3.70 \(\int x (d+i c d x) (a+b \tan ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=211 \[ \frac{b^2 d \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^2}+\frac{5 d \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^2}-\frac{2 i b d \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^2}+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i b d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{a b d x}{c}+\frac{b^2 d \log \left (c^2 x^2+1\right )}{2 c^2}-\frac{i b^2 d \tan ^{-1}(c x)}{3 c^2}+\frac{i b^2 d x}{3 c}-\frac{b^2 d x \tan ^{-1}(c x)}{c} \]

[Out]

-((a*b*d*x)/c) + ((I/3)*b^2*d*x)/c - ((I/3)*b^2*d*ArcTan[c*x])/c^2 - (b^2*d*x*ArcTan[c*x])/c - (I/3)*b*d*x^2*(
a + b*ArcTan[c*x]) + (5*d*(a + b*ArcTan[c*x])^2)/(6*c^2) + (d*x^2*(a + b*ArcTan[c*x])^2)/2 + (I/3)*c*d*x^3*(a
+ b*ArcTan[c*x])^2 - (((2*I)/3)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (b^2*d*Log[1 + c^2*x^2])/(2*
c^2) + (b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^2)

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Rubi [A]  time = 0.358016, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4876, 4852, 4916, 4846, 260, 4884, 321, 203, 4920, 4854, 2402, 2315} \[ \frac{b^2 d \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^2}+\frac{5 d \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^2}-\frac{2 i b d \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^2}+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i b d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{a b d x}{c}+\frac{b^2 d \log \left (c^2 x^2+1\right )}{2 c^2}-\frac{i b^2 d \tan ^{-1}(c x)}{3 c^2}+\frac{i b^2 d x}{3 c}-\frac{b^2 d x \tan ^{-1}(c x)}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*b*d*x)/c) + ((I/3)*b^2*d*x)/c - ((I/3)*b^2*d*ArcTan[c*x])/c^2 - (b^2*d*x*ArcTan[c*x])/c - (I/3)*b*d*x^2*(
a + b*ArcTan[c*x]) + (5*d*(a + b*ArcTan[c*x])^2)/(6*c^2) + (d*x^2*(a + b*ArcTan[c*x])^2)/2 + (I/3)*c*d*x^3*(a
+ b*ArcTan[c*x])^2 - (((2*I)/3)*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (b^2*d*Log[1 + c^2*x^2])/(2*
c^2) + (b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/(3*c^2)

Rule 4876

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || 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 4916

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcTan[c*x])^p)
/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 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 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 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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
 c, n, m, p, x]

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])

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]

Rubi steps

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

Mathematica [A]  time = 0.496672, size = 208, normalized size = 0.99 \[ \frac{d \left (-2 b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+2 i a^2 c^3 x^3+3 a^2 c^2 x^2-2 i a b c^2 x^2+2 i a b \log \left (c^2 x^2+1\right )+2 b \tan ^{-1}(c x) \left (a \left (2 i c^3 x^3+3 c^2 x^2+3\right )-i b \left (c^2 x^2-3 i c x+1\right )-2 i b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-6 a b c x+3 b^2 \log \left (c^2 x^2+1\right )+b^2 \left (2 i c^3 x^3+3 c^2 x^2+1\right ) \tan ^{-1}(c x)^2+2 i b^2 c x\right )}{6 c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d*(-6*a*b*c*x + (2*I)*b^2*c*x + 3*a^2*c^2*x^2 - (2*I)*a*b*c^2*x^2 + (2*I)*a^2*c^3*x^3 + b^2*(1 + 3*c^2*x^2 +
(2*I)*c^3*x^3)*ArcTan[c*x]^2 + 2*b*ArcTan[c*x]*((-I)*b*(1 - (3*I)*c*x + c^2*x^2) + a*(3 + 3*c^2*x^2 + (2*I)*c^
3*x^3) - (2*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + (2*I)*a*b*Log[1 + c^2*x^2] + 3*b^2*Log[1 + c^2*x^2] - 2*b^2
*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/(6*c^2)

