\(\int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx\) [87]

Optimal result

Integrand size = 22, antiderivative size = 226 \[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=-\frac {7}{2} i a b d^3 x-b^2 d^3 x-\frac {1}{12} i b^2 c d^3 x^2+\frac {b^2 d^3 \arctan (c x)}{c}-\frac {7}{2} i b^2 d^3 x \arctan (c x)+b c d^3 x^2 (a+b \arctan (c x))+\frac {1}{6} i b c^2 d^3 x^3 (a+b \arctan (c x))-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}+\frac {4 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{c}+\frac {11 i b^2 d^3 \log \left (1+c^2 x^2\right )}{6 c}-\frac {2 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c} \]

[Out]

-7/2*I*a*b*d^3*x-b^2*d^3*x-1/12*I*b^2*c*d^3*x^2+b^2*d^3*arctan(c*x)/c-7/2*I*b^2*d^3*x*arctan(c*x)+b*c*d^3*x^2*
(a+b*arctan(c*x))+1/6*I*b*c^2*d^3*x^3*(a+b*arctan(c*x))-1/4*I*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/c+4*b*d^3*(a
+b*arctan(c*x))*ln(2/(1-I*c*x))/c+11/6*I*b^2*d^3*ln(c^2*x^2+1)/c-2*I*b^2*d^3*polylog(2,1-2/(1-I*c*x))/c

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {4974, 4930, 266, 4946, 327, 209, 272, 45, 1600, 4964, 2449, 2352} \[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\frac {1}{6} i b c^2 d^3 x^3 (a+b \arctan (c x))+b c d^3 x^2 (a+b \arctan (c x))-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}+\frac {4 b d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{c}-\frac {7}{2} i a b d^3 x+\frac {b^2 d^3 \arctan (c x)}{c}-\frac {7}{2} i b^2 d^3 x \arctan (c x)+\frac {11 i b^2 d^3 \log \left (c^2 x^2+1\right )}{6 c}-\frac {2 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c}-\frac {1}{12} i b^2 c d^3 x^2-b^2 d^3 x \]

[In]

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

[Out]

((-7*I)/2)*a*b*d^3*x - b^2*d^3*x - (I/12)*b^2*c*d^3*x^2 + (b^2*d^3*ArcTan[c*x])/c - ((7*I)/2)*b^2*d^3*x*ArcTan
[c*x] + b*c*d^3*x^2*(a + b*ArcTan[c*x]) + (I/6)*b*c^2*d^3*x^3*(a + b*ArcTan[c*x]) - ((I/4)*d^3*(1 + I*c*x)^4*(
a + b*ArcTan[c*x])^2)/c + (4*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 - I*c*x)])/c + (((11*I)/6)*b^2*d^3*Log[1 + c^2
*x^2])/c - ((2*I)*b^2*d^3*PolyLog[2, 1 - 2/(1 - I*c*x)])/c

Rule 45

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 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 272

