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

Optimal. Leaf size=106 \[ -\frac{1}{42} a^3 c^2 x^6+\frac{4 c^2 \log \left (a^2 x^2+1\right )}{105 a^3}+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{9}{140} a c^2 x^4-\frac{4 c^2 x^2}{105 a}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x) \]

[Out]

(-4*c^2*x^2)/(105*a) - (9*a*c^2*x^4)/140 - (a^3*c^2*x^6)/42 + (c^2*x^3*ArcTan[a*x])/3 + (2*a^2*c^2*x^5*ArcTan[
a*x])/5 + (a^4*c^2*x^7*ArcTan[a*x])/7 + (4*c^2*Log[1 + a^2*x^2])/(105*a^3)

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Rubi [A]  time = 0.171692, antiderivative size = 106, 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, 266, 43} \[ -\frac{1}{42} a^3 c^2 x^6+\frac{4 c^2 \log \left (a^2 x^2+1\right )}{105 a^3}+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{9}{140} a c^2 x^4-\frac{4 c^2 x^2}{105 a}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*c^2*x^2)/(105*a) - (9*a*c^2*x^4)/140 - (a^3*c^2*x^6)/42 + (c^2*x^3*ArcTan[a*x])/3 + (2*a^2*c^2*x^5*ArcTan[
a*x])/5 + (a^4*c^2*x^7*ArcTan[a*x])/7 + (4*c^2*Log[1 + a^2*x^2])/(105*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 )^2 \tan ^{-1}(a x) \, dx &=\int \left (c^2 x^2 \tan ^{-1}(a x)+2 a^2 c^2 x^4 \tan ^{-1}(a x)+a^4 c^2 x^6 \tan ^{-1}(a x)\right ) \, dx\\ &=c^2 \int x^2 \tan ^{-1}(a x) \, dx+\left (2 a^2 c^2\right ) \int x^4 \tan ^{-1}(a x) \, dx+\left (a^4 c^2\right ) \int x^6 \tan ^{-1}(a x) \, dx\\ &=\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)-\frac{1}{3} \left (a c^2\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{5} \left (2 a^3 c^2\right ) \int \frac{x^5}{1+a^2 x^2} \, dx-\frac{1}{7} \left (a^5 c^2\right ) \int \frac{x^7}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)-\frac{1}{6} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{5} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{14} \left (a^5 c^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)-\frac{1}{6} \left (a c^2\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}{5} \left (a^3 c^2\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 (a^5 c^2\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{4 c^2 x^2}{105 a}-\frac{9}{140} a c^2 x^4-\frac{1}{42} a^3 c^2 x^6+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)+\frac{4 c^2 \log \left (1+a^2 x^2\right )}{105 a^3}\\ \end{align*}

Mathematica [A]  time = 0.063396, size = 106, normalized size = 1. \[ -\frac{1}{42} a^3 c^2 x^6+\frac{4 c^2 \log \left (a^2 x^2+1\right )}{105 a^3}+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac{9}{140} a c^2 x^4-\frac{4 c^2 x^2}{105 a}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*c^2*x^2)/(105*a) - (9*a*c^2*x^4)/140 - (a^3*c^2*x^6)/42 + (c^2*x^3*ArcTan[a*x])/3 + (2*a^2*c^2*x^5*ArcTan[
a*x])/5 + (a^4*c^2*x^7*ArcTan[a*x])/7 + (4*c^2*Log[1 + a^2*x^2])/(105*a^3)

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Maple [A]  time = 0.024, size = 93, normalized size = 0.9 \begin{align*} -{\frac{4\,{c}^{2}{x}^{2}}{105\,a}}-{\frac{9\,a{c}^{2}{x}^{4}}{140}}-{\frac{{a}^{3}{c}^{2}{x}^{6}}{42}}+{\frac{{c}^{2}{x}^{3}\arctan \left ( ax \right ) }{3}}+{\frac{2\,{a}^{2}{c}^{2}{x}^{5}\arctan \left ( ax \right ) }{5}}+{\frac{{a}^{4}{c}^{2}{x}^{7}\arctan \left ( ax \right ) }{7}}+{\frac{4\,{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{105\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/105*c^2*x^2/a-9/140*a*c^2*x^4-1/42*a^3*c^2*x^6+1/3*c^2*x^3*arctan(a*x)+2/5*a^2*c^2*x^5*arctan(a*x)+1/7*a^4*
c^2*x^7*arctan(a*x)+4/105*c^2*ln(a^2*x^2+1)/a^3

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Maxima [A]  time = 0.974115, size = 128, normalized size = 1.21 \begin{align*} -\frac{1}{420} \, a{\left (\frac{10 \, a^{4} c^{2} x^{6} + 27 \, a^{2} c^{2} x^{4} + 16 \, c^{2} x^{2}}{a^{2}} - \frac{16 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )} + \frac{1}{105} \,{\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} 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)^2*arctan(a*x),x, algorithm="maxima")

[Out]

-1/420*a*((10*a^4*c^2*x^6 + 27*a^2*c^2*x^4 + 16*c^2*x^2)/a^2 - 16*c^2*log(a^2*x^2 + 1)/a^4) + 1/105*(15*a^4*c^
2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*arctan(a*x)

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Fricas [A]  time = 1.6932, size = 211, normalized size = 1.99 \begin{align*} -\frac{10 \, a^{6} c^{2} x^{6} + 27 \, a^{4} c^{2} x^{4} + 16 \, a^{2} c^{2} x^{2} - 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \,{\left (15 \, a^{7} c^{2} x^{7} + 42 \, a^{5} c^{2} x^{5} + 35 \, a^{3} c^{2} x^{3}\right )} \arctan \left (a x\right )}{420 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(10*a^6*c^2*x^6 + 27*a^4*c^2*x^4 + 16*a^2*c^2*x^2 - 16*c^2*log(a^2*x^2 + 1) - 4*(15*a^7*c^2*x^7 + 42*a^
5*c^2*x^5 + 35*a^3*c^2*x^3)*arctan(a*x))/a^3

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Sympy [A]  time = 2.47349, size = 105, normalized size = 0.99 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{7} \operatorname{atan}{\left (a x \right )}}{7} - \frac{a^{3} c^{2} x^{6}}{42} + \frac{2 a^{2} c^{2} x^{5} \operatorname{atan}{\left (a x \right )}}{5} - \frac{9 a c^{2} x^{4}}{140} + \frac{c^{2} x^{3} \operatorname{atan}{\left (a x \right )}}{3} - \frac{4 c^{2} x^{2}}{105 a} + \frac{4 c^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{105 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)**2*atan(a*x),x)

[Out]

Piecewise((a**4*c**2*x**7*atan(a*x)/7 - a**3*c**2*x**6/42 + 2*a**2*c**2*x**5*atan(a*x)/5 - 9*a*c**2*x**4/140 +
c**2*x**3*atan(a*x)/3 - 4*c**2*x**2/(105*a) + 4*c**2*log(x**2 + a**(-2))/(105*a**3), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.17053, size = 128, normalized size = 1.21 \begin{align*} \frac{1}{105} \,{\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \arctan \left (a x\right ) + \frac{4 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{105 \, a^{3}} - \frac{10 \, a^{9} c^{2} x^{6} + 27 \, a^{7} c^{2} x^{4} + 16 \, a^{5} c^{2} x^{2}}{420 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*arctan(a*x) + 4/105*c^2*log(a^2*x^2 + 1)/a^3 - 1/420*(10*
a^9*c^2*x^6 + 27*a^7*c^2*x^4 + 16*a^5*c^2*x^2)/a^6