\(\int \frac {(c+a^2 c x^2)^3 \arctan (a x)}{x} \, dx\) [169]

Optimal result

Integrand size = 20, antiderivative size = 132 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=-\frac {11}{12} a c^3 x-\frac {7}{36} a^3 c^3 x^3-\frac {1}{30} a^5 c^3 x^5+\frac {11}{12} c^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x) \]

[Out]

-11/12*a*c^3*x-7/36*a^3*c^3*x^3-1/30*a^5*c^3*x^5+11/12*c^3*arctan(a*x)+3/2*a^2*c^3*x^2*arctan(a*x)+3/4*a^4*c^3
*x^4*arctan(a*x)+1/6*a^6*c^3*x^6*arctan(a*x)+1/2*I*c^3*polylog(2,-I*a*x)-1/2*I*c^3*polylog(2,I*a*x)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5068, 4940, 2438, 4946, 327, 209, 308} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=\frac {1}{6} a^6 c^3 x^6 \arctan (a x)-\frac {1}{30} a^5 c^3 x^5+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)-\frac {7}{36} a^3 c^3 x^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {11}{12} c^3 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x)-\frac {11}{12} a c^3 x \]

[In]

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

[Out]

(-11*a*c^3*x)/12 - (7*a^3*c^3*x^3)/36 - (a^5*c^3*x^5)/30 + (11*c^3*ArcTan[a*x])/12 + (3*a^2*c^3*x^2*ArcTan[a*x
])/2 + (3*a^4*c^3*x^4*ArcTan[a*x])/4 + (a^6*c^3*x^6*ArcTan[a*x])/6 + (I/2)*c^3*PolyLog[2, (-I)*a*x] - (I/2)*c^
3*PolyLog[2, I*a*x]

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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*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 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 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 5068

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3 \arctan (a x)}{x}+3 a^2 c^3 x \arctan (a x)+3 a^4 c^3 x^3 \arctan (a x)+a^6 c^3 x^5 \arctan (a x)\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)}{x} \, dx+\left (3 a^2 c^3\right ) \int x \arctan (a x) \, dx+\left (3 a^4 c^3\right ) \int x^3 \arctan (a x) \, dx+\left (a^6 c^3\right ) \int x^5 \arctan (a x) \, dx \\ & = \frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} \left (i c^3\right ) \int \frac {\log (1-i a x)}{x} \, dx-\frac {1}{2} \left (i c^3\right ) \int \frac {\log (1+i a x)}{x} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{6} \left (a^7 c^3\right ) \int \frac {x^6}{1+a^2 x^2} \, dx \\ & = -\frac {3}{2} a c^3 x+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x)+\frac {1}{2} \left (3 a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{6} \left (a^7 c^3\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = -\frac {11}{12} a c^3 x-\frac {7}{36} a^3 c^3 x^3-\frac {1}{30} a^5 c^3 x^5+\frac {3}{2} c^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x)+\frac {1}{6} \left (a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx \\ & = -\frac {11}{12} a c^3 x-\frac {7}{36} a^3 c^3 x^3-\frac {1}{30} a^5 c^3 x^5+\frac {11}{12} c^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=-\frac {11}{12} a c^3 x-\frac {7}{36} a^3 c^3 x^3-\frac {1}{30} a^5 c^3 x^5+\frac {11}{12} c^3 \arctan (a x)+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)+\frac {1}{2} i c^3 \operatorname {PolyLog}(2,-i a x)-\frac {1}{2} i c^3 \operatorname {PolyLog}(2,i a x) \]

[In]

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

[Out]

(-11*a*c^3*x)/12 - (7*a^3*c^3*x^3)/36 - (a^5*c^3*x^5)/30 + (11*c^3*ArcTan[a*x])/12 + (3*a^2*c^3*x^2*ArcTan[a*x
])/2 + (3*a^4*c^3*x^4*ArcTan[a*x])/4 + (a^6*c^3*x^6*ArcTan[a*x])/6 + (I/2)*c^3*PolyLog[2, (-I)*a*x] - (I/2)*c^
3*PolyLog[2, I*a*x]