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 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^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}+\frac {(i b) \int \left (-7 d^4 (a+b \arctan (c x))-4 i c d^4 x (a+b \arctan (c x))+c^2 d^4 x^2 (a+b \arctan (c x))-\frac {8 i \left (i d^4-c d^4 x\right ) (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{2 d} \\ & = -\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}+\frac {(4 b) \int \frac {\left (i d^4-c d^4 x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{d}-\frac {1}{2} \left (7 i b d^3\right ) \int (a+b \arctan (c x)) \, dx+\left (2 b c d^3\right ) \int x (a+b \arctan (c x)) \, dx+\frac {1}{2} \left (i b c^2 d^3\right ) \int x^2 (a+b \arctan (c x)) \, dx \\ & = -\frac {7}{2} i a b d^3 x+b c d^3 x^2 (a+b \arctan (c x))+\frac {1}{6} i b c^2 d^3 x^3 (a+b \arctan (c x))-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}+\frac {(4 b) \int \frac {a+b \arctan (c x)}{-\frac {i}{d^4}-\frac {c x}{d^4}} \, dx}{d}-\frac {1}{2} \left (7 i b^2 d^3\right ) \int \arctan (c x) \, dx-\left (b^2 c^2 d^3\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {1}{6} \left (i b^2 c^3 d^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx \\ & = -\frac {7}{2} i a b d^3 x-b^2 d^3 x-\frac {7}{2} i b^2 d^3 x \arctan (c x)+b c d^3 x^2 (a+b \arctan (c x))+\frac {1}{6} i b c^2 d^3 x^3 (a+b \arctan (c x))-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}+\frac {4 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{c}+\left (b^2 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx-\left (4 b^2 d^3\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx+\frac {1}{2} \left (7 i b^2 c d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx-\frac {1}{12} \left (i b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {7}{2} i a b d^3 x-b^2 d^3 x+\frac {b^2 d^3 \arctan (c x)}{c}-\frac {7}{2} i b^2 d^3 x \arctan (c x)+b c d^3 x^2 (a+b \arctan (c x))+\frac {1}{6} i b c^2 d^3 x^3 (a+b \arctan (c x))-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}+\frac {4 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{c}+\frac {7 i b^2 d^3 \log \left (1+c^2 x^2\right )}{4 c}-\frac {\left (4 i b^2 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{c}-\frac {1}{12} \left (i b^2 c^3 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = -\frac {7}{2} i a b d^3 x-b^2 d^3 x-\frac {1}{12} i b^2 c d^3 x^2+\frac {b^2 d^3 \arctan (c x)}{c}-\frac {7}{2} i b^2 d^3 x \arctan (c x)+b c d^3 x^2 (a+b \arctan (c x))+\frac {1}{6} i b c^2 d^3 x^3 (a+b \arctan (c x))-\frac {i d^3 (1+i c x)^4 (a+b \arctan (c x))^2}{4 c}+\frac {4 b d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{c}+\frac {11 i b^2 d^3 \log \left (1+c^2 x^2\right )}{6 c}-\frac {2 i b^2 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{c} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.65 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.18 \[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=-\frac {i d^3 \left (b^2+12 i a^2 c x+42 a b c x-12 i b^2 c x-18 a^2 c^2 x^2+12 i a b c^2 x^2+b^2 c^2 x^2-12 i a^2 c^3 x^3-2 a b c^3 x^3+3 a^2 c^4 x^4+3 b^2 (-i+c x)^4 \arctan (c x)^2+2 b \arctan (c x) \left (b \left (6 i+21 c x+6 i c^2 x^2-c^3 x^3\right )+3 a \left (-7+4 i c x-6 c^2 x^2-4 i c^3 x^3+c^4 x^4\right )+24 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-24 i a b \log \left (1+c^2 x^2\right )-22 b^2 \log \left (1+c^2 x^2\right )+24 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{12 c} \]

[In]

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

[Out]

((-1/12*I)*d^3*(b^2 + (12*I)*a^2*c*x + 42*a*b*c*x - (12*I)*b^2*c*x - 18*a^2*c^2*x^2 + (12*I)*a*b*c^2*x^2 + b^2
*c^2*x^2 - (12*I)*a^2*c^3*x^3 - 2*a*b*c^3*x^3 + 3*a^2*c^4*x^4 + 3*b^2*(-I + c*x)^4*ArcTan[c*x]^2 + 2*b*ArcTan[
c*x]*(b*(6*I + 21*c*x + (6*I)*c^2*x^2 - c^3*x^3) + 3*a*(-7 + (4*I)*c*x - 6*c^2*x^2 - (4*I)*c^3*x^3 + c^4*x^4)
+ (24*I)*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - (24*I)*a*b*Log[1 + c^2*x^2] - 22*b^2*Log[1 + c^2*x^2] + 24*b^2*Po
lyLog[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. 411 vs. \(2 (204 ) = 408\).

