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

Optimal. Leaf size=289 \[ -\frac {c^2 \left (a^2 x^2+1\right )}{20 a}-\frac {c^2 \log \left (a^2 x^2+1\right )}{2 a}+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{20 a}-\frac {2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{5 a}+\frac {1}{10} c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)+\frac {4 c^2 \text {Li}_3\left (1-\frac {2}{i a x+1}\right )}{5 a}+\frac {8 i c^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)}{5 a}+\frac {8 i c^2 \tan ^{-1}(a x)^3}{15 a}+\frac {8}{15} c^2 x \tan ^{-1}(a x)^3+c^2 x \tan ^{-1}(a x)+\frac {8 c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{5 a} \]

[Out]

-1/20*c^2*(a^2*x^2+1)/a+c^2*x*arctan(a*x)+1/10*c^2*x*(a^2*x^2+1)*arctan(a*x)-2/5*c^2*(a^2*x^2+1)*arctan(a*x)^2
/a-3/20*c^2*(a^2*x^2+1)^2*arctan(a*x)^2/a+8/15*I*c^2*arctan(a*x)^3/a+8/15*c^2*x*arctan(a*x)^3+4/15*c^2*x*(a^2*
x^2+1)*arctan(a*x)^3+1/5*c^2*x*(a^2*x^2+1)^2*arctan(a*x)^3+8/5*c^2*arctan(a*x)^2*ln(2/(1+I*a*x))/a-1/2*c^2*ln(
a^2*x^2+1)/a+8/5*I*c^2*arctan(a*x)*polylog(2,1-2/(1+I*a*x))/a+4/5*c^2*polylog(3,1-2/(1+I*a*x))/a

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Rubi [A]  time = 0.25, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {4880, 4846, 4920, 4854, 4884, 4994, 6610, 260, 4878} \[ \frac {4 c^2 \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{5 a}+\frac {8 i c^2 \tan ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a}-\frac {c^2 \left (a^2 x^2+1\right )}{20 a}-\frac {c^2 \log \left (a^2 x^2+1\right )}{2 a}+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3-\frac {3 c^2 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2}{20 a}-\frac {2 c^2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2}{5 a}+\frac {1}{10} c^2 x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)+\frac {8 i c^2 \tan ^{-1}(a x)^3}{15 a}+\frac {8}{15} c^2 x \tan ^{-1}(a x)^3+c^2 x \tan ^{-1}(a x)+\frac {8 c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{5 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*(1 + a^2*x^2))/(20*a) + c^2*x*ArcTan[a*x] + (c^2*x*(1 + a^2*x^2)*ArcTan[a*x])/10 - (2*c^2*(1 + a^2*x^2)*
ArcTan[a*x]^2)/(5*a) - (3*c^2*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/(20*a) + (((8*I)/15)*c^2*ArcTan[a*x]^3)/a + (8*c^
2*x*ArcTan[a*x]^3)/15 + (4*c^2*x*(1 + a^2*x^2)*ArcTan[a*x]^3)/15 + (c^2*x*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/5 + (
8*c^2*ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/(5*a) - (c^2*Log[1 + a^2*x^2])/(2*a) + (((8*I)/5)*c^2*ArcTan[a*x]*Poly
Log[2, 1 - 2/(1 + I*a*x)])/a + (4*c^2*PolyLog[3, 1 - 2/(1 + I*a*x)])/(5*a)

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 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 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 4878

