\(\int \frac {x^2 (a+b \arctan (c x))}{d+i c d x} \, dx\) [44]

Optimal result

Integrand size = 23, antiderivative size = 156 \[ \int \frac {x^2 (a+b \arctan (c x))}{d+i c d x} \, dx=\frac {a x}{c^2 d}+\frac {i b x}{2 c^2 d}-\frac {i b \arctan (c x)}{2 c^3 d}+\frac {b x \arctan (c x)}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))}{2 c d}-\frac {i (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4986, 4946, 327, 209, 4930, 266, 4964, 2449, 2352} \[ \int \frac {x^2 (a+b \arctan (c x))}{d+i c d x} \, dx=-\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^3 d}-\frac {i x^2 (a+b \arctan (c x))}{2 c d}+\frac {a x}{c^2 d}-\frac {i b \arctan (c x)}{2 c^3 d}+\frac {b x \arctan (c x)}{c^2 d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c^3 d}+\frac {i b x}{2 c^2 d}-\frac {b \log \left (c^2 x^2+1\right )}{2 c^3 d} \]

[In]

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

[Out]

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

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 4986

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[f/e,
Int[(f*x)^(m - 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f/e), Int[(f*x)^(m - 1)*((a + b*ArcTan[c*x])^p/(d +
e*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0] && GtQ[m, 0]

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 (a+b \arctan (c x))}{d+i c d x} \, dx=-\frac {-2 a c x-i b c x+i a c^2 x^2+2 b \arctan (c x)^2+i \arctan (c x) \left (-2 i a+b+2 i b c x+b c^2 x^2+2 b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-i a \log \left (1+c^2 x^2\right )+b \log \left (1+c^2 x^2\right )+b \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{2 c^3 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.71

method	result	size
derivativedivides	\(\frac {\frac {a c x}{d}-\frac {i a \,c^{2} x^{2}}{2 d}+\frac {i b c x}{2 d}-\frac {a \arctan \left (c x \right )}{d}+\frac {b \arctan \left (c x \right ) c x}{d}+\frac {i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d}-\frac {i b \arctan \left (\frac {c x}{2}\right )}{8 d}+\frac {b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d}+\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d}-\frac {b \ln \left (c x -i\right )^{2}}{4 d}+\frac {b}{2 d}+\frac {i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 d}-\frac {b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 d}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 d}-\frac {i b \arctan \left (c x \right ) c^{2} x^{2}}{2 d}-\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 d}-\frac {3 i b \arctan \left (c x \right )}{4 d}}{c^{3}}\)	\(267\)
default	\(\frac {\frac {a c x}{d}-\frac {i a \,c^{2} x^{2}}{2 d}+\frac {i b c x}{2 d}-\frac {a \arctan \left (c x \right )}{d}+\frac {b \arctan \left (c x \right ) c x}{d}+\frac {i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d}-\frac {i b \arctan \left (\frac {c x}{2}\right )}{8 d}+\frac {b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d}+\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d}-\frac {b \ln \left (c x -i\right )^{2}}{4 d}+\frac {b}{2 d}+\frac {i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 d}-\frac {b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 d}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 d}-\frac {i b \arctan \left (c x \right ) c^{2} x^{2}}{2 d}-\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 d}-\frac {3 i b \arctan \left (c x \right )}{4 d}}{c^{3}}\)	\(267\)
risch	\(\frac {b \ln \left (i c x +1\right )^{2}}{4 c^{3} d}-\frac {b \left (\frac {1}{2} c \,x^{2}+i x \right ) \ln \left (i c x +1\right )}{2 c^{2} d}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 c^{3} d}+\frac {a x}{d \,c^{2}}-\frac {a \arctan \left (c x \right )}{c^{3} d}+\frac {i b x}{2 c^{2} d}-\frac {i a \,x^{2}}{2 c d}+\frac {x^{2} b \ln \left (-i c x +1\right )}{4 c d}+\frac {i a}{2 c^{3} d}-\frac {\ln \left (-i c x +1\right ) b}{4 c^{3} d}+\frac {i b x \ln \left (-i c x +1\right )}{2 c^{2} d}+\frac {b}{8 c^{3} d}+\frac {b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 c^{3} d}-\frac {b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 c^{3} d}+\frac {b \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 c^{3} d}-\frac {3 i b \arctan \left (c x \right )}{4 c^{3} d}-\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 c^{3} d}\)	\(290\)
parts	\(\frac {i b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 c^{3} d}+\frac {a x}{d \,c^{2}}+\frac {i b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 c^{3} d}-\frac {a \arctan \left (c x \right )}{c^{3} d}+\frac {b x \arctan \left (c x \right )}{c^{2} d}+\frac {i a \ln \left (c^{2} x^{2}+1\right )}{2 c^{3} d}+\frac {i b x}{2 c^{2} d}+\frac {b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{3} d}+\frac {b \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{3} d}-\frac {b \ln \left (c x -i\right )^{2}}{4 c^{3} d}-\frac {i a \,x^{2}}{2 c d}+\frac {b}{2 c^{3} d}-\frac {b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 c^{3} d}-\frac {i b \arctan \left (\frac {c x}{2}\right )}{8 c^{3} d}-\frac {3 i b \arctan \left (c x \right )}{4 c^{3} d}+\frac {i b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{3} d}-\frac {3 b \ln \left (c^{2} x^{2}+1\right )}{8 c^{3} d}-\frac {i b \arctan \left (c x \right ) x^{2}}{2 c d}\)	\(308\)