Time = 1.52 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.82

method	result	size
derivativedivides	\(\frac {-\frac {i d^{3} a^{2} \left (i c x +1\right )^{4}}{4}+b^{2} d^{3} \left (-\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )^{2}+\frac {3 i \arctan \left (c x \right )^{2} c^{2} x^{2}}{2}+\arctan \left (c x \right )^{2} c x -\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \left (-7 c x \arctan \left (c x \right )+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+4 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-2 i \arctan \left (c x \right ) c^{2} x^{2}+4 \arctan \left (c x \right )^{2}-2 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right )^{2}-\ln \left (c x +i\right )^{2}-2 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+2 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-2 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-2 i \arctan \left (c x \right )-\frac {c^{2} x^{2}}{6}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{3}+2 i c x \right )}{2}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )+\frac {3 i \arctan \left (c x \right ) c^{2} x^{2}}{2}+c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right )}{4}+\frac {i \left (-7 c x +\frac {c^{3} x^{3}}{3}-2 i c^{2} x^{2}+4 i \ln \left (c^{2} x^{2}+1\right )+8 \arctan \left (c x \right )\right )}{4}\right )}{c}\)	\(412\)
default	\(\frac {-\frac {i d^{3} a^{2} \left (i c x +1\right )^{4}}{4}+b^{2} d^{3} \left (-\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )^{2}+\frac {3 i \arctan \left (c x \right )^{2} c^{2} x^{2}}{2}+\arctan \left (c x \right )^{2} c x -\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \left (-7 c x \arctan \left (c x \right )+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+4 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-2 i \arctan \left (c x \right ) c^{2} x^{2}+4 \arctan \left (c x \right )^{2}-2 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right )^{2}-\ln \left (c x +i\right )^{2}-2 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+2 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-2 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-2 i \arctan \left (c x \right )-\frac {c^{2} x^{2}}{6}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{3}+2 i c x \right )}{2}\right )+2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )+\frac {3 i \arctan \left (c x \right ) c^{2} x^{2}}{2}+c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right )}{4}+\frac {i \left (-7 c x +\frac {c^{3} x^{3}}{3}-2 i c^{2} x^{2}+4 i \ln \left (c^{2} x^{2}+1\right )+8 \arctan \left (c x \right )\right )}{4}\right )}{c}\)	\(412\)
parts	\(-\frac {i d^{3} a^{2} \left (i c x +1\right )^{4}}{4 c}+\frac {b^{2} d^{3} \left (-\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )^{2}+\frac {3 i \arctan \left (c x \right )^{2} c^{2} x^{2}}{2}+\arctan \left (c x \right )^{2} c x -\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \left (-7 c x \arctan \left (c x \right )+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+4 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )-2 i \arctan \left (c x \right ) c^{2} x^{2}+4 \arctan \left (c x \right )^{2}-2 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )+2 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )+\ln \left (c x -i\right )^{2}-\ln \left (c x +i\right )^{2}-2 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )+2 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-2 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-2 i \arctan \left (c x \right )-\frac {c^{2} x^{2}}{6}+\frac {11 \ln \left (c^{2} x^{2}+1\right )}{3}+2 i c x \right )}{2}\right )}{c}+\frac {2 a \,d^{3} b \left (-\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}-c^{3} x^{3} \arctan \left (c x \right )+\frac {3 i \arctan \left (c x \right ) c^{2} x^{2}}{2}+c x \arctan \left (c x \right )-\frac {i \arctan \left (c x \right )}{4}+\frac {i \left (-7 c x +\frac {c^{3} x^{3}}{3}-2 i c^{2} x^{2}+4 i \ln \left (c^{2} x^{2}+1\right )+8 \arctan \left (c x \right )\right )}{4}\right )}{c}\)	\(417\)
risch	\(-b^{2} d^{3} x -\frac {2 b \ln \left (c^{2} x^{2}+1\right ) a \,d^{3}}{c}+\frac {47 b^{2} d^{3} \arctan \left (c x \right )}{32 c}+\frac {14 a b \,d^{3}}{3 c}+x \,d^{3} a^{2}-\frac {7 i a b \,d^{3} x}{2}+\frac {7 d^{3} b^{2} \ln \left (-i c x +1\right ) x}{4}+a b c \,d^{3} x^{2}-a^{2} c^{2} d^{3} x^{3}+i \ln \left (-i c x +1\right ) x a b \,d^{3}+\frac {i d^{3} \left (c x -i\right )^{4} b^{2} \ln \left (i c x +1\right )^{2}}{16 c}-\frac {\ln \left (-i c x +1\right )^{2} x \,b^{2} d^{3}}{4}-\frac {13 i b^{2} d^{3}}{12 c}+\frac {15 i d^{3} a^{2}}{4 c}+\frac {i d^{3} c^{3} b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}-\frac {3 i d^{3} c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}+\frac {2 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^{3}}{c}-\frac {2 i b^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d^{3}}{c}+\frac {7 i b \arctan \left (c x \right ) a \,d^{3}}{2 c}-\frac {3 d^{3} c a b \ln \left (-i c x +1\right ) x^{2}}{2}+\frac {d^{3} c^{3} b a \ln \left (-i c x +1\right ) x^{4}}{4}+\frac {i d^{3} c^{2} a b \,x^{3}}{6}+\frac {i b^{2} d^{3} c \ln \left (-i c x +1\right ) x^{2}}{2}-i d^{3} c^{2} a b \ln \left (-i c x +1\right ) x^{3}+\left (-\frac {i d^{3} \left (c x -i\right )^{4} b^{2} \ln \left (-i c x +1\right )}{8 c}-\frac {b \,d^{3} \left (3 a \,c^{4} x^{4}-12 i a \,c^{3} x^{3}-b \,c^{3} x^{3}+6 i b \,c^{2} x^{2}-18 c^{2} x^{2} a +12 i x a c -24 i b \ln \left (-i c x +1\right )+21 x b c \right )}{12 c}\right ) \ln \left (i c x +1\right )-\frac {15 i d^{3} b^{2} \ln \left (-i c x +1\right )^{2}}{16 c}-\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right ) x^{3}}{12}+\frac {d^{3} c^{2} b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{4}-\frac {i a^{2} c^{3} d^{3} x^{4}}{4}+\frac {3 i d^{3} c \,x^{2} a^{2}}{2}+\frac {397 i b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right )}{192 c}-\frac {15 i b^{2} \ln \left (-i c x +1\right ) d^{3}}{32 c}+\frac {2 i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right ) d^{3}}{c}-\frac {i b^{2} c \,d^{3} x^{2}}{12}\)	\(709\)

