\(\int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx\) [282]

Optimal result

Integrand size = 22, antiderivative size = 166 \[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {x}{3 a^4 c}-\frac {\arctan (a x)}{3 a^5 c}-\frac {x^2 \arctan (a x)}{3 a^3 c}-\frac {4 i \arctan (a x)^2}{3 a^5 c}-\frac {x \arctan (a x)^2}{a^4 c}+\frac {x^3 \arctan (a x)^2}{3 a^2 c}+\frac {\arctan (a x)^3}{3 a^5 c}-\frac {8 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^5 c}-\frac {4 i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a^5 c} \]

[Out]

1/3*x/a^4/c-1/3*arctan(a*x)/a^5/c-1/3*x^2*arctan(a*x)/a^3/c-4/3*I*arctan(a*x)^2/a^5/c-x*arctan(a*x)^2/a^4/c+1/
3*x^3*arctan(a*x)^2/a^2/c+1/3*arctan(a*x)^3/a^5/c-8/3*arctan(a*x)*ln(2/(1+I*a*x))/a^5/c-4/3*I*polylog(2,1-2/(1
+I*a*x))/a^5/c

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5036, 4946, 327, 209, 5040, 4964, 2449, 2352, 4930, 5004} \[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {\arctan (a x)^3}{3 a^5 c}-\frac {4 i \arctan (a x)^2}{3 a^5 c}-\frac {\arctan (a x)}{3 a^5 c}-\frac {8 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^5 c}-\frac {4 i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{3 a^5 c}-\frac {x \arctan (a x)^2}{a^4 c}+\frac {x}{3 a^4 c}-\frac {x^2 \arctan (a x)}{3 a^3 c}+\frac {x^3 \arctan (a x)^2}{3 a^2 c} \]

[In]

Int[(x^4*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]

[Out]

x/(3*a^4*c) - ArcTan[a*x]/(3*a^5*c) - (x^2*ArcTan[a*x])/(3*a^3*c) - (((4*I)/3)*ArcTan[a*x]^2)/(a^5*c) - (x*Arc
Tan[a*x]^2)/(a^4*c) + (x^3*ArcTan[a*x]^2)/(3*a^2*c) + ArcTan[a*x]^3/(3*a^5*c) - (8*ArcTan[a*x]*Log[2/(1 + I*a*
x)])/(3*a^5*c) - (((4*I)/3)*PolyLog[2, 1 - 2/(1 + I*a*x)])/(a^5*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 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 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}& = -\frac {\int \frac {x^2 \arctan (a x)^2}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x^2 \arctan (a x)^2 \, dx}{a^2 c} \\ & = \frac {x^3 \arctan (a x)^2}{3 a^2 c}+\frac {\int \frac {\arctan (a x)^2}{c+a^2 c x^2} \, dx}{a^4}-\frac {\int \arctan (a x)^2 \, dx}{a^4 c}-\frac {2 \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx}{3 a c} \\ & = -\frac {x \arctan (a x)^2}{a^4 c}+\frac {x^3 \arctan (a x)^2}{3 a^2 c}+\frac {\arctan (a x)^3}{3 a^5 c}-\frac {2 \int x \arctan (a x) \, dx}{3 a^3 c}+\frac {2 \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{3 a^3 c}+\frac {2 \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{a^3 c} \\ & = -\frac {x^2 \arctan (a x)}{3 a^3 c}-\frac {4 i \arctan (a x)^2}{3 a^5 c}-\frac {x \arctan (a x)^2}{a^4 c}+\frac {x^3 \arctan (a x)^2}{3 a^2 c}+\frac {\arctan (a x)^3}{3 a^5 c}-\frac {2 \int \frac {\arctan (a x)}{i-a x} \, dx}{3 a^4 c}-\frac {2 \int \frac {\arctan (a x)}{i-a x} \, dx}{a^4 c}+\frac {\int \frac {x^2}{1+a^2 x^2} \, dx}{3 a^2 c} \\ & = \frac {x}{3 a^4 c}-\frac {x^2 \arctan (a x)}{3 a^3 c}-\frac {4 i \arctan (a x)^2}{3 a^5 c}-\frac {x \arctan (a x)^2}{a^4 c}+\frac {x^3 \arctan (a x)^2}{3 a^2 c}+\frac {\arctan (a x)^3}{3 a^5 c}-\frac {8 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^5 c}-\frac {\int \frac {1}{1+a^2 x^2} \, dx}{3 a^4 c}+\frac {2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^4 c}+\frac {2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^4 c} \\ & = \frac {x}{3 a^4 c}-\frac {\arctan (a x)}{3 a^5 c}-\frac {x^2 \arctan (a x)}{3 a^3 c}-\frac {4 i \arctan (a x)^2}{3 a^5 c}-\frac {x \arctan (a x)^2}{a^4 c}+\frac {x^3 \arctan (a x)^2}{3 a^2 c}+\frac {\arctan (a x)^3}{3 a^5 c}-\frac {8 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^5 c}-\frac {(2 i) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^5 c}-\frac {(2 i) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^5 c} \\ & = \frac {x}{3 a^4 c}-\frac {\arctan (a x)}{3 a^5 c}-\frac {x^2 \arctan (a x)}{3 a^3 c}-\frac {4 i \arctan (a x)^2}{3 a^5 c}-\frac {x \arctan (a x)^2}{a^4 c}+\frac {x^3 \arctan (a x)^2}{3 a^2 c}+\frac {\arctan (a x)^3}{3 a^5 c}-\frac {8 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^5 c}-\frac {4 i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a^5 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.54 \[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {a x+\left (4 i-3 a x+a^3 x^3\right ) \arctan (a x)^2+\arctan (a x)^3-\arctan (a x) \left (1+a^2 x^2+8 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+4 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )}{3 a^5 c} \]

