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

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

[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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Rubi [A]  time = 0.17, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4948, 4852, 266, 43} \[ -\frac {1}{42} a^3 c^2 x^6+\frac {4 c^2 \log \left (a^2 x^2+1\right )}{105 a^3}+\frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)-\frac {9}{140} a c^2 x^4-\frac {4 c^2 x^2}{105 a}+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x) \]

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 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 )^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*}

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Mathematica [A]  time = 0.06, size = 106, normalized size = 1.00 \[ \frac {1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)-\frac {1}{42} a^3 c^2 x^6+\frac {2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)+\frac {4 c^2 \log \left (a^2 x^2+1\right )}{105 a^3}-\frac {9}{140} a c^2 x^4+\frac {1}{3} c^2 x^3 \tan ^{-1}(a x)-\frac {4 c^2 x^2}{105 a} \]

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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fricas [A]  time = 0.44, size = 94, normalized size = 0.89 \[ -\frac {10 \, a^{6} c^{2} x^{6} + 27 \, a^{4} c^{2} x^{4} + 16 \, a^{2} c^{2} x^{2} - 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \, {\left (15 \, a^{7} c^{2} x^{7} + 42 \, a^{5} c^{2} x^{5} + 35 \, a^{3} c^{2} x^{3}\right )} \arctan \left (a x\right )}{420 \, a^{3}} \]

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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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)^2*arctan(a*x),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 0.03, size = 93, normalized size = 0.88 \[ -\frac {4 c^{2} x^{2}}{105 a}-\frac {9 a \,c^{2} x^{4}}{140}-\frac {a^{3} c^{2} x^{6}}{42}+\frac {c^{2} x^{3} \arctan \left (a x \right )}{3}+\frac {2 a^{2} c^{2} x^{5} \arctan \left (a x \right )}{5}+\frac {a^{4} c^{2} x^{7} \arctan \left (a x \right )}{7}+\frac {4 c^{2} \ln \left (a^{2} x^{2}+1\right )}{105 a^{3}} \]

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 = 0.33, size = 95, normalized size = 0.90 \[ -\frac {1}{420} \, a {\left (\frac {10 \, a^{4} c^{2} x^{6} + 27 \, a^{2} c^{2} x^{4} + 16 \, c^{2} x^{2}}{a^{2}} - \frac {16 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )} + \frac {1}{105} \, {\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \arctan \left (a x\right ) \]

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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mupad [B]  time = 0.54, size = 81, normalized size = 0.76 \[ \frac {c^2\,\left (16\,\ln \left (a^2\,x^2+1\right )-16\,a^2\,x^2-27\,a^4\,x^4-10\,a^6\,x^6+140\,a^3\,x^3\,\mathrm {atan}\left (a\,x\right )+168\,a^5\,x^5\,\mathrm {atan}\left (a\,x\right )+60\,a^7\,x^7\,\mathrm {atan}\left (a\,x\right )\right )}{420\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*(16*log(a^2*x^2 + 1) - 16*a^2*x^2 - 27*a^4*x^4 - 10*a^6*x^6 + 140*a^3*x^3*atan(a*x) + 168*a^5*x^5*atan(a*
x) + 60*a^7*x^7*atan(a*x)))/(420*a^3)

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sympy [A]  time = 1.83, size = 105, normalized size = 0.99 \[ \begin {cases} \frac {a^{4} c^{2} x^{7} \operatorname {atan}{\left (a x \right )}}{7} - \frac {a^{3} c^{2} x^{6}}{42} + \frac {2 a^{2} c^{2} x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {9 a c^{2} x^{4}}{140} + \frac {c^{2} x^{3} \operatorname {atan}{\left (a x \right )}}{3} - \frac {4 c^{2} x^{2}}{105 a} + \frac {4 c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{105 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]

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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