\(\int x (d+i c d x) (a+b \arctan (c x))^2 \, dx\) [70]

Optimal result

Integrand size = 21, antiderivative size = 211 \[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=-\frac {a b d x}{c}+\frac {i b^2 d x}{3 c}-\frac {i b^2 d \arctan (c x)}{3 c^2}-\frac {b^2 d x \arctan (c x)}{c}-\frac {1}{3} i b d x^2 (a+b \arctan (c x))+\frac {5 d (a+b \arctan (c x))^2}{6 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2-\frac {2 i b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^2}+\frac {b^2 d \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^2} \]

[Out]

-a*b*d*x/c+1/3*I*b^2*d*x/c-1/3*I*b^2*d*arctan(c*x)/c^2-b^2*d*x*arctan(c*x)/c-1/3*I*b*d*x^2*(a+b*arctan(c*x))+5
/6*d*(a+b*arctan(c*x))^2/c^2+1/2*d*x^2*(a+b*arctan(c*x))^2+1/3*I*c*d*x^3*(a+b*arctan(c*x))^2-2/3*I*b*d*(a+b*ar
ctan(c*x))*ln(2/(1+I*c*x))/c^2+1/2*b^2*d*ln(c^2*x^2+1)/c^2+1/3*b^2*d*polylog(2,1-2/(1+I*c*x))/c^2

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4996, 4946, 5036, 4930, 266, 5004, 327, 209, 5040, 4964, 2449, 2352} \[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {5 d (a+b \arctan (c x))^2}{6 c^2}-\frac {2 i b d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^2}+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2+\frac {1}{2} d x^2 (a+b \arctan (c x))^2-\frac {1}{3} i b d x^2 (a+b \arctan (c x))-\frac {a b d x}{c}-\frac {i b^2 d \arctan (c x)}{3 c^2}-\frac {b^2 d x \arctan (c x)}{c}+\frac {b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^2}+\frac {b^2 d \log \left (c^2 x^2+1\right )}{2 c^2}+\frac {i b^2 d x}{3 c} \]

[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 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 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 4996

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 5004

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 5036

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 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[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]

Rubi steps \begin{align*} \text {integral}& = \int \left (d x (a+b \arctan (c x))^2+i c d x^2 (a+b \arctan (c x))^2\right ) \, dx \\ & = d \int x (a+b \arctan (c x))^2 \, dx+(i c d) \int x^2 (a+b \arctan (c x))^2 \, dx \\ & = \frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2-(b c d) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{3} \left (2 i b c^2 d\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2-\frac {1}{3} (2 i b d) \int x (a+b \arctan (c x)) \, dx+\frac {1}{3} (2 i b d) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {(b d) \int (a+b \arctan (c x)) \, dx}{c}+\frac {(b d) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c} \\ & = -\frac {a b d x}{c}-\frac {1}{3} i b d x^2 (a+b \arctan (c x))+\frac {5 d (a+b \arctan (c x))^2}{6 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2-\frac {(2 i b d) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c}-\frac {\left (b^2 d\right ) \int \arctan (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 \arctan (c x)}{c}-\frac {1}{3} i b d x^2 (a+b \arctan (c x))+\frac {5 d (a+b \arctan (c x))^2}{6 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2-\frac {2 i b d (a+b \arctan (c x)) \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 \arctan (c x)}{3 c^2}-\frac {b^2 d x \arctan (c x)}{c}-\frac {1}{3} i b d x^2 (a+b \arctan (c x))+\frac {5 d (a+b \arctan (c x))^2}{6 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2-\frac {2 i b d (a+b \arctan (c x)) \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 ) \text {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 \arctan (c x)}{3 c^2}-\frac {b^2 d x \arctan (c x)}{c}-\frac {1}{3} i b d x^2 (a+b \arctan (c x))+\frac {5 d (a+b \arctan (c x))^2}{6 c^2}+\frac {1}{2} d x^2 (a+b \arctan (c x))^2+\frac {1}{3} i c d x^3 (a+b \arctan (c x))^2-\frac {2 i b d (a+b \arctan (c x)) \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 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.99 \[ \int x (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {d \left (-6 a b c x+2 i b^2 c x+3 a^2 c^2 x^2-2 i a b c^2 x^2+2 i a^2 c^3 x^3+b^2 \left (1+3 c^2 x^2+2 i c^3 x^3\right ) \arctan (c x)^2+2 b \arctan (c x) \left (-i b \left (1-3 i c x+c^2 x^2\right )+a \left (3+3 c^2 x^2+2 i c^3 x^3\right )-2 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )+2 i a b \log \left (1+c^2 x^2\right )+3 b^2 \log \left (1+c^2 x^2\right )-2 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{6 c^2} \]

[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)

