3.2.66 \(\int x^2 (c+a^2 c x^2)^3 \text {ArcTan}(a x) \, dx\) [166]

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

[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 = 0.16, 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 = {5068, 4946, 272, 45} \begin {gather*} \frac {1}{9} a^6 c^3 x^9 \text {ArcTan}(a x)-\frac {1}{72} a^5 c^3 x^8+\frac {3}{7} a^4 c^3 x^7 \text {ArcTan}(a x)-\frac {10}{189} a^3 c^3 x^6+\frac {3}{5} a^2 c^3 x^5 \text {ArcTan}(a x)+\frac {8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}+\frac {1}{3} c^3 x^3 \text {ArcTan}(a x)-\frac {89 a c^3 x^4}{1260}-\frac {8 c^3 x^2}{315 a} \end {gather*}

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 45

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 272

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 4946

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

Rule 5068

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 ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{10} \left (3 a^3 c^3\right ) \text {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 ) \text {Subst}\left (\int \frac {x^3}{1+a^2 x} \, dx,x,x^2\right )-\frac {1}{18} \left (a^7 c^3\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.03, size = 95, normalized size = 0.70 \begin {gather*} \frac {c^3 \left (-a^2 x^2 \left (192+534 a^2 x^2+400 a^4 x^4+105 a^6 x^6\right )+24 a^3 x^3 \left (105+189 a^2 x^2+135 a^4 x^4+35 a^6 x^6\right ) \text {ArcTan}(a x)+192 \log \left (1+a^2 x^2\right )\right )}{7560 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(-(a^2*x^2*(192 + 534*a^2*x^2 + 400*a^4*x^4 + 105*a^6*x^6)) + 24*a^3*x^3*(105 + 189*a^2*x^2 + 135*a^4*x^4
+ 35*a^6*x^6)*ArcTan[a*x] + 192*Log[1 + a^2*x^2]))/(7560*a^3)

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Maple [A]
time = 0.14, size = 116, normalized size = 0.85

method	result	size
derivativedivides	\(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{9} x^{9}}{9}+\frac {3 c^{3} \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )}{5}+\frac {a^{3} c^{3} x^{3} \arctan \left (a x \right )}{3}-\frac {c^{3} \left (\frac {35 a^{8} x^{8}}{8}+\frac {50 a^{6} x^{6}}{3}+\frac {89 a^{4} x^{4}}{4}+8 a^{2} x^{2}-8 \ln \left (a^{2} x^{2}+1\right )\right )}{315}}{a^{3}}\)	\(116\)
default	\(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{9} x^{9}}{9}+\frac {3 c^{3} \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )}{5}+\frac {a^{3} c^{3} x^{3} \arctan \left (a x \right )}{3}-\frac {c^{3} \left (\frac {35 a^{8} x^{8}}{8}+\frac {50 a^{6} x^{6}}{3}+\frac {89 a^{4} x^{4}}{4}+8 a^{2} x^{2}-8 \ln \left (a^{2} x^{2}+1\right )\right )}{315}}{a^{3}}\)	\(116\)
risch	\(-\frac {i c^{3} x^{3} \left (35 a^{6} x^{6}+135 a^{4} x^{4}+189 a^{2} x^{2}+105\right ) \ln \left (i a x +1\right )}{630}+\frac {i c^{3} a^{6} x^{9} \ln \left (-i a x +1\right )}{18}-\frac {a^{5} c^{3} x^{8}}{72}+\frac {3 i c^{3} a^{4} x^{7} \ln \left (-i a x +1\right )}{14}-\frac {10 a^{3} c^{3} x^{6}}{189}+\frac {3 i c^{3} a^{2} x^{5} \ln \left (-i a x +1\right )}{10}-\frac {89 a \,c^{3} x^{4}}{1260}+\frac {i c^{3} x^{3} \ln \left (-i a x +1\right )}{6}-\frac {8 c^{3} x^{2}}{315 a}+\frac {8 c^{3} \ln \left (-a^{2} x^{2}-1\right )}{315 a^{3}}\)	\(183\)
meijerg	\(\frac {c^{3} \left (\frac {a^{2} x^{2} \left (-15 a^{6} x^{6}+20 a^{4} x^{4}-30 a^{2} x^{2}+60\right )}{270}+\frac {4 a^{10} x^{10} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{9 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{9}\right )}{4 a^{3}}+\frac {3 c^{3} \left (-\frac {a^{2} x^{2} \left (4 a^{4} x^{4}-6 a^{2} x^{2}+12\right )}{42}+\frac {4 a^{8} x^{8} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{7}\right )}{4 a^{3}}+\frac {3 c^{3} \left (\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{15}+\frac {4 a^{6} x^{6} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{5}\right )}{4 a^{3}}+\frac {c^{3} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 a^{4} x^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{4 a^{3}}\)	\(280\)

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,method=_RETURNVERBOSE)

[Out]

1/a^3*(1/9*c^3*arctan(a*x)*a^9*x^9+3/7*c^3*arctan(a*x)*a^7*x^7+3/5*a^5*c^3*x^5*arctan(a*x)+1/3*a^3*c^3*x^3*arc
tan(a*x)-1/315*c^3*(35/8*a^8*x^8+50/3*a^6*x^6+89/4*a^4*x^4+8*a^2*x^2-8*ln(a^2*x^2+1)))

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Maxima [A]
time = 0.29, size = 118, normalized size = 0.87 \begin {gather*} \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 {gather*}

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 = 2.59, size = 116, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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 = 0.59, size = 138, normalized size = 1.01 \begin {gather*} \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 {gather*}

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

[Out]

sage0*x

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Mupad [B]
time = 0.44, size = 108, normalized size = 0.79 \begin {gather*} \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} \end {gather*}

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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