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

Optimal. Leaf size=111 \[ -\frac{1}{56} a^3 c^2 x^7+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{c^2 x}{24 a^3}-\frac{c^2 \tan ^{-1}(a x)}{24 a^4}-\frac{1}{24} a c^2 x^5-\frac{c^2 x^3}{72 a}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x) \]

[Out]

(c^2*x)/(24*a^3) - (c^2*x^3)/(72*a) - (a*c^2*x^5)/24 - (a^3*c^2*x^7)/56 - (c^2*ArcTan[a*x])/(24*a^4) + (c^2*x^
4*ArcTan[a*x])/4 + (a^2*c^2*x^6*ArcTan[a*x])/3 + (a^4*c^2*x^8*ArcTan[a*x])/8

________________________________________________________________________________________

Rubi [A]  time = 0.155648, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4948, 4852, 302, 203} \[ -\frac{1}{56} a^3 c^2 x^7+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{c^2 x}{24 a^3}-\frac{c^2 \tan ^{-1}(a x)}{24 a^4}-\frac{1}{24} a c^2 x^5-\frac{c^2 x^3}{72 a}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x)/(24*a^3) - (c^2*x^3)/(72*a) - (a*c^2*x^5)/24 - (a^3*c^2*x^7)/56 - (c^2*ArcTan[a*x])/(24*a^4) + (c^2*x^
4*ArcTan[a*x])/4 + (a^2*c^2*x^6*ArcTan[a*x])/3 + (a^4*c^2*x^8*ArcTan[a*x])/8

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 )^2 \tan ^{-1}(a x) \, dx &=\int \left (c^2 x^3 \tan ^{-1}(a x)+2 a^2 c^2 x^5 \tan ^{-1}(a x)+a^4 c^2 x^7 \tan ^{-1}(a x)\right ) \, dx\\ &=c^2 \int x^3 \tan ^{-1}(a x) \, dx+\left (2 a^2 c^2\right ) \int x^5 \tan ^{-1}(a x) \, dx+\left (a^4 c^2\right ) \int x^7 \tan ^{-1}(a x) \, dx\\ &=\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac{1}{4} \left (a c^2\right ) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{3} \left (a^3 c^2\right ) \int \frac{x^6}{1+a^2 x^2} \, dx-\frac{1}{8} \left (a^5 c^2\right ) \int \frac{x^8}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac{1}{4} \left (a c^2\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}{3} \left (a^3 c^2\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 (a^5 c^2\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{c^2 x}{24 a^3}-\frac{c^2 x^3}{72 a}-\frac{1}{24} a c^2 x^5-\frac{1}{56} a^3 c^2 x^7+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{8 a^3}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{4 a^3}+\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{3 a^3}\\ &=\frac{c^2 x}{24 a^3}-\frac{c^2 x^3}{72 a}-\frac{1}{24} a c^2 x^5-\frac{1}{56} a^3 c^2 x^7-\frac{c^2 \tan ^{-1}(a x)}{24 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0953528, size = 111, normalized size = 1. \[ -\frac{1}{56} a^3 c^2 x^7+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{c^2 x}{24 a^3}-\frac{c^2 \tan ^{-1}(a x)}{24 a^4}-\frac{1}{24} a c^2 x^5-\frac{c^2 x^3}{72 a}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x)/(24*a^3) - (c^2*x^3)/(72*a) - (a*c^2*x^5)/24 - (a^3*c^2*x^7)/56 - (c^2*ArcTan[a*x])/(24*a^4) + (c^2*x^
4*ArcTan[a*x])/4 + (a^2*c^2*x^6*ArcTan[a*x])/3 + (a^4*c^2*x^8*ArcTan[a*x])/8

________________________________________________________________________________________

Maple [A]  time = 0.025, size = 96, normalized size = 0.9 \begin{align*}{\frac{{c}^{2}x}{24\,{a}^{3}}}-{\frac{{c}^{2}{x}^{3}}{72\,a}}-{\frac{a{c}^{2}{x}^{5}}{24}}-{\frac{{a}^{3}{c}^{2}{x}^{7}}{56}}-{\frac{{c}^{2}\arctan \left ( ax \right ) }{24\,{a}^{4}}}+{\frac{{c}^{2}{x}^{4}\arctan \left ( ax \right ) }{4}}+{\frac{{a}^{2}{c}^{2}{x}^{6}\arctan \left ( ax \right ) }{3}}+{\frac{{a}^{4}{c}^{2}{x}^{8}\arctan \left ( ax \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*c^2*x/a^3-1/72*c^2*x^3/a-1/24*a*c^2*x^5-1/56*a^3*c^2*x^7-1/24*c^2*arctan(a*x)/a^4+1/4*c^2*x^4*arctan(a*x)
+1/3*a^2*c^2*x^6*arctan(a*x)+1/8*a^4*c^2*x^8*arctan(a*x)