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.50

method	result	size
parts	\(a^{2} d \left (\frac {1}{3} i c \,x^{3}+\frac {1}{2} x^{2}\right )+\frac {d \,b^{2} \left (\frac {i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{3}+\frac {i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\arctan \left (c x \right )^{2}}{2}-c x \arctan \left (c x \right )-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right )^{2}}{12}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right )^{2}}{12}-\frac {i \arctan \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i c x}{3}\right )}{c^{2}}+\frac {2 a b d \left (\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {i c^{2} x^{2}}{6}-\frac {c x}{2}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\arctan \left (c x \right )}{2}\right )}{c^{2}}\)	\(316\)
derivativedivides	\(\frac {a^{2} d \left (\frac {1}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d \,b^{2} \left (\frac {i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{3}+\frac {i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\arctan \left (c x \right )^{2}}{2}-c x \arctan \left (c x \right )-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right )^{2}}{12}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right )^{2}}{12}-\frac {i \arctan \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i c x}{3}\right )+2 a b d \left (\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {i c^{2} x^{2}}{6}-\frac {c x}{2}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\arctan \left (c x \right )}{2}\right )}{c^{2}}\)	\(319\)
default	\(\frac {a^{2} d \left (\frac {1}{3} i c^{3} x^{3}+\frac {1}{2} c^{2} x^{2}\right )+d \,b^{2} \left (\frac {i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )^{2}}{2}-\frac {i \arctan \left (c x \right ) c^{2} x^{2}}{3}+\frac {i \arctan \left (c x \right )^{2} c^{3} x^{3}}{3}+\frac {\arctan \left (c x \right )^{2}}{2}-c x \arctan \left (c x \right )-\frac {\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{6}+\frac {\ln \left (c x -i\right )^{2}}{12}+\frac {\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{6}-\frac {\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{6}-\frac {\ln \left (c x +i\right )^{2}}{12}-\frac {i \arctan \left (c x \right )}{3}+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}+\frac {i c x}{3}\right )+2 a b d \left (\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{3}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{2}-\frac {i c^{2} x^{2}}{6}-\frac {c x}{2}+\frac {i \ln \left (c^{2} x^{2}+1\right )}{6}+\frac {\arctan \left (c x \right )}{2}\right )}{c^{2}}\)	\(319\)
risch	\(\frac {a^{2} d \,x^{2}}{2}+\frac {73 b^{2} d \ln \left (c^{2} x^{2}+1\right )}{144 c^{2}}-\frac {a b d x}{c}-\frac {b^{2} d}{3 c^{2}}-\frac {i b d \,x^{2} a}{3}+\frac {d \,b^{2} \ln \left (-i c x +1\right ) x^{2}}{6}-\frac {i d \,b^{2} \ln \left (-i c x +1\right ) x}{2 c}+\frac {i b d a \ln \left (c^{2} x^{2}+1\right )}{3 c^{2}}-\frac {d c a b \ln \left (-i c x +1\right ) x^{3}}{3}-\frac {i d c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}+\frac {i d a b \ln \left (-i c x +1\right ) x^{2}}{2}-\frac {i d \,b^{2} \left (2 c^{3} x^{3}-3 i c^{2} x^{2}-i\right ) \ln \left (i c x +1\right )^{2}}{24 c^{2}}+\frac {b d a \arctan \left (c x \right )}{c^{2}}+\frac {5 d \,a^{2}}{6 c^{2}}+\frac {i b^{2} d x}{3 c}-\frac {25 i b^{2} d \arctan \left (c x \right )}{72 c^{2}}-\frac {4 i d a b}{3 c^{2}}+\frac {b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{2}}-\frac {b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{2}}+\frac {i a^{2} c d \,x^{3}}{3}+\frac {b^{2} d \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{2}}-\frac {d \,b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}-\frac {5 d \,b^{2} \ln \left (-i c x +1\right )^{2}}{24 c^{2}}-\frac {d \,b^{2} \ln \left (-i c x +1\right )}{72 c^{2}}+\left (\frac {i d \,b^{2} \left (2 c \,x^{3}-3 i x^{2}\right ) \ln \left (-i c x +1\right )}{12}+\frac {b d \left (4 a \,c^{3} x^{3}-6 i a \,c^{2} x^{2}-2 b \,c^{2} x^{2}+6 i b c x +5 b \ln \left (-i c x +1\right )\right )}{12 c^{2}}\right ) \ln \left (i c x +1\right )\)	\(485\)

[In]

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

[Out]

a^2*d*(1/3*I*x^3*c+1/2*x^2)+d*b^2/c^2*(1/3*I*arctan(c*x)*ln(c^2*x^2+1)+1/2*c^2*x^2*arctan(c*x)^2-1/3*I*arctan(
c*x)*c^2*x^2+1/3*I*arctan(c*x)^2*c^3*x^3+1/2*arctan(c*x)^2-c*x*arctan(c*x)-1/6*ln(c*x-I)*ln(c^2*x^2+1)+1/6*dil
og(-1/2*I*(c*x+I))+1/6*ln(c*x-I)*ln(-1/2*I*(c*x+I))+1/12*ln(c*x-I)^2+1/6*ln(c*x+I)*ln(c^2*x^2+1)-1/6*dilog(1/2
*I*(c*x-I))-1/6*ln(c*x+I)*ln(1/2*I*(c*x-I))-1/12*ln(c*x+I)^2-1/3*I*arctan(c*x)+1/2*ln(c^2*x^2+1)+1/3*I*c*x)+2*
a*b*d/c^2*(1/3*I*arctan(c*x)*c^3*x^3+1/2*c^2*x^2*arctan(c*x)-1/6*I*c^2*x^2-1/2*c*x+1/6*I*ln(c^2*x^2+1)+1/2*arc
tan(c*x))

Fricas [F]

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

[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 + 2*(-3*I*a*b - 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)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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