\(\int \frac {(d+i c d x)^2 (a+b \arctan (c x))}{x^2} \, dx\) [15]

Optimal result

Integrand size = 23, antiderivative size = 89 \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))}{x^2} \, dx=-a c^2 d^2 x-b c^2 d^2 x \arctan (c x)-\frac {d^2 (a+b \arctan (c x))}{x}+2 i a c d^2 \log (x)+b c d^2 \log (x)-b c d^2 \operatorname {PolyLog}(2,-i c x)+b c d^2 \operatorname {PolyLog}(2,i c x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4996, 4930, 266, 4946, 272, 36, 29, 31, 4940, 2438} \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))}{x^2} \, dx=-\frac {d^2 (a+b \arctan (c x))}{x}-a c^2 d^2 x+2 i a c d^2 \log (x)-b c^2 d^2 x \arctan (c x)-b c d^2 \operatorname {PolyLog}(2,-i c x)+b c d^2 \operatorname {PolyLog}(2,i c x)+b c d^2 \log (x) \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 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 2438

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

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 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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 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])

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.89 \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))}{x^2} \, dx=-\frac {d^2 \left (a+a c^2 x^2+b \arctan (c x)+b c^2 x^2 \arctan (c x)-2 i a c x \log (x)-b c x \log (c x)+b c x \operatorname {PolyLog}(2,-i c x)-b c x \operatorname {PolyLog}(2,i c x)\right )}{x} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.26

method	result	size
parts	\(a \,d^{2} \left (-c^{2} x +2 i c \ln \left (x \right )-\frac {1}{x}\right )+b \,d^{2} c \left (-c x \arctan \left (c x \right )+2 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{c x}-\ln \left (c x \right ) \ln \left (i c x +1\right )+\ln \left (c x \right ) \ln \left (-i c x +1\right )-\operatorname {dilog}\left (i c x +1\right )+\operatorname {dilog}\left (-i c x +1\right )+\ln \left (c x \right )\right )\)	\(112\)
derivativedivides	\(c \left (a \,d^{2} \left (-c x +2 i \ln \left (c x \right )-\frac {1}{c x}\right )+b \,d^{2} \left (-c x \arctan \left (c x \right )+2 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{c x}-\ln \left (c x \right ) \ln \left (i c x +1\right )+\ln \left (c x \right ) \ln \left (-i c x +1\right )-\operatorname {dilog}\left (i c x +1\right )+\operatorname {dilog}\left (-i c x +1\right )+\ln \left (c x \right )\right )\right )\)	\(115\)
default	\(c \left (a \,d^{2} \left (-c x +2 i \ln \left (c x \right )-\frac {1}{c x}\right )+b \,d^{2} \left (-c x \arctan \left (c x \right )+2 i \arctan \left (c x \right ) \ln \left (c x \right )-\frac {\arctan \left (c x \right )}{c x}-\ln \left (c x \right ) \ln \left (i c x +1\right )+\ln \left (c x \right ) \ln \left (-i c x +1\right )-\operatorname {dilog}\left (i c x +1\right )+\operatorname {dilog}\left (-i c x +1\right )+\ln \left (c x \right )\right )\right )\)	\(115\)
risch	\(\frac {i b \,d^{2} \ln \left (i c x +1\right )}{2 x}-b c \,d^{2}-\frac {i d^{2} b \ln \left (-i c x +1\right )}{2 x}+c \,d^{2} b \operatorname {dilog}\left (-i c x +1\right )+\frac {c \,d^{2} b \ln \left (-i c x \right )}{2}+2 i c \,d^{2} a \ln \left (-i c x \right )+\frac {i b \,c^{2} d^{2} \ln \left (i c x +1\right ) x}{2}-a \,c^{2} d^{2} x -\frac {i c^{2} d^{2} b \ln \left (-i c x +1\right ) x}{2}-\frac {d^{2} a}{x}-i c \,d^{2} a -b c \,d^{2} \operatorname {dilog}\left (i c x +1\right )+\frac {b c \,d^{2} \ln \left (i c x \right )}{2}\)	\(179\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.58 \[ \int \frac {(d+i c d x)^2 (a+b \arctan (c x))}{x^2} \, dx=\left \{\begin {array}{cl} -\frac {a\,d^2}{x} & \text {\ if\ \ }c=0\\ \frac {b\,d^2\,\left (c^2\,\ln \left (x\right )-\frac {c^2\,\ln \left (c^2\,x^2+1\right )}{2}\right )}{c}+b\,c\,d^2\,\left ({\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )-{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\right )+\frac {b\,c\,d^2\,\ln \left (c^2\,x^2+1\right )}{2}-\frac {a\,d^2\,\left (c^2\,x^2+1-c\,x\,\ln \left (x\right )\,2{}\mathrm {i}\right )}{x}-\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{x}-b\,c^2\,d^2\,x\,\mathrm {atan}\left (c\,x\right ) & \text {\ if\ \ }c\neq 0 \end {array}\right . \]

[In]

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

[Out]

piecewise(c == 0, -(a*d^2)/x, c ~= 0, (b*d^2*(c^2*log(x) - (c^2*log(c^2*x^2 + 1))/2))/c + b*c*d^2*(dilog(- c*x
*1i + 1) - dilog(c*x*1i + 1)) + (b*c*d^2*log(c^2*x^2 + 1))/2 - (a*d^2*(c^2*x^2 - c*x*log(x)*2i + 1))/x - (b*d^
2*atan(c*x))/x - b*c^2*d^2*x*atan(c*x))