________________________________________________________________________________________

Maxima [A]  time = 1.44119, size = 132, normalized size = 1.19 \begin{align*} -\frac{1}{504} \, a{\left (\frac{21 \, c^{2} \arctan \left (a x\right )}{a^{5}} + \frac{9 \, a^{6} c^{2} x^{7} + 21 \, a^{4} c^{2} x^{5} + 7 \, a^{2} c^{2} x^{3} - 21 \, c^{2} x}{a^{4}}\right )} + \frac{1}{24} \,{\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} 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)^2*arctan(a*x),x, algorithm="maxima")

[Out]

-1/504*a*(21*c^2*arctan(a*x)/a^5 + (9*a^6*c^2*x^7 + 21*a^4*c^2*x^5 + 7*a^2*c^2*x^3 - 21*c^2*x)/a^4) + 1/24*(3*
a^4*c^2*x^8 + 8*a^2*c^2*x^6 + 6*c^2*x^4)*arctan(a*x)

________________________________________________________________________________________

Fricas [A]  time = 1.56244, size = 196, normalized size = 1.77 \begin{align*} -\frac{9 \, a^{7} c^{2} x^{7} + 21 \, a^{5} c^{2} x^{5} + 7 \, a^{3} c^{2} x^{3} - 21 \, a c^{2} x - 21 \,{\left (3 \, a^{8} c^{2} x^{8} + 8 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - c^{2}\right )} \arctan \left (a x\right )}{504 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/504*(9*a^7*c^2*x^7 + 21*a^5*c^2*x^5 + 7*a^3*c^2*x^3 - 21*a*c^2*x - 21*(3*a^8*c^2*x^8 + 8*a^6*c^2*x^6 + 6*a^
4*c^2*x^4 - c^2)*arctan(a*x))/a^4

________________________________________________________________________________________

Sympy [A]  time = 3.45755, size = 104, normalized size = 0.94 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{8} \operatorname{atan}{\left (a x \right )}}{8} - \frac{a^{3} c^{2} x^{7}}{56} + \frac{a^{2} c^{2} x^{6} \operatorname{atan}{\left (a x \right )}}{3} - \frac{a c^{2} x^{5}}{24} + \frac{c^{2} x^{4} \operatorname{atan}{\left (a x \right )}}{4} - \frac{c^{2} x^{3}}{72 a} + \frac{c^{2} x}{24 a^{3}} - \frac{c^{2} \operatorname{atan}{\left (a x \right )}}{24 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)**2*atan(a*x),x)

[Out]

Piecewise((a**4*c**2*x**8*atan(a*x)/8 - a**3*c**2*x**7/56 + a**2*c**2*x**6*atan(a*x)/3 - a*c**2*x**5/24 + c**2
*x**4*atan(a*x)/4 - c**2*x**3/(72*a) + c**2*x/(24*a**3) - c**2*atan(a*x)/(24*a**4), Ne(a, 0)), (0, True))

________________________________________________________________________________________

Giac [A]  time = 1.14705, size = 132, normalized size = 1.19 \begin{align*} \frac{1}{24} \,{\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} x^{4}\right )} \arctan \left (a x\right ) - \frac{c^{2} \arctan \left (a x\right )}{24 \, a^{4}} - \frac{9 \, a^{17} c^{2} x^{7} + 21 \, a^{15} c^{2} x^{5} + 7 \, a^{13} c^{2} x^{3} - 21 \, a^{11} c^{2} x}{504 \, a^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*a^4*c^2*x^8 + 8*a^2*c^2*x^6 + 6*c^2*x^4)*arctan(a*x) - 1/24*c^2*arctan(a*x)/a^4 - 1/504*(9*a^17*c^2*x^
7 + 21*a^15*c^2*x^5 + 7*a^13*c^2*x^3 - 21*a^11*c^2*x)/a^14