\(\int (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx\) [79]

Optimal result

Integrand size = 22, antiderivative size = 192 \[ \int (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-2 i a b d^2 x-\frac {1}{3} b^2 d^2 x+\frac {b^2 d^2 \arctan (c x)}{3 c}-2 i b^2 d^2 x \arctan (c x)+\frac {1}{3} b c d^2 x^2 (a+b \arctan (c x))-\frac {i d^2 (1+i c x)^3 (a+b \arctan (c x))^2}{3 c}+\frac {8 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{3 c}+\frac {i b^2 d^2 \log \left (1+c^2 x^2\right )}{c}-\frac {4 i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{3 c} \]

[Out]

-2*I*a*b*d^2*x-1/3*b^2*d^2*x+1/3*b^2*d^2*arctan(c*x)/c-2*I*b^2*d^2*x*arctan(c*x)+1/3*b*c*d^2*x^2*(a+b*arctan(c
*x))-1/3*I*d^2*(1+I*c*x)^3*(a+b*arctan(c*x))^2/c+8/3*b*d^2*(a+b*arctan(c*x))*ln(2/(1-I*c*x))/c+I*b^2*d^2*ln(c^
2*x^2+1)/c-4/3*I*b^2*d^2*polylog(2,1-2/(1-I*c*x))/c

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {4974, 4930, 266, 4946, 327, 209, 1600, 4964, 2449, 2352} \[ \int (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\frac {1}{3} b c d^2 x^2 (a+b \arctan (c x))-\frac {i d^2 (1+i c x)^3 (a+b \arctan (c x))^2}{3 c}+\frac {8 b d^2 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{3 c}-2 i a b d^2 x+\frac {b^2 d^2 \arctan (c x)}{3 c}-2 i b^2 d^2 x \arctan (c x)+\frac {i b^2 d^2 \log \left (c^2 x^2+1\right )}{c}-\frac {4 i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{3 c}-\frac {1}{3} b^2 d^2 x \]

[In]

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

[Out]

(-2*I)*a*b*d^2*x - (b^2*d^2*x)/3 + (b^2*d^2*ArcTan[c*x])/(3*c) - (2*I)*b^2*d^2*x*ArcTan[c*x] + (b*c*d^2*x^2*(a
+ b*ArcTan[c*x]))/3 - ((I/3)*d^2*(1 + I*c*x)^3*(a + b*ArcTan[c*x])^2)/c + (8*b*d^2*(a + b*ArcTan[c*x])*Log[2/
(1 - I*c*x)])/(3*c) + (I*b^2*d^2*Log[1 + c^2*x^2])/c - (((4*I)/3)*b^2*d^2*PolyLog[2, 1 - 2/(1 - I*c*x)])/c

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

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 327

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 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2352

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

Rule 2449

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 4930

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

Rule 4946

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

Rule 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[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 4974