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Maple [B]  time = 0.093, size = 416, normalized size = 2. \begin{align*}{\frac{{\frac{i}{3}}dab\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{2}}}+{\frac{d{b}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,{c}^{2}}}+{\frac{2\,i}{3}}cdab\arctan \left ( cx \right ){x}^{3}-{\frac{i}{3}}dab{x}^{2}+{\frac{i}{3}}cd{a}^{2}{x}^{3}-{\frac{i}{3}}d{b}^{2}\arctan \left ( cx \right ){x}^{2}+{\frac{dab\arctan \left ( cx \right ) }{{c}^{2}}}+{\frac{d{b}^{2}\ln \left ( cx+i \right ) \ln \left ({c}^{2}{x}^{2}+1 \right ) }{6\,{c}^{2}}}-{\frac{d{b}^{2}\ln \left ( cx+i \right ) \ln \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) }{6\,{c}^{2}}}-{\frac{d{b}^{2}\ln \left ( cx-i \right ) \ln \left ({c}^{2}{x}^{2}+1 \right ) }{6\,{c}^{2}}}+{\frac{d{b}^{2}\ln \left ( cx-i \right ) \ln \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{6\,{c}^{2}}}+dab\arctan \left ( cx \right ){x}^{2}+{\frac{{\frac{i}{3}}d{b}^{2}\arctan \left ( cx \right ) \ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{2}}}+{\frac{d{a}^{2}{x}^{2}}{2}}+{\frac{d{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{2\,{c}^{2}}}+{\frac{d{b}^{2} \left ( \ln \left ( cx-i \right ) \right ) ^{2}}{12\,{c}^{2}}}+{\frac{d{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{x}^{2}}{2}}+{\frac{d{b}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{6\,{c}^{2}}}-{\frac{d{b}^{2} \left ( \ln \left ( cx+i \right ) \right ) ^{2}}{12\,{c}^{2}}}-{\frac{d{b}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) }{6\,{c}^{2}}}+{\frac{{\frac{i}{3}}d{b}^{2}x}{c}}-{\frac{{\frac{i}{3}}d{b}^{2}\arctan \left ( cx \right ) }{{c}^{2}}}-{\frac{dabx}{c}}-{\frac{d{b}^{2}x\arctan \left ( cx \right ) }{c}}+{\frac{i}{3}}cd{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{x}^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*I/c^2*d*a*b*ln(c^2*x^2+1)+1/2*b^2*d*ln(c^2*x^2+1)/c^2+2/3*I*c*d*a*b*arctan(c*x)*x^3-1/3*I*d*a*b*x^2+1/3*I*
c*d*a^2*x^3-1/3*I*d*b^2*arctan(c*x)*x^2+1/c^2*d*a*b*arctan(c*x)+1/6/c^2*d*b^2*ln(c*x+I)*ln(c^2*x^2+1)-1/6/c^2*
d*b^2*ln(c*x+I)*ln(1/2*I*(c*x-I))-1/6/c^2*d*b^2*ln(c*x-I)*ln(c^2*x^2+1)+1/6/c^2*d*b^2*ln(c*x-I)*ln(-1/2*I*(c*x
+I))+d*a*b*arctan(c*x)*x^2+1/3*I/c^2*d*b^2*arctan(c*x)*ln(c^2*x^2+1)+1/2*d*a^2*x^2+1/2/c^2*d*b^2*arctan(c*x)^2
+1/12/c^2*d*b^2*ln(c*x-I)^2+1/2*d*b^2*arctan(c*x)^2*x^2+1/6/c^2*d*b^2*dilog(-1/2*I*(c*x+I))-1/12/c^2*d*b^2*ln(
c*x+I)^2-1/6/c^2*d*b^2*dilog(1/2*I*(c*x-I))+1/3*I*b^2*d*x/c-1/3*I*b^2*d*arctan(c*x)/c^2-a*b*d*x/c-b^2*d*x*arct
an(c*x)/c+1/3*I*c*d*b^2*arctan(c*x)^2*x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} i \, a^{2} c d x^{3} + \frac{1}{2} \, b^{2} d x^{2} \arctan \left (c x\right )^{2} + \frac{1}{3} i \,{\left (2 \, x^{3} \arctan \left (c x\right ) - c{\left (\frac{x^{2}}{c^{2}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} a b c d + \frac{1}{48} i \,{\left (4 \, x^{3} \arctan \left (c x\right )^{2} - x^{3} \log \left (c^{2} x^{2} + 1\right )^{2} + 48 \, \int \frac{4 \, c^{2} x^{4} \log \left (c^{2} x^{2} + 1\right ) - 8 \, c x^{3} \arctan \left (c x\right ) + 36 \,{\left (c^{2} x^{4} + x^{2}\right )} \arctan \left (c x\right )^{2} + 3 \,{\left (c^{2} x^{4} + x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{48 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x}\right )} b^{2} c d + \frac{1}{2} \, a^{2} d x^{2} +{\left (x^{2} \arctan \left (c x\right ) - c{\left (\frac{x}{c^{2}} - \frac{\arctan \left (c x\right )}{c^{3}}\right )}\right )} a b d - \frac{1}{2} \,{\left (2 \, c{\left (\frac{x}{c^{2}} - \frac{\arctan \left (c x\right )}{c^{3}}\right )} \arctan \left (c x\right ) + \frac{\arctan \left (c x\right )^{2} - \log \left (c^{2} x^{2} + 1\right )}{c^{2}}\right )} b^{2} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*I*a^2*c*d*x^3 + 1/2*b^2*d*x^2*arctan(c*x)^2 + 1/3*I*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4
))*a*b*c*d + 1/48*I*(4*x^3*arctan(c*x)^2 - x^3*log(c^2*x^2 + 1)^2 + 48*integrate(1/48*(4*c^2*x^4*log(c^2*x^2 +
 1) - 8*c*x^3*arctan(c*x) + 36*(c^2*x^4 + x^2)*arctan(c*x)^2 + 3*(c^2*x^4 + x^2)*log(c^2*x^2 + 1)^2)/(c^2*x^2
+ 1), x))*b^2*c*d + 1/2*a^2*d*x^2 + (x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*d - 1/2*(2*c*(x/c^2 -
arctan(c*x)/c^3)*arctan(c*x) + (arctan(c*x)^2 - log(c^2*x^2 + 1))/c^2)*b^2*d