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

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 4884

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

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 4994

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

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin {align*} \int \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx &=-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{20 a}+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac {1}{10} (3 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x) \, dx+\frac {1}{5} (4 c) \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3 \, dx\\ &=-\frac {c^2 \left (1+a^2 x^2\right )}{20 a}+\frac {1}{10} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)-\frac {2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{5 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{20 a}+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac {1}{5} c^2 \int \tan ^{-1}(a x) \, dx+\frac {1}{15} \left (8 c^2\right ) \int \tan ^{-1}(a x)^3 \, dx+\frac {1}{5} \left (4 c^2\right ) \int \tan ^{-1}(a x) \, dx\\ &=-\frac {c^2 \left (1+a^2 x^2\right )}{20 a}+c^2 x \tan ^{-1}(a x)+\frac {1}{10} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)-\frac {2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{5 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{20 a}+\frac {8}{15} c^2 x \tan ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3-\frac {1}{5} \left (a c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{5} \left (4 a c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{5} \left (8 a c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 \left (1+a^2 x^2\right )}{20 a}+c^2 x \tan ^{-1}(a x)+\frac {1}{10} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)-\frac {2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{5 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{20 a}+\frac {8 i c^2 \tan ^{-1}(a x)^3}{15 a}+\frac {8}{15} c^2 x \tan ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3-\frac {c^2 \log \left (1+a^2 x^2\right )}{2 a}+\frac {1}{5} \left (8 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{i-a x} \, dx\\ &=-\frac {c^2 \left (1+a^2 x^2\right )}{20 a}+c^2 x \tan ^{-1}(a x)+\frac {1}{10} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)-\frac {2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{5 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{20 a}+\frac {8 i c^2 \tan ^{-1}(a x)^3}{15 a}+\frac {8}{15} c^2 x \tan ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac {8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a}-\frac {c^2 \log \left (1+a^2 x^2\right )}{2 a}-\frac {1}{5} \left (16 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 \left (1+a^2 x^2\right )}{20 a}+c^2 x \tan ^{-1}(a x)+\frac {1}{10} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)-\frac {2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{5 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{20 a}+\frac {8 i c^2 \tan ^{-1}(a x)^3}{15 a}+\frac {8}{15} c^2 x \tan ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac {8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a}-\frac {c^2 \log \left (1+a^2 x^2\right )}{2 a}+\frac {8 i c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{5 a}-\frac {1}{5} \left (8 i c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {c^2 \left (1+a^2 x^2\right )}{20 a}+c^2 x \tan ^{-1}(a x)+\frac {1}{10} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)-\frac {2 c^2 \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2}{5 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^2}{20 a}+\frac {8 i c^2 \tan ^{-1}(a x)^3}{15 a}+\frac {8}{15} c^2 x \tan ^{-1}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \tan ^{-1}(a x)^3+\frac {8 c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a}-\frac {c^2 \log \left (1+a^2 x^2\right )}{2 a}+\frac {8 i c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{5 a}+\frac {4 c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{5 a}\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 195, normalized size = 0.67 \[ \frac {c^2 \left (12 a^5 x^5 \tan ^{-1}(a x)^3-9 a^4 x^4 \tan ^{-1}(a x)^2+40 a^3 x^3 \tan ^{-1}(a x)^3+6 a^3 x^3 \tan ^{-1}(a x)-3 a^2 x^2-30 \log \left (a^2 x^2+1\right )-42 a^2 x^2 \tan ^{-1}(a x)^2-96 i \tan ^{-1}(a x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+48 \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )+60 a x \tan ^{-1}(a x)^3+66 a x \tan ^{-1}(a x)-32 i \tan ^{-1}(a x)^3-33 \tan ^{-1}(a x)^2+96 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-3\right )}{60 a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*(-3 - 3*a^2*x^2 + 66*a*x*ArcTan[a*x] + 6*a^3*x^3*ArcTan[a*x] - 33*ArcTan[a*x]^2 - 42*a^2*x^2*ArcTan[a*x]^
2 - 9*a^4*x^4*ArcTan[a*x]^2 - (32*I)*ArcTan[a*x]^3 + 60*a*x*ArcTan[a*x]^3 + 40*a^3*x^3*ArcTan[a*x]^3 + 12*a^5*
x^5*ArcTan[a*x]^3 + 96*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - 30*Log[1 + a^2*x^2] - (96*I)*ArcTan[a*x]
*PolyLog[2, -E^((2*I)*ArcTan[a*x])] + 48*PolyLog[3, -E^((2*I)*ArcTan[a*x])]))/(60*a)

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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