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

Rubi steps \begin{align*} \text {integral}& = -\frac {i d^2 (1+i c x)^3 (a+b \arctan (c x))^2}{3 c}+\frac {(2 i b) \int \left (-3 d^3 (a+b \arctan (c x))-i c d^3 x (a+b \arctan (c x))-\frac {4 i \left (i d^3-c d^3 x\right ) (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{3 d} \\ & = -\frac {i d^2 (1+i c x)^3 (a+b \arctan (c x))^2}{3 c}+\frac {(8 b) \int \frac {\left (i d^3-c d^3 x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 d}-\left (2 i b d^2\right ) \int (a+b \arctan (c x)) \, dx+\frac {1}{3} \left (2 b c d^2\right ) \int x (a+b \arctan (c x)) \, dx \\ & = -2 i a b d^2 x+\frac {1}{3} b c d^2 x^2 (a+b \arctan (c x))-\frac {i d^2 (1+i c x)^3 (a+b \arctan (c x))^2}{3 c}+\frac {(8 b) \int \frac {a+b \arctan (c x)}{-\frac {i}{d^3}-\frac {c x}{d^3}} \, dx}{3 d}-\left (2 i b^2 d^2\right ) \int \arctan (c x) \, dx-\frac {1}{3} \left (b^2 c^2 d^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx \\ & = -2 i a b d^2 x-\frac {1}{3} b^2 d^2 x-2 i b^2 d^2 x \arctan (c x)+\frac {1}{3} b c d^2 x^2 (a+b \arctan (c x))-\frac {i d^2 (1+i c x)^3 (a+b \arctan (c x))^2}{3 c}+\frac {8 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{3 c}+\frac {1}{3} \left (b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx-\frac {1}{3} \left (8 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx+\left (2 i b^2 c d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx \\ & = -2 i a b d^2 x-\frac {1}{3} b^2 d^2 x+\frac {b^2 d^2 \arctan (c x)}{3 c}-2 i b^2 d^2 x \arctan (c x)+\frac {1}{3} b c d^2 x^2 (a+b \arctan (c x))-\frac {i d^2 (1+i c x)^3 (a+b \arctan (c x))^2}{3 c}+\frac {8 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{3 c}+\frac {i b^2 d^2 \log \left (1+c^2 x^2\right )}{c}-\frac {\left (8 i b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{3 c} \\ & = -2 i a b d^2 x-\frac {1}{3} b^2 d^2 x+\frac {b^2 d^2 \arctan (c x)}{3 c}-2 i b^2 d^2 x \arctan (c x)+\frac {1}{3} b c d^2 x^2 (a+b \arctan (c x))-\frac {i d^2 (1+i c x)^3 (a+b \arctan (c x))^2}{3 c}+\frac {8 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{3 c}+\frac {i b^2 d^2 \log \left (1+c^2 x^2\right )}{c}-\frac {4 i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{3 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.07 \[ \int (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=-\frac {d^2 \left (-3 a^2 c x+6 i a b c x+b^2 c x-3 i a^2 c^2 x^2-a b c^2 x^2+a^2 c^3 x^3+b^2 (-i+c x)^3 \arctan (c x)^2-b \arctan (c x) \left (b \left (1-6 i c x+c^2 x^2\right )+a \left (6 i+6 c x+6 i c^2 x^2-2 c^3 x^3\right )+8 b \log \left (1+e^{2 i \arctan (c x)}\right )\right )+4 a b \log \left (1+c^2 x^2\right )-3 i b^2 \log \left (1+c^2 x^2\right )+4 i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{3 c} \]

[In]

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

[Out]

-1/3*(d^2*(-3*a^2*c*x + (6*I)*a*b*c*x + b^2*c*x - (3*I)*a^2*c^2*x^2 - a*b*c^2*x^2 + a^2*c^3*x^3 + b^2*(-I + c*
x)^3*ArcTan[c*x]^2 - b*ArcTan[c*x]*(b*(1 - (6*I)*c*x + c^2*x^2) + a*(6*I + 6*c*x + (6*I)*c^2*x^2 - 2*c^3*x^3)
+ 8*b*Log[1 + E^((2*I)*ArcTan[c*x])]) + 4*a*b*Log[1 + c^2*x^2] - (3*I)*b^2*Log[1 + c^2*x^2] + (4*I)*b^2*PolyLo
g[2, -E^((2*I)*ArcTan[c*x])]))/c

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (172 ) = 344\).