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{6}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+c^{3} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c^{3} \left (\frac {2 a^{5} x^{5}}{5}+\frac {7 a^{3} x^{3}}{3}+11 a x -11 \arctan \left (a x \right )-6 i \ln \left (a x \right ) \ln \left (i a x +1\right )+6 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-6 i \operatorname {dilog}\left (i a x +1\right )+6 i \operatorname {dilog}\left (-i a x +1\right )\right )}{12}\)	\(143\)
default	\(\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{6}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+c^{3} \arctan \left (a x \right ) \ln \left (a x \right )-\frac {c^{3} \left (\frac {2 a^{5} x^{5}}{5}+\frac {7 a^{3} x^{3}}{3}+11 a x -11 \arctan \left (a x \right )-6 i \ln \left (a x \right ) \ln \left (i a x +1\right )+6 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-6 i \operatorname {dilog}\left (i a x +1\right )+6 i \operatorname {dilog}\left (-i a x +1\right )\right )}{12}\)	\(143\)
parts	\(\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )}{6}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )}{4}+\frac {3 a^{2} c^{3} x^{2} \arctan \left (a x \right )}{2}+c^{3} \arctan \left (a x \right ) \ln \left (x \right )-\frac {c^{3} a \left (\frac {2 a^{4} x^{5}}{5}+\frac {7 a^{2} x^{3}}{3}+11 x -\frac {11 \arctan \left (a x \right )}{a}-\frac {6 i \ln \left (x \right ) \left (\ln \left (i a x +1\right )-\ln \left (-i a x +1\right )\right )}{a}-\frac {6 i \left (\operatorname {dilog}\left (i a x +1\right )-\operatorname {dilog}\left (-i a x +1\right )\right )}{a}\right )}{12}\)	\(144\)
risch	\(-\frac {11 a \,c^{3} x}{12}+\frac {3 i c^{3} \ln \left (-i a x +1\right ) x^{4} a^{4}}{8}+\frac {3 i c^{3} \ln \left (-i a x +1\right ) x^{2} a^{2}}{4}+\frac {i c^{3} \ln \left (-i a x +1\right ) x^{6} a^{6}}{12}-\frac {i c^{3} \operatorname {dilog}\left (-i a x +1\right )}{2}+\frac {11 c^{3} \arctan \left (a x \right )}{12}-\frac {a^{5} c^{3} x^{5}}{30}-\frac {7 a^{3} c^{3} x^{3}}{36}-\frac {3 i c^{3} \ln \left (i a x +1\right ) x^{4} a^{4}}{8}-\frac {3 i c^{3} \ln \left (i a x +1\right ) x^{2} a^{2}}{4}-\frac {i c^{3} \ln \left (i a x +1\right ) x^{6} a^{6}}{12}+\frac {i c^{3} \operatorname {dilog}\left (i a x +1\right )}{2}\)	\(188\)
meijerg	\(\frac {c^{3} \left (-\frac {2 a x \left (21 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{315}+\frac {2 a x \left (7 a^{6} x^{6}+7\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{21 \sqrt {a^{2} x^{2}}}\right )}{4}+\frac {3 c^{3} \left (\frac {a x \left (-5 a^{2} x^{2}+15\right )}{15}-\frac {a x \left (-5 a^{4} x^{4}+5\right ) \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}\right )}{4}+\frac {3 c^{3} \left (-2 a x +\frac {2 \left (3 a^{2} x^{2}+3\right ) \arctan \left (a x \right )}{3}\right )}{4}+\frac {c^{3} \left (-\frac {2 i a x \operatorname {polylog}\left (2, i \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}+\frac {2 i a x \operatorname {polylog}\left (2, -i \sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}\right )}{4}\)	\(204\)

[In]

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

[Out]