[In]

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

[Out]

1/c^3*(1/d*a*c*x-1/2*I/d*a*c^2*x^2+1/2*I/d*b*c*x-1/d*a*arctan(c*x)+1/d*b*arctan(c*x)*c*x+I/d*b*arctan(c*x)*ln(
c*x-I)-1/8*I/d*b*arctan(1/2*c*x)+1/2/d*b*ln(-1/2*I*(c*x+I))*ln(c*x-I)+1/2/d*b*dilog(-1/2*I*(c*x+I))-1/4/d*b*ln
(c*x-I)^2+1/2/d*b+1/4*I/d*b*arctan(1/2*c*x-1/2*I)-1/16/d*b*ln(c^4*x^4+10*c^2*x^2+9)+1/2*I/d*a*ln(c^2*x^2+1)+1/
8*I/d*b*arctan(1/6*c^3*x^3+7/6*c*x)-1/2*I/d*b*arctan(c*x)*c^2*x^2-3/8/d*b*ln(c^2*x^2+1)-3/4*I/d*b*arctan(c*x))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2 (a+b \arctan (c x))}{d+i c d x} \, dx=- \frac {i \left (\int \frac {2 b \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {4 a c^{3} x^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac {b c^{2} x^{2}}{c^{2} x^{2} + 1}\, dx + \int \frac {4 i a c^{2} x^{2}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {2 i b c x}{c^{2} x^{2} + 1}\right )\, dx + \int \left (- \frac {i b c^{3} x^{3}}{c^{2} x^{2} + 1}\right )\, dx + \int \frac {2 b c^{2} x^{2} \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {2 i b c x \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {2 i b c^{3} x^{3} \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\right )\, dx\right )}{4 c^{2} d} + \frac {\left (b c^{2} x^{2} + 2 i b c x - 2 b \log {\left (i c x + 1 \right )}\right ) \log {\left (- i c x + 1 \right )}}{4 c^{3} d} \]

[In]

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

[Out]

-I*(Integral(2*b*log(I*c*x + 1)/(c**2*x**2 + 1), x) + Integral(4*a*c**3*x**3/(c**2*x**2 + 1), x) + Integral(b*
c**2*x**2/(c**2*x**2 + 1), x) + Integral(4*I*a*c**2*x**2/(c**2*x**2 + 1), x) + Integral(-2*I*b*c*x/(c**2*x**2
+ 1), x) + Integral(-I*b*c**3*x**3/(c**2*x**2 + 1), x) + Integral(2*b*c**2*x**2*log(I*c*x + 1)/(c**2*x**2 + 1)
, x) + Integral(2*I*b*c*x*log(I*c*x + 1)/(c**2*x**2 + 1), x) + Integral(-2*I*b*c**3*x**3*log(I*c*x + 1)/(c**2*
x**2 + 1), x))/(4*c**2*d) + (b*c**2*x**2 + 2*I*b*c*x - 2*b*log(I*c*x + 1))*log(-I*c*x + 1)/(4*c**3*d)

Maxima [F]

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

[In]

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

[Out]

-1/2*a*((I*c*x^2 - 2*x)/(c^2*d) - 2*I*log(I*c*x + 1)/(c^3*d)) - 1/8*(32*I*c^6*d*integrate(1/8*x^3*arctan(c*x)/
(c^4*d*x^2 + c^2*d), x) + 16*c^6*d*integrate(1/8*x^3*log(c^2*x^2 + 1)/(c^4*d*x^2 + c^2*d), x) - 32*c^5*d*integ
rate(1/8*x^2*arctan(c*x)/(c^4*d*x^2 + c^2*d), x) + 16*I*c^5*d*integrate(1/8*x^2*log(c^2*x^2 + 1)/(c^4*d*x^2 +
c^2*d), x) - 32*I*c^4*d*integrate(1/8*x*arctan(c*x)/(c^4*d*x^2 + c^2*d), x) - 16*c^4*d*integrate(1/8*x*log(c^2
*x^2 + 1)/(c^4*d*x^2 + c^2*d), x) + 16*I*c^3*d*integrate(1/8*log(c^2*x^2 + 1)/(c^4*d*x^2 + c^2*d), x) + c^2*x^
2 + 2*I*c*x - 2*(-I*c^2*x^2 + 2*c*x + I)*arctan(c*x) + 2*arctan(c*x)^2 - (c^2*x^2 + 2*I*c*x + 1)*log(c^2*x^2 +
1) + log(c^2*x^2 + 1)^2 + 2*log(8*c^4*d*x^2 + 8*c^2*d))*b/(c^3*d)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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