[In]

int((d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/c*(-1/4*I*d^3*a^2*(1+I*c*x)^4+b^2*d^3*(-1/4*I*arctan(c*x)^2*c^4*x^4-c^3*x^3*arctan(c*x)^2+3/2*I*arctan(c*x)^
2*c^2*x^2+arctan(c*x)^2*c*x-1/4*I*arctan(c*x)^2+1/2*I*(-7*c*x*arctan(c*x)+1/3*c^3*x^3*arctan(c*x)+4*I*arctan(c
*x)*ln(c^2*x^2+1)-2*I*arctan(c*x)*c^2*x^2+4*arctan(c*x)^2-2*ln(c*x-I)*ln(c^2*x^2+1)+2*ln(c*x+I)*ln(c^2*x^2+1)+
2*ln(c*x-I)*ln(-1/2*I*(c*x+I))+ln(c*x-I)^2-ln(c*x+I)^2-2*ln(c*x+I)*ln(1/2*I*(c*x-I))+2*dilog(-1/2*I*(c*x+I))-2
*dilog(1/2*I*(c*x-I))-2*I*arctan(c*x)-1/6*c^2*x^2+11/3*ln(c^2*x^2+1)+2*I*c*x))+2*a*d^3*b*(-1/4*I*arctan(c*x)*c
^4*x^4-c^3*x^3*arctan(c*x)+3/2*I*arctan(c*x)*c^2*x^2+c*x*arctan(c*x)-1/4*I*arctan(c*x)+1/4*I*(-7*c*x+1/3*c^3*x
^3-2*I*c^2*x^2+4*I*ln(c^2*x^2+1)+8*arctan(c*x))))

