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

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

[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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Rubi [A]  time = 0.21, antiderivative size = 141, 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, 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 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])

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 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^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*}

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Mathematica [A]  time = 0.14, size = 141, normalized size = 1.00 \[ \frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)-\frac {1}{90} a^5 c^3 x^9+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac {c^3 \tan ^{-1}(a x)}{40 a^4}-\frac {11}{280} a^3 c^3 x^7+\frac {c^3 x}{40 a^3}+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)-\frac {9}{200} a c^3 x^5+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)-\frac {c^3 x^3}{120 a} \]

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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fricas [A]  time = 0.57, size = 113, normalized size = 0.80 \[ -\frac {140 \, a^{9} c^{3} x^{9} + 495 \, a^{7} c^{3} x^{7} + 567 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3} - 315 \, a c^{3} x - 315 \, {\left (4 \, a^{10} c^{3} x^{10} + 15 \, a^{8} c^{3} x^{8} + 20 \, a^{6} c^{3} x^{6} + 10 \, a^{4} c^{3} x^{4} - c^{3}\right )} \arctan \left (a x\right )}{12600 \, a^{4}} \]

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

[Out]

sage0*x

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maple [A]  time = 0.02, size = 122, normalized size = 0.87 \[ \frac {c^{3} x}{40 a^{3}}-\frac {c^{3} x^{3}}{120 a}-\frac {9 a \,c^{3} x^{5}}{200}-\frac {11 a^{3} c^{3} x^{7}}{280}-\frac {a^{5} c^{3} x^{9}}{90}-\frac {c^{3} \arctan \left (a x \right )}{40 a^{4}}+\frac {c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {a^{2} c^{3} x^{6} \arctan \left (a x \right )}{2}+\frac {3 a^{4} c^{3} x^{8} \arctan \left (a x \right )}{8}+\frac {a^{6} c^{3} x^{10} \arctan \left (a x \right )}{10} \]

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 = 0.42, size = 120, normalized size = 0.85 \[ -\frac {1}{12600} \, a {\left (\frac {315 \, c^{3} \arctan \left (a x\right )}{a^{5}} + \frac {140 \, a^{8} c^{3} x^{9} + 495 \, a^{6} c^{3} x^{7} + 567 \, a^{4} c^{3} x^{5} + 105 \, a^{2} c^{3} x^{3} - 315 \, c^{3} x}{a^{4}}\right )} + \frac {1}{40} \, {\left (4 \, a^{6} c^{3} x^{10} + 15 \, a^{4} c^{3} x^{8} + 20 \, a^{2} c^{3} x^{6} + 10 \, c^{3} x^{4}\right )} \arctan \left (a x\right ) \]

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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mupad [B]  time = 0.44, size = 111, normalized size = 0.79 \[ \mathrm {atan}\left (a\,x\right )\,\left (\frac {a^6\,c^3\,x^{10}}{10}+\frac {3\,a^4\,c^3\,x^8}{8}+\frac {a^2\,c^3\,x^6}{2}+\frac {c^3\,x^4}{4}\right )+\frac {c^3\,x}{40\,a^3}-\frac {9\,a\,c^3\,x^5}{200}-\frac {c^3\,\mathrm {atan}\left (a\,x\right )}{40\,a^4}-\frac {c^3\,x^3}{120\,a}-\frac {11\,a^3\,c^3\,x^7}{280}-\frac {a^5\,c^3\,x^9}{90} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.83, size = 138, normalized size = 0.98 \[ \begin {cases} \frac {a^{6} c^{3} x^{10} \operatorname {atan}{\left (a x \right )}}{10} - \frac {a^{5} c^{3} x^{9}}{90} + \frac {3 a^{4} c^{3} x^{8} \operatorname {atan}{\left (a x \right )}}{8} - \frac {11 a^{3} c^{3} x^{7}}{280} + \frac {a^{2} c^{3} x^{6} \operatorname {atan}{\left (a x \right )}}{2} - \frac {9 a c^{3} x^{5}}{200} + \frac {c^{3} x^{4} \operatorname {atan}{\left (a x \right )}}{4} - \frac {c^{3} x^{3}}{120 a} + \frac {c^{3} x}{40 a^{3}} - \frac {c^{3} \operatorname {atan}{\left (a x \right )}}{40 a^{4}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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