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

Optimal. Leaf size=242 \[ -\frac{4 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{15 a^2}-\frac{c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{10 a}-\frac{2 c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{15 a}+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^3}{6 a^2}+\frac{c^2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{20 a^2}+\frac{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{15 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{8 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^2}-\frac{1}{60} a c^2 x^3-\frac{11 c^2 x}{60 a}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a} \]

[Out]

(-11*c^2*x)/(60*a) - (a*c^2*x^3)/60 + (2*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a^2) + (c^2*(1 + a^2*x^2)^2*ArcTan
[a*x])/(20*a^2) - (((4*I)/15)*c^2*ArcTan[a*x]^2)/a^2 - (4*c^2*x*ArcTan[a*x]^2)/(15*a) - (2*c^2*x*(1 + a^2*x^2)
*ArcTan[a*x]^2)/(15*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(10*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/(6
*a^2) - (8*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^2) - (((4*I)/15)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2

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Rubi [A]  time = 0.188007, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4930, 4880, 4846, 4920, 4854, 2402, 2315, 8} \[ -\frac{4 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{15 a^2}-\frac{c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{10 a}-\frac{2 c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{15 a}+\frac{c^2 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^3}{6 a^2}+\frac{c^2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)}{20 a^2}+\frac{2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{15 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{8 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^2}-\frac{1}{60} a c^2 x^3-\frac{11 c^2 x}{60 a}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*c^2*x)/(60*a) - (a*c^2*x^3)/60 + (2*c^2*(1 + a^2*x^2)*ArcTan[a*x])/(15*a^2) + (c^2*(1 + a^2*x^2)^2*ArcTan
[a*x])/(20*a^2) - (((4*I)/15)*c^2*ArcTan[a*x]^2)/a^2 - (4*c^2*x*ArcTan[a*x]^2)/(15*a) - (2*c^2*x*(1 + a^2*x^2)
*ArcTan[a*x]^2)/(15*a) - (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(10*a) + (c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^3)/(6
*a^2) - (8*c^2*ArcTan[a*x]*Log[2/(1 + I*a*x)])/(15*a^2) - (((4*I)/15)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^2

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)^
(q + 1)*(a + b*ArcTan[c*x])^p)/(2*e*(q + 1)), x] - Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rule 4880

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> -Simp[(b*p*(d + e*x^2)^q
*(a + b*ArcTan[c*x])^(p - 1))/(2*c*q*(2*q + 1)), x] + (Dist[(2*d*q)/(2*q + 1), Int[(d + e*x^2)^(q - 1)*(a + b*
ArcTan[c*x])^p, x], x] + Dist[(b^2*d*p*(p - 1))/(2*q*(2*q + 1)), Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^(
p - 2), x], x] + Simp[(x*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p)/(2*q + 1), x]) /; FreeQ[{a, b, c, d, e}, x] && E
qQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]

Rule 4846

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

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int x \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx &=\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{\int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx}{2 a}\\ &=\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{c \int \left (c+a^2 c x^2\right ) \, dx}{20 a}-\frac{(2 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx}{5 a}\\ &=-\frac{c^2 x}{20 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{\left (2 c^2\right ) \int 1 \, dx}{15 a}-\frac{\left (4 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx}{15 a}\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}+\frac{1}{15} \left (8 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{\left (8 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{15 a}\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{8 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^2}+\frac{\left (8 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{15 a}\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{8 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^2}-\frac{\left (8 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{15 a^2}\\ &=-\frac{11 c^2 x}{60 a}-\frac{1}{60} a c^2 x^3+\frac{2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{15 a^2}+\frac{c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)}{20 a^2}-\frac{4 i c^2 \tan ^{-1}(a x)^2}{15 a^2}-\frac{4 c^2 x \tan ^{-1}(a x)^2}{15 a}-\frac{2 c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{15 a}-\frac{c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{10 a}+\frac{c^2 \left (1+a^2 x^2\right )^3 \tan ^{-1}(a x)^3}{6 a^2}-\frac{8 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^2}-\frac{4 i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{15 a^2}\\ \end{align*}