1/6*a^6*c^3*x^6*arctan(a*x)+3/4*a^4*c^3*x^4*arctan(a*x)+3/2*a^2*c^3*x^2*arctan(a*x)+c^3*arctan(a*x)*ln(a*x)-1/
12*c^3*(2/5*a^5*x^5+7/3*a^3*x^3+11*a*x-11*arctan(a*x)-6*I*ln(a*x)*ln(1+I*a*x)+6*I*ln(a*x)*ln(1-I*a*x)-6*I*dilo
g(1+I*a*x)+6*I*dilog(1-I*a*x))

Fricas [F]

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

[In]

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

[Out]

integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)/x, x)

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=c^{3} \left (\int \frac {\operatorname {atan}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(atan(a*x)/x, x) + Integral(3*a**2*x*atan(a*x), x) + Integral(3*a**4*x**3*atan(a*x), x) + Integr
al(a**6*x**5*atan(a*x), x))

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=-\frac {1}{30} \, a^{5} c^{3} x^{5} - \frac {7}{36} \, a^{3} c^{3} x^{3} - \frac {11}{12} \, a c^{3} x - \frac {1}{4} \, \pi c^{3} \log \left (a^{2} x^{2} + 1\right ) + c^{3} \arctan \left (a x\right ) \log \left (a x\right ) - \frac {1}{2} i \, c^{3} {\rm Li}_2\left (i \, a x + 1\right ) + \frac {1}{2} i \, c^{3} {\rm Li}_2\left (-i \, a x + 1\right ) + \frac {1}{12} \, {\left (2 \, a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} + 18 \, a^{2} c^{3} x^{2} + 11 \, c^{3}\right )} \arctan \left (a x\right ) \]

[In]

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

[Out]

-1/30*a^5*c^3*x^5 - 7/36*a^3*c^3*x^3 - 11/12*a*c^3*x - 1/4*pi*c^3*log(a^2*x^2 + 1) + c^3*arctan(a*x)*log(a*x)
- 1/2*I*c^3*dilog(I*a*x + 1) + 1/2*I*c^3*dilog(-I*a*x + 1) + 1/12*(2*a^6*c^3*x^6 + 9*a^4*c^3*x^4 + 18*a^2*c^3*
x^2 + 11*c^3)*arctan(a*x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.18 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)}{x} \, dx=\left \{\begin {array}{cl} 0 & \text {\ if\ \ }a=0\\ 3\,a^2\,c^3\,\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{2\,a^2}+\frac {x^2}{2}\right )-\frac {a^5\,c^3\,\left (\frac {x}{a^4}-\frac {\mathrm {atan}\left (a\,x\right )}{a^5}+\frac {x^5}{5}-\frac {x^3}{3\,a^2}\right )}{6}-\frac {3\,a\,c^3\,x}{2}-\frac {c^3\,\left (3\,\mathrm {atan}\left (a\,x\right )-3\,a\,x+a^3\,x^3\right )}{4}+\frac {3\,a^4\,c^3\,x^4\,\mathrm {atan}\left (a\,x\right )}{4}+\frac {a^6\,c^3\,x^6\,\mathrm {atan}\left (a\,x\right )}{6}-\frac {c^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1-a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {c^3\,{\mathrm {Li}}_{\mathrm {2}}\left (1+a\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

[In]

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

[Out]

piecewise(a == 0, 0, a ~= 0, - (c^3*dilog(- a*x*1i + 1)*1i)/2 + (c^3*dilog(a*x*1i + 1)*1i)/2 - (c^3*(3*atan(a*
x) - 3*a*x + a^3*x^3))/4 - (a^5*c^3*(x/a^4 - atan(a*x)/a^5 + x^5/5 - x^3/(3*a^2)))/6 - (3*a*c^3*x)/2 + 3*a^2*c
^3*atan(a*x)*(1/(2*a^2) + x^2/2) + (3*a^4*c^3*x^4*atan(a*x))/4 + (a^6*c^3*x^6*atan(a*x))/6)