Fricas [F]

\[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

1/16*(I*b^2*c^3*d^3*x^4 + 4*b^2*c^2*d^3*x^3 - 6*I*b^2*c*d^3*x^2 - 4*b^2*d^3*x)*log(-(c*x + I)/(c*x - I))^2 + i
ntegral(1/4*(-4*I*a^2*c^5*d^3*x^5 - 12*a^2*c^4*d^3*x^4 + 8*I*a^2*c^3*d^3*x^3 - 8*a^2*c^2*d^3*x^2 + 12*I*a^2*c*
d^3*x + 4*a^2*d^3 + (4*a*b*c^5*d^3*x^5 + (-12*I*a*b - b^2)*c^4*d^3*x^4 - 4*(2*a*b - I*b^2)*c^3*d^3*x^3 - 2*(4*
I*a*b - 3*b^2)*c^2*d^3*x^2 - 4*(3*a*b + I*b^2)*c*d^3*x + 4*I*a*b*d^3)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1)
, x)

Sympy [F(-1)]

Timed out. \[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

-1/4*I*a^2*c^3*d^3*x^4 - 4*b^2*c^5*d^3*integrate(1/16*x^5*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 2*b
^2*c^5*d^3*integrate(1/16*x^5*arctan(c*x)/(c^2*x^2 + 1), x) - a^2*c^2*d^3*x^3 - 36*b^2*c^4*d^3*integrate(1/16*
x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) - 3*b^2*c^4*d^3*integrate(1/16*x^4*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) -
5*b^2*c^4*d^3*integrate(1/16*x^4*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) - 1/6*I*(3*x^4*arctan(c*x) - c*((c^2*x^3 -
3*x)/c^4 + 3*arctan(c*x)/c^5))*a*b*c^3*d^3 + 8*b^2*c^3*d^3*integrate(1/16*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c
^2*x^2 + 1), x) + 20*b^2*c^3*d^3*integrate(1/16*x^3*arctan(c*x)/(c^2*x^2 + 1), x) - (2*x^3*arctan(c*x) - c*(x^
2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*c^2*d^3 + 3/2*I*a^2*c*d^3*x^2 - 24*b^2*c^2*d^3*integrate(1/16*x^2*arctan(c*
x)^2/(c^2*x^2 + 1), x) - 2*b^2*c^2*d^3*integrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 10*b^2*c^2*d^
3*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3*I*(x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*
a*b*c*d^3 + 1/4*b^2*d^3*arctan(c*x)^3/c + 12*b^2*c*d^3*integrate(1/16*x*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2
+ 1), x) - 8*b^2*c*d^3*integrate(1/16*x*arctan(c*x)/(c^2*x^2 + 1), x) + a^2*d^3*x + b^2*d^3*integrate(1/16*log
(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + (2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*d^3/c + 1/16*(-I*b^2*c^3*d^3*x^
4 - 4*b^2*c^2*d^3*x^3 + 6*I*b^2*c*d^3*x^2 + 4*b^2*d^3*x)*arctan(c*x)^2 + 1/16*(b^2*c^3*d^3*x^4 - 4*I*b^2*c^2*d
^3*x^3 - 6*b^2*c*d^3*x^2 + 4*I*b^2*d^3*x)*arctan(c*x)*log(c^2*x^2 + 1) - 1/64*(-I*b^2*c^3*d^3*x^4 - 4*b^2*c^2*
d^3*x^3 + 6*I*b^2*c*d^3*x^2 + 4*b^2*d^3*x)*log(c^2*x^2 + 1)^2 - I*integrate(1/16*(12*(b^2*c^5*d^3*x^5 - 2*b^2*
c^3*d^3*x^3 - 3*b^2*c*d^3*x)*arctan(c*x)^2 + (b^2*c^5*d^3*x^5 - 2*b^2*c^3*d^3*x^3 - 3*b^2*c*d^3*x)*log(c^2*x^2
+ 1)^2 - 10*(b^2*c^4*d^3*x^4 - 2*b^2*c^2*d^3*x^2)*arctan(c*x) + (b^2*c^5*d^3*x^5 - 10*b^2*c^3*d^3*x^3 + 4*b^2
*c*d^3*x - 4*(3*b^2*c^4*d^3*x^4 + 2*b^2*c^2*d^3*x^2 - b^2*d^3)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x
)

Giac [F]

\[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int (d+i c d x)^3 (a+b \arctan (c x))^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3 \,d x \]

[In]

int((a + b*atan(c*x))^2*(d + c*d*x*1i)^3,x)

[Out]

int((a + b*atan(c*x))^2*(d + c*d*x*1i)^3, x)