Time = 1.20 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.84

method	result	size
derivativedivides	\(\frac {-\frac {i a^{2} d^{2} \left (i c x +1\right )^{3}}{3}+b^{2} d^{2} \left (-\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}+i \arctan \left (c x \right )^{2} c^{2} x^{2}+\arctan \left (c x \right )^{2} c x -\frac {i \arctan \left (c x \right )^{2}}{3}+\frac {2 i \left (-3 c x \arctan \left (c x \right )+\frac {i c x}{2}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+2 \arctan \left (c x \right )^{2}-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\frac {\ln \left (c x -i\right )^{2}}{2}-\frac {\ln \left (c x +i\right )^{2}}{2}-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{2}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {i \arctan \left (c x \right )}{2}\right )}{3}\right )+2 a \,d^{2} b \left (-\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+i \arctan \left (c x \right ) c^{2} x^{2}+c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right )}{3}+\frac {i \left (-3 c x -\frac {i c^{2} x^{2}}{2}+2 i \ln \left (c^{2} x^{2}+1\right )+4 \arctan \left (c x \right )\right )}{3}\right )}{c}\)	\(354\)
default	\(\frac {-\frac {i a^{2} d^{2} \left (i c x +1\right )^{3}}{3}+b^{2} d^{2} \left (-\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}+i \arctan \left (c x \right )^{2} c^{2} x^{2}+\arctan \left (c x \right )^{2} c x -\frac {i \arctan \left (c x \right )^{2}}{3}+\frac {2 i \left (-3 c x \arctan \left (c x \right )+\frac {i c x}{2}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+2 \arctan \left (c x \right )^{2}-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\frac {\ln \left (c x -i\right )^{2}}{2}-\frac {\ln \left (c x +i\right )^{2}}{2}-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{2}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {i \arctan \left (c x \right )}{2}\right )}{3}\right )+2 a \,d^{2} b \left (-\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+i \arctan \left (c x \right ) c^{2} x^{2}+c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right )}{3}+\frac {i \left (-3 c x -\frac {i c^{2} x^{2}}{2}+2 i \ln \left (c^{2} x^{2}+1\right )+4 \arctan \left (c x \right )\right )}{3}\right )}{c}\)	\(354\)
parts	\(-\frac {i a^{2} d^{2} \left (i c x +1\right )^{3}}{3 c}+\frac {b^{2} d^{2} \left (-\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}+i \arctan \left (c x \right )^{2} c^{2} x^{2}+\arctan \left (c x \right )^{2} c x -\frac {i \arctan \left (c x \right )^{2}}{3}+\frac {2 i \left (-3 c x \arctan \left (c x \right )+\frac {i c x}{2}+2 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )+2 \arctan \left (c x \right )^{2}-\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )+\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\frac {\ln \left (c x -i\right )^{2}}{2}-\frac {\ln \left (c x +i\right )^{2}}{2}-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{2}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {i \arctan \left (c x \right )}{2}\right )}{3}\right )}{c}+\frac {2 a \,d^{2} b \left (-\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+i \arctan \left (c x \right ) c^{2} x^{2}+c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right )}{3}+\frac {i \left (-3 c x -\frac {i c^{2} x^{2}}{2}+2 i \ln \left (c^{2} x^{2}+1\right )+4 \arctan \left (c x \right )\right )}{3}\right )}{c}\)	\(359\)
risch	\(-\frac {b^{2} d^{2} x}{3}+\frac {7 a b \,d^{2}}{3 c}+x \,d^{2} a^{2}+i \ln \left (-i c x +1\right ) x a b \,d^{2}+\frac {4 i b^{2} \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d^{2}}{3 c}-\frac {4 i b^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d^{2}}{3 c}-\frac {4 a b \,d^{2} \ln \left (c^{2} x^{2}+1\right )}{3 c}+\frac {13 b^{2} d^{2} \arctan \left (c x \right )}{18 c}+\frac {43 i b^{2} d^{2} \ln \left (c^{2} x^{2}+1\right )}{36 c}+\frac {4 i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right ) d^{2}}{3 c}-\frac {7 i b^{2} \ln \left (-i c x +1\right ) d^{2}}{18 c}-\frac {7 i \ln \left (-i c x +1\right )^{2} b^{2} d^{2}}{12 c}+\frac {7 i a^{2} d^{2}}{3 c}-\frac {\ln \left (-i c x +1\right )^{2} x \,b^{2} d^{2}}{4}+\frac {a b c \,d^{2} x^{2}}{3}+d^{2} b^{2} \ln \left (-i c x +1\right ) x -2 i a b \,d^{2} x -\frac {a^{2} c^{2} d^{2} x^{3}}{3}+\frac {2 i a b \,d^{2} \arctan \left (c x \right )}{c}+\frac {i d^{2} c \,b^{2} \ln \left (-i c x +1\right ) x^{2}}{6}-\frac {i d^{2} c \ln \left (-i c x +1\right )^{2} x^{2} b^{2}}{4}-d^{2} c \ln \left (-i c x +1\right ) x^{2} a b +\left (-\frac {d^{2} \left (c x -i\right )^{3} b^{2} \ln \left (-i c x +1\right )}{6 c}+\frac {i b \,d^{2} \left (2 a \,c^{3} x^{3}-6 i a \,c^{2} x^{2}-b \,c^{2} x^{2}+6 i b c x -6 c x a +8 b \ln \left (-i c x +1\right )\right )}{6 c}\right ) \ln \left (i c x +1\right )-\frac {i d^{2} c^{2} b a \ln \left (-i c x +1\right ) x^{3}}{3}+\frac {d^{2} \left (c x -i\right )^{3} b^{2} \ln \left (i c x +1\right )^{2}}{12 c}+\frac {d^{2} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}+i a^{2} c \,d^{2} x^{2}-\frac {i b^{2} d^{2}}{3 c}\)	\(578\)

[In]

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

[Out]

