\(\int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx\) [174]

Optimal result

Integrand size = 20, antiderivative size = 113 \[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=-\frac {x}{2 a^3 c}+\frac {\arctan (a x)}{2 a^4 c}+\frac {x^2 \arctan (a x)}{2 a^2 c}+\frac {i \arctan (a x)^2}{2 a^4 c}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5036, 4946, 327, 209, 5040, 4964, 2449, 2352} \[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=\frac {i \arctan (a x)^2}{2 a^4 c}+\frac {\arctan (a x)}{2 a^4 c}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a^4 c}-\frac {x}{2 a^3 c}+\frac {x^2 \arctan (a x)}{2 a^2 c} \]

[In]

Int[(x^3*ArcTan[a*x])/(c + a^2*c*x^2),x]

[Out]

-1/2*x/(a^3*c) + ArcTan[a*x]/(2*a^4*c) + (x^2*ArcTan[a*x])/(2*a^2*c) + ((I/2)*ArcTan[a*x]^2)/(a^4*c) + (ArcTan
[a*x]*Log[2/(1 + I*a*x)])/(a^4*c) + ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^4*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 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 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 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}& = -\frac {\int \frac {x \arctan (a x)}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x \arctan (a x) \, dx}{a^2 c} \\ & = \frac {x^2 \arctan (a x)}{2 a^2 c}+\frac {i \arctan (a x)^2}{2 a^4 c}+\frac {\int \frac {\arctan (a x)}{i-a x} \, dx}{a^3 c}-\frac {\int \frac {x^2}{1+a^2 x^2} \, dx}{2 a c} \\ & = -\frac {x}{2 a^3 c}+\frac {x^2 \arctan (a x)}{2 a^2 c}+\frac {i \arctan (a x)^2}{2 a^4 c}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\int \frac {1}{1+a^2 x^2} \, dx}{2 a^3 c}-\frac {\int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c} \\ & = -\frac {x}{2 a^3 c}+\frac {\arctan (a x)}{2 a^4 c}+\frac {x^2 \arctan (a x)}{2 a^2 c}+\frac {i \arctan (a x)^2}{2 a^4 c}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {i \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^4 c} \\ & = -\frac {x}{2 a^3 c}+\frac {\arctan (a x)}{2 a^4 c}+\frac {x^2 \arctan (a x)}{2 a^2 c}+\frac {i \arctan (a x)^2}{2 a^4 c}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=-\frac {x}{2 a^3 c}+\frac {\arctan (a x)}{2 a^4 c}+\frac {x^2 \arctan (a x)}{2 a^2 c}+\frac {i \arctan (a x)^2}{2 a^4 c}+\frac {\arctan (a x) \log \left (\frac {2 i}{i-a x}\right )}{a^4 c}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i+a x}{i-a x}\right )}{2 a^4 c} \]

[In]

Integrate[(x^3*ArcTan[a*x])/(c + a^2*c*x^2),x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.35

method	result	size
parts	\(\frac {x^{2} \arctan \left (a x \right )}{2 a^{2} c}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c \,a^{4}}-\frac {a \left (\frac {x}{a^{4}}-\frac {\arctan \left (a x \right )}{a^{5}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 a^{6}}\right )}{2 c}\)	\(152\)
derivativedivides	\(\frac {\frac {\arctan \left (a x \right ) a^{2} x^{2}}{2 c}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {a x -\arctan \left (a x \right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c}}{a^{4}}\)	\(183\)
default	\(\frac {\frac {\arctan \left (a x \right ) a^{2} x^{2}}{2 c}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {a x -\arctan \left (a x \right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c}}{a^{4}}\)	\(183\)
risch	\(\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c \,a^{4}}+\frac {i \ln \left (-i a x +1\right ) x^{2}}{4 c \,a^{2}}+\frac {i \ln \left (-i a x +1\right )}{4 c \,a^{4}}-\frac {x}{2 a^{3} c}-\frac {i \ln \left (i a x +1\right ) x^{2}}{4 c \,a^{2}}-\frac {i \ln \left (i a x +1\right )}{4 c \,a^{4}}-\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c \,a^{4}}+\frac {i \ln \left (i a x +1\right )^{2}}{8 c \,a^{4}}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c \,a^{4}}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c \,a^{4}}-\frac {i \ln \left (-i a x +1\right )^{2}}{8 c \,a^{4}}\)	\(207\)

[In]

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

[Out]

1/2*x^2*arctan(a*x)/a^2/c-1/2/c*arctan(a*x)/a^4*ln(a^2*x^2+1)-1/2/c*a*(1/a^4*x-1/a^5*arctan(a*x)-1/4/a^6*sum(1
/_alpha*(2*ln(x-_alpha)*ln(a^2*x^2+1)-a^2*(1/a^2/_alpha*ln(x-_alpha)^2+2*_alpha*ln(x-_alpha)*ln(1/2*(x+_alpha)
/_alpha)+2*_alpha*dilog(1/2*(x+_alpha)/_alpha))),_alpha=RootOf(_Z^2*a^2+1)))

Fricas [F]

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

[In]

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

[Out]

integral(x^3*arctan(a*x)/(a^2*c*x^2 + c), x)

Sympy [F]

\[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

integrate(x**3*atan(a*x)/(a**2*c*x**2+c),x)

[Out]

Integral(x**3*atan(a*x)/(a**2*x**2 + 1), x)/c

Maxima [F]

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

[In]

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

[Out]

integrate(x^3*arctan(a*x)/(a^2*c*x^2 + c), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int((x^3*atan(a*x))/(c + a^2*c*x^2),x)

[Out]

int((x^3*atan(a*x))/(c + a^2*c*x^2), x)