[In]

Integrate[(x^4*ArcTan[a*x]^2)/(c + a^2*c*x^2),x]

[Out]

(a*x + (4*I - 3*a*x + a^3*x^3)*ArcTan[a*x]^2 + ArcTan[a*x]^3 - ArcTan[a*x]*(1 + a^2*x^2 + 8*Log[1 + E^((2*I)*A
rcTan[a*x])]) + (4*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])])/(3*a^5*c)

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.35

method	result	size
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{3 c}-\frac {\arctan \left (a x \right )^{2} a x}{c}+\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\arctan \left (a x \right )^{3}\right )}{3 c}}{a^{5}}\)	\(224\)
default	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{3 c}-\frac {\arctan \left (a x \right )^{2} a x}{c}+\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+\arctan \left (a x \right )^{3}\right )}{3 c}}{a^{5}}\)	\(224\)
parts	\(\frac {x^{3} \arctan \left (a x \right )^{2}}{3 a^{2} c}-\frac {x \arctan \left (a x \right )^{2}}{a^{4} c}+\frac {\arctan \left (a x \right )^{3}}{a^{5} c}-\frac {2 \left (\frac {\arctan \left (a x \right )^{3}}{3 a^{5}}+\frac {\frac {a^{2} \arctan \left (a x \right ) x^{2}}{2}-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a x}{2}+\frac {\arctan \left (a x \right )}{2}-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{3 a^{5}}\right )}{c}\)	\(236\)

[In]

int(x^4*arctan(a*x)^2/(a^2*c*x^2+c),x,method=_RETURNVERBOSE)

[Out]

1/a^5*(1/3/c*arctan(a*x)^2*a^3*x^3-1/c*arctan(a*x)^2*a*x+1/c*arctan(a*x)^3-2/3/c*(1/2*a^2*arctan(a*x)*x^2-2*ar
ctan(a*x)*ln(a^2*x^2+1)-1/2*a*x+1/2*arctan(a*x)-I*(ln(a*x-I)*ln(a^2*x^2+1)-dilog(-1/2*I*(I+a*x))-ln(a*x-I)*ln(
-1/2*I*(I+a*x))-1/2*ln(a*x-I)^2)+I*(ln(I+a*x)*ln(a^2*x^2+1)-dilog(1/2*I*(a*x-I))-ln(I+a*x)*ln(1/2*I*(a*x-I))-1
/2*ln(I+a*x)^2)+arctan(a*x)^3))

Fricas [F]

\[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c} \,d x } \]

[In]

integrate(x^4*arctan(a*x)^2/(a^2*c*x^2+c),x, algorithm="fricas")

[Out]

integral(x^4*arctan(a*x)^2/(a^2*c*x^2 + c), x)

Sympy [F]

\[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]

[In]

integrate(x**4*atan(a*x)**2/(a**2*c*x**2+c),x)

[Out]

Integral(x**4*atan(a*x)**2/(a**2*x**2 + 1), x)/c

Maxima [F]

\[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c} \,d x } \]

[In]

integrate(x^4*arctan(a*x)^2/(a^2*c*x^2+c),x, algorithm="maxima")

[Out]

1/48*(4*(432*a^4*integrate(1/48*x^4*arctan(a*x)^2/(a^6*c*x^2 + a^4*c), x) + 36*a^4*integrate(1/48*x^4*log(a^2*
x^2 + 1)^2/(a^6*c*x^2 + a^4*c), x) + 48*a^4*integrate(1/48*x^4*log(a^2*x^2 + 1)/(a^6*c*x^2 + a^4*c), x) - 96*a
^3*integrate(1/48*x^3*arctan(a*x)/(a^6*c*x^2 + a^4*c), x) - 144*a^2*integrate(1/48*x^2*log(a^2*x^2 + 1)/(a^6*c
*x^2 + a^4*c), x) + 288*a*integrate(1/48*x*arctan(a*x)/(a^6*c*x^2 + a^4*c), x) - arctan(a*x)^3/(a^5*c) - 36*in
tegrate(1/48*log(a^2*x^2 + 1)^2/(a^6*c*x^2 + a^4*c), x))*a^5*c + 4*(a^3*x^3 - 3*a*x)*arctan(a*x)^2 + 8*arctan(
a*x)^3 - (a^3*x^3 - 3*a*x)*log(a^2*x^2 + 1)^2)/(a^5*c)

Giac [F]

\[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c} \,d x } \]

[In]

integrate(x^4*arctan(a*x)^2/(a^2*c*x^2+c),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \arctan (a x)^2}{c+a^2 c x^2} \, dx=\int \frac {x^4\,{\mathrm {atan}\left (a\,x\right )}^2}{c\,a^2\,x^2+c} \,d x \]

[In]

int((x^4*atan(a*x)^2)/(c + a^2*c*x^2),x)

[Out]

int((x^4*atan(a*x)^2)/(c + a^2*c*x^2), x)