sage0*x

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maple [C]  time = 5.53, size = 2691, normalized size = 9.31 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/10*c^2*x*arctan(a*x)+c^2*x*arctan(a*x)^3-3/20*I*a*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x
)^2*x^2+4/5*I/a*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^2*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2+2/5
*I/a*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*ar
ctan(a*x)^2-1/20*a^2*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan
(a*x)^2*x^3+1/10*a^2*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan
(a*x)^2*x^3+1/20*a^2*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)
^2/(a^2*x^2+1)+I)*arctan(a*x)^2*x^3-1/10*a^2*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)*csgn(I*(1+I*a*x)^4/(a^2*
x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*arctan(a*x)^2*x^3-11/20/a*c^2*arctan(a*x)^2+1/a*c^2*ln((1+I*a*x)^2/(
a^2*x^2+1)+1)+4/5/a*c^2*polylog(3,-(1+I*a*x)^2/(a^2*x^2+1))-1/20*a^2*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)
^2)^3*arctan(a*x)^2*x^3+1/20*a^2*c^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^3*arct
an(a*x)^2*x^3+3/20*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a
*x)^2*x-3/10*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2*
x-3/20*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)
+I)*arctan(a*x)^2*x+3/10*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*
x)^2/(a^2*x^2+1)+I)^2*arctan(a*x)^2*x+1/20*I/a*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2-2/
5*I/a*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))^3*arctan(a*x)^2-2/5*I/a*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1
+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2+7/20*I/a*c^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(
a^2*x^2+1)+I)^3*arctan(a*x)^2+3/20*I*a*c^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^
3*arctan(a*x)^2*x^2-2/5*I/a*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*arcta
n(a*x)^2+2/5*I/a*c^2*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*csgn(I/((1+I*a*x)^2/(a
^2*x^2+1)+1)^2)*arctan(a*x)^2-7/10*I/a*c^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^
2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)*arctan(a*x)^2+7/20*I/a*c^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*
x)^2/(a^2*x^2+1)+I)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*arctan(a*x)^2+1/20*I/a*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2
*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*arctan(a*x)^2-1/10*I/a*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2
+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2*arctan(a*x)^2+1/10*a^2*c^2*arctan(a*x)*x^3-3/20*a^3*c^2*arctan
(a*x)^2*x^4-7/10*a*c^2*arctan(a*x)^2*x^2+1/5*a^4*c^2*arctan(a*x)^3*x^5+2/3*a^2*c^2*arctan(a*x)^3*x^3-4/5/a*c^2
*arctan(a*x)^2*ln(a^2*x^2+1)+8/5/a*c^2*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2))+8/5/a*c^2*ln(2)*arctan(a*
x)^2-8/15*I/a*c^2*arctan(a*x)^3-I/a*c^2*arctan(a*x)-3/20*c^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)
^2/(a^2*x^2+1)+I)^3*arctan(a*x)^2*x+3/20*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^3*arctan(a*x)^2*x-8/5*I/
a*c^2*arctan(a*x)*polylog(2,-(1+I*a*x)^2/(a^2*x^2+1))-1/20/a*c^2-1/20*c^2*x^2*a-2/5*I/a*c^2*Pi*csgn(I*(1+I*a*x
)^2/(a^2*x^2+1))*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^2*x^2+1)+1)^2)*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)
+1)^2)*arctan(a*x)^2-3/20*I*a*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^
2)*arctan(a*x)^2*x^2+3/10*I*a*c^2*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)
^2*arctan(a*x)^2*x^2-3/10*I*a*c^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*csgn(I*
(1+I*a*x)^2/(a^2*x^2+1)+I)*arctan(a*x)^2*x^2+3/20*I*a*c^2*Pi*csgn(I*(1+I*a*x)^4/(a^2*x^2+1)^2+2*I*(1+I*a*x)^2/
(a^2*x^2+1)+I)*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)+I)^2*arctan(a*x)^2*x^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 140 \, a^{6} c^{2} \int \frac {x^{6} \arctan \left (a x\right )^{3}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 15 \, a^{6} c^{2} \int \frac {x^{6} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a^{6} c^{2} \int \frac {x^{6} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} - 12 \, a^{5} c^{2} \int \frac {x^{5} \arctan \left (a x\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 3 \, a^{5} c^{2} \int \frac {x^{5} \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 420 \, a^{4} c^{2} \int \frac {x^{4} \arctan \left (a x\right )^{3}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 45 \, a^{4} c^{2} \int \frac {x^{4} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 40 \, a^{4} c^{2} \int \frac {x^{4} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} - 40 \, a^{3} c^{2} \int \frac {x^{3} \arctan \left (a x\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 10 \, a^{3} c^{2} \int \frac {x^{3} \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac {7 \, c^{2} \arctan \left (a x\right )^{4}}{32 \, a} + 420 \, a^{2} c^{2} \int \frac {x^{2} \arctan \left (a x\right )^{3}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 45 \, a^{2} c^{2} \int \frac {x^{2} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 60 \, a^{2} c^{2} \int \frac {x^{2} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac {1}{120} \, {\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \arctan \left (a x\right )^{3} - 60 \, a c^{2} \int \frac {x \arctan \left (a x\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 15 \, a c^{2} \int \frac {x \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} - \frac {1}{160} \, {\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + 15 \, c^{2} \int \frac {\arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

140*a^6*c^2*integrate(1/160*x^6*arctan(a*x)^3/(a^2*x^2 + 1), x) + 15*a^6*c^2*integrate(1/160*x^6*arctan(a*x)*l
og(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 12*a^6*c^2*integrate(1/160*x^6*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1
), x) - 12*a^5*c^2*integrate(1/160*x^5*arctan(a*x)^2/(a^2*x^2 + 1), x) + 3*a^5*c^2*integrate(1/160*x^5*log(a^2
*x^2 + 1)^2/(a^2*x^2 + 1), x) + 420*a^4*c^2*integrate(1/160*x^4*arctan(a*x)^3/(a^2*x^2 + 1), x) + 45*a^4*c^2*i
ntegrate(1/160*x^4*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 40*a^4*c^2*integrate(1/160*x^4*arctan(a*
x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 40*a^3*c^2*integrate(1/160*x^3*arctan(a*x)^2/(a^2*x^2 + 1), x) + 10*a^
3*c^2*integrate(1/160*x^3*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 7/32*c^2*arctan(a*x)^4/a + 420*a^2*c^2*integr
ate(1/160*x^2*arctan(a*x)^3/(a^2*x^2 + 1), x) + 45*a^2*c^2*integrate(1/160*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/
(a^2*x^2 + 1), x) + 60*a^2*c^2*integrate(1/160*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) + 1/120*(3*a
^4*c^2*x^5 + 10*a^2*c^2*x^3 + 15*c^2*x)*arctan(a*x)^3 - 60*a*c^2*integrate(1/160*x*arctan(a*x)^2/(a^2*x^2 + 1)
, x) + 15*a*c^2*integrate(1/160*x*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) - 1/160*(3*a^4*c^2*x^5 + 10*a^2*c^2*x^3
+ 15*c^2*x)*arctan(a*x)*log(a^2*x^2 + 1)^2 + 15*c^2*integrate(1/160*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^2 +
1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a*x)^3*(c + a^2*c*x^2)^2,x)

[Out]

int(atan(a*x)^3*(c + a^2*c*x^2)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int 2 a^{2} x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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