Mathematica [A]  time = 0.743275, size = 131, normalized size = 0.54 \[ \frac{c^2 \left (16 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-a x \left (a^2 x^2+11\right )+10 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^3-2 \left (3 a^5 x^5+10 a^3 x^3+15 a x-8 i\right ) \tan ^{-1}(a x)^2+\tan ^{-1}(a x) \left (3 a^4 x^4+14 a^2 x^2-32 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+11\right )\right )}{60 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*(-(a*x*(11 + a^2*x^2)) - 2*(-8*I + 15*a*x + 10*a^3*x^3 + 3*a^5*x^5)*ArcTan[a*x]^2 + 10*(1 + a^2*x^2)^3*Ar
cTan[a*x]^3 + ArcTan[a*x]*(11 + 14*a^2*x^2 + 3*a^4*x^4 - 32*Log[1 + E^((2*I)*ArcTan[a*x])]) + (16*I)*PolyLog[2
, -E^((2*I)*ArcTan[a*x])]))/(60*a^2)

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Maple [A]  time = 0.11, size = 368, normalized size = 1.5 \begin{align*}{\frac{{a}^{4}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{6}}{6}}+{\frac{{a}^{2}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{4}}{2}}+{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{2}}{2}}-{\frac{{a}^{3}{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{5}}{10}}-{\frac{a{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}}{3}}-{\frac{{c}^{2}x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,a}}+{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{6\,{a}^{2}}}+{\frac{{a}^{2}{c}^{2}\arctan \left ( ax \right ){x}^{4}}{20}}+{\frac{7\,{c}^{2}\arctan \left ( ax \right ){x}^{2}}{30}}+{\frac{4\,{c}^{2}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{15\,{a}^{2}}}-{\frac{a{c}^{2}{x}^{3}}{60}}-{\frac{11\,{c}^{2}x}{60\,a}}+{\frac{11\,{c}^{2}\arctan \left ( ax \right ) }{60\,{a}^{2}}}-{\frac{{\frac{i}{15}}{c}^{2} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{2}}}+{\frac{{\frac{i}{15}}{c}^{2} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{2}}}+{\frac{{\frac{2\,i}{15}}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{2}}}-{\frac{{\frac{2\,i}{15}}{c}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{2}}}+{\frac{{\frac{2\,i}{15}}{c}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{2}}}+{\frac{{\frac{2\,i}{15}}{c}^{2}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}}-{\frac{{\frac{2\,i}{15}}{c}^{2}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{2}}}-{\frac{{\frac{2\,i}{15}}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a^4*c^2*arctan(a*x)^3*x^6+1/2*a^2*c^2*arctan(a*x)^3*x^4+1/2*c^2*arctan(a*x)^3*x^2-1/10*a^3*c^2*arctan(a*x)
^2*x^5-1/3*a*c^2*arctan(a*x)^2*x^3-1/2*c^2*x*arctan(a*x)^2/a+1/6/a^2*c^2*arctan(a*x)^3+1/20*a^2*c^2*arctan(a*x
)*x^4+7/30*c^2*arctan(a*x)*x^2+4/15/a^2*c^2*arctan(a*x)*ln(a^2*x^2+1)-1/60*a*c^2*x^3-11/60*c^2*x/a+11/60/a^2*c
^2*arctan(a*x)-1/15*I/a^2*c^2*ln(a*x-I)^2+1/15*I/a^2*c^2*ln(a*x+I)^2+2/15*I/a^2*c^2*ln(a*x+I)*ln(1/2*I*(a*x-I)
)-2/15*I/a^2*c^2*dilog(-1/2*I*(a*x+I))+2/15*I/a^2*c^2*dilog(1/2*I*(a*x-I))+2/15*I/a^2*c^2*ln(a*x-I)*ln(a^2*x^2
+1)-2/15*I/a^2*c^2*ln(a*x-I)*ln(-1/2*I*(a*x+I))-2/15*I/a^2*c^2*ln(a*x+I)*ln(a^2*x^2+1)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{3} + c^{2} x\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x)*arctan(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int x \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{3} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{5} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a**2*c*x**2+c)**2*atan(a*x)**3,x)

[Out]

c**2*(Integral(x*atan(a*x)**3, x) + Integral(2*a**2*x**3*atan(a*x)**3, x) + Integral(a**4*x**5*atan(a*x)**3, x
))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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