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{24} \,{\left (-2 i \, b^{2} c d x^{3} - 3 \, b^{2} d x^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2} +{\rm integral}\left (\frac{6 i \, a^{2} c^{3} d x^{4} + 6 \, a^{2} c^{2} d x^{3} + 6 i \, a^{2} c d x^{2} + 6 \, a^{2} d x -{\left (6 \, a b c^{3} d x^{4} -{\left (6 i \, a b + 2 \, b^{2}\right )} c^{2} d x^{3} + 3 \,{\left (2 \, a b + i \, b^{2}\right )} c d x^{2} - 6 i \, a b d x\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{6 \,{\left (c^{2} x^{2} + 1\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(-2*I*b^2*c*d*x^3 - 3*b^2*d*x^2)*log(-(c*x + I)/(c*x - I))^2 + integral(1/6*(6*I*a^2*c^3*d*x^4 + 6*a^2*c^
2*d*x^3 + 6*I*a^2*c*d*x^2 + 6*a^2*d*x - (6*a*b*c^3*d*x^4 - (6*I*a*b + 2*b^2)*c^2*d*x^3 + 3*(2*a*b + I*b^2)*c*d
*x^2 - 6*I*a*b*d*x)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d+I*c*d*x)*(a+b*atan(c*x))**2,x)

[Out]

Exception raised: AttributeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((I*c*d*x + d)*(b*arctan(c*x) + a)^2*x, x)