \left (a x\right )}{7560 \, a^{3}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 0.03, size = 119, normalized size = 0.88 \[ -\frac {8 c^{3} x^{2}}{315 a}-\frac {89 a \,c^{3} x^{4}}{1260}-\frac {10 a^{3} c^{3} x^{6}}{189}-\frac {a^{5} c^{3} x^{8}}{72}+\frac {c^{3} x^{3} \arctan \left (a x \right )}{3}+\frac {3 a^{2} c^{3} x^{5} \arctan \left (a x \right )}{5}+\frac {3 a^{4} c^{3} x^{7} \arctan \left (a x \right )}{7}+\frac {a^{6} c^{3} x^{9} \arctan \left (a x \right )}{9}+\frac {8 c^{3} \ln \left (a^{2} x^{2}+1\right )}{315 a^{3}} \]

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 = 0.32, size = 118, normalized size = 0.87 \[ \frac {1}{7560} \, a {\left (\frac {192 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{4}} - \frac {105 \, a^{6} c^{3} x^{8} + 400 \, a^{4} c^{3} x^{6} + 534 \, a^{2} c^{3} x^{4} + 192 \, c^{3} x^{2}}{a^{2}}\right )} + \frac {1}{315} \, {\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right ) \]

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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mupad [B]  time = 0.44, size = 108, normalized size = 0.79 \[ \mathrm {atan}\left (a\,x\right )\,\left (\frac {a^6\,c^3\,x^9}{9}+\frac {3\,a^4\,c^3\,x^7}{7}+\frac {3\,a^2\,c^3\,x^5}{5}+\frac {c^3\,x^3}{3}\right )-\frac {89\,a\,c^3\,x^4}{1260}+\frac {8\,c^3\,\ln \left (a^2\,x^2+1\right )}{315\,a^3}-\frac {8\,c^3\,x^2}{315\,a}-\frac {10\,a^3\,c^3\,x^6}{189}-\frac {a^5\,c^3\,x^8}{72} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.97, size = 138, normalized size = 1.01 \[ \begin {cases} \frac {a^{6} c^{3} x^{9} \operatorname {atan}{\left (a x \right )}}{9} - \frac {a^{5} c^{3} x^{8}}{72} + \frac {3 a^{4} c^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{7} - \frac {10 a^{3} c^{3} x^{6}}{189} + \frac {3 a^{2} c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {89 a c^{3} x^{4}}{1260} + \frac {c^{3} x^{3} \operatorname {atan}{\left (a x \right )}}{3} - \frac {8 c^{3} x^{2}}{315 a} + \frac {8 c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{315 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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