1/c*(-1/3*I*a^2*d^2*(1+I*c*x)^3+b^2*d^2*(-1/3*c^3*x^3*arctan(c*x)^2+I*arctan(c*x)^2*c^2*x^2+arctan(c*x)^2*c*x-
1/3*I*arctan(c*x)^2+2/3*I*(-3*c*x*arctan(c*x)+1/2*I*c*x+2*I*arctan(c*x)*ln(c^2*x^2+1)+2*arctan(c*x)^2-ln(c*x-I
)*ln(c^2*x^2+1)+ln(c*x+I)*ln(c^2*x^2+1)+ln(c*x-I)*ln(-1/2*I*(c*x+I))+1/2*ln(c*x-I)^2-1/2*ln(c*x+I)^2-ln(c*x+I)
*ln(1/2*I*(c*x-I))+dilog(-1/2*I*(c*x+I))-dilog(1/2*I*(c*x-I))-1/2*I*arctan(c*x)*c^2*x^2+3/2*ln(c^2*x^2+1)-1/2*
I*arctan(c*x)))+2*a*d^2*b*(-1/3*c^3*x^3*arctan(c*x)+I*arctan(c*x)*c^2*x^2+c*x*arctan(c*x)-1/3*I*arctan(c*x)+1/
3*I*(-3*c*x-1/2*I*c^2*x^2+2*I*ln(c^2*x^2+1)+4*arctan(c*x))))

Fricas [F]

\[ \int (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

1/12*(b^2*c^2*d^2*x^3 - 3*I*b^2*c*d^2*x^2 - 3*b^2*d^2*x)*log(-(c*x + I)/(c*x - I))^2 + integral(-1/3*(3*a^2*c^
4*d^2*x^4 - 6*I*a^2*c^3*d^2*x^3 - 6*I*a^2*c*d^2*x - 3*a^2*d^2 - (-3*I*a*b*c^4*d^2*x^4 - (6*a*b - I*b^2)*c^3*d^
2*x^3 + 3*b^2*c^2*d^2*x^2 - 3*(2*a*b + I*b^2)*c*d^2*x + 3*I*a*b*d^2)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1),
x)

Sympy [F(-1)]

Timed out. \[ \int (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

-1/3*a^2*c^2*d^2*x^3 - 36*b^2*c^4*d^2*integrate(1/48*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) - 3*b^2*c^4*d^2*integ
rate(1/48*x^4*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 4*b^2*c^4*d^2*integrate(1/48*x^4*log(c^2*x^2 + 1)/(c^2*x^
2 + 1), x) + 24*b^2*c^3*d^2*integrate(1/48*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 32*b^2*c^3*d^2
*integrate(1/48*x^3*arctan(c*x)/(c^2*x^2 + 1), x) - 1/3*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4
))*a*b*c^2*d^2 + I*a^2*c*d^2*x^2 + 24*b^2*c^2*d^2*integrate(1/48*x^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 2*I*
(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*c*d^2 + 1/4*b^2*d^2*arctan(c*x)^3/c + 24*b^2*c*d^2*integra
te(1/48*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 24*b^2*c*d^2*integrate(1/48*x*arctan(c*x)/(c^2*x^2
+ 1), x) + a^2*d^2*x + 3*b^2*d^2*integrate(1/48*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + (2*c*x*arctan(c*x) - lo
g(c^2*x^2 + 1))*a*b*d^2/c - 1/12*(b^2*c^2*d^2*x^3 - 3*I*b^2*c*d^2*x^2 - 3*b^2*d^2*x)*arctan(c*x)^2 + 1/12*(-I*
b^2*c^2*d^2*x^3 - 3*b^2*c*d^2*x^2 + 3*I*b^2*d^2*x)*arctan(c*x)*log(c^2*x^2 + 1) + 1/48*(b^2*c^2*d^2*x^3 - 3*I*
b^2*c*d^2*x^2 - 3*b^2*d^2*x)*log(c^2*x^2 + 1)^2 + I*integrate(1/24*(36*(b^2*c^3*d^2*x^3 + b^2*c*d^2*x)*arctan(
c*x)^2 + 3*(b^2*c^3*d^2*x^3 + b^2*c*d^2*x)*log(c^2*x^2 + 1)^2 + 4*(b^2*c^4*d^2*x^4 - 6*b^2*c^2*d^2*x^2)*arctan
(c*x) + 2*(4*b^2*c^3*d^2*x^3 - 3*b^2*c*d^2*x + 3*(b^2*c^4*d^2*x^4 - b^2*d^2)*arctan(c*x))*log(c^2*x^2 + 1))/(c
^2*x^2 + 1), x)

Giac [F]

\[ \int (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int (d+i c d x)^2 (a+b \arctan (c x))^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2 \,d x \]

[In]

int((a + b*atan(c*x))^2*(d + c*d*x*1i)^2,x)

[Out]

int((a + b*atan(c*x))^2*(d + c*d*x*1i)^2, x)