Integrand size = 20, antiderivative size = 133 \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {x}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{4 a^4 c^2}+\frac {\arctan (a x)}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^2}{2 a^4 c^2}-\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c^2} \]
-1/4*x/a^3/c^2/(a^2*x^2+1)-1/4*arctan(a*x)/a^4/c^2+1/2*arctan(a*x)/a^4/c^2 /(a^2*x^2+1)-1/2*I*arctan(a*x)^2/a^4/c^2-arctan(a*x)*ln(2/(1+I*a*x))/a^4/c ^2-1/2*I*polylog(2,1-2/(1+I*a*x))/a^4/c^2
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.58 \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {4 i \arctan (a x)^2+2 \arctan (a x) \left (\cos (2 \arctan (a x))-4 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+4 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-\sin (2 \arctan (a x))}{8 a^4 c^2} \]
((4*I)*ArcTan[a*x]^2 + 2*ArcTan[a*x]*(Cos[2*ArcTan[a*x]] - 4*Log[1 + E^((2 *I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - Sin[2*ArcT an[a*x]])/(8*a^4*c^2)
Time = 0.65 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5499, 27, 5455, 5379, 2849, 2752, 5465, 215, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3 \arctan (a x)}{\left (a^2 c x^2+c\right )^2} \, dx\) |
| \(\Big \downarrow \) 5499 |
| \(\displaystyle \frac {\int \frac {x \arctan (a x)}{c \left (a^2 x^2+1\right )}dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)}{c^2 \left (a^2 x^2+1\right )^2}dx}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2 c^2}-\frac {\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a^2 c^2}\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle -\frac {\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a^2 c^2}+\frac {-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2 c^2}\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle -\frac {\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a^2 c^2}+\frac {-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2 c^2}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle -\frac {\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a^2 c^2}+\frac {-\frac {\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2 c^2}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle -\frac {\int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx}{a^2 c^2}+\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}}{a^2 c^2}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}}{a^2 c^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle -\frac {\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}}{a^2 c^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}}{a^2 c^2}+\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}}{a^2 c^2}\) |
-((-1/2*ArcTan[a*x]/(a^2*(1 + a^2*x^2)) + (x/(2*(1 + a^2*x^2)) + ArcTan[a* x]/(2*a))/(2*a))/(a^2*c^2)) + (((-1/2*I)*ArcTan[a*x]^2)/a^2 - ((ArcTan[a*x ]*Log[2/(1 + I*a*x)])/a + ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a)/a)/(a^2 *c^2)
3.2.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-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]
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] + Simp[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 ]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/e Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar cTan[c*x])^p, x], x] - Simp[d/e Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTan [c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ [p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.45 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.29
| method | result | size |
| parts | \(\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{2} a^{4}}+\frac {\arctan \left (a x \right )}{2 a^{4} c^{2} \left (a^{2} x^{2}+1\right )}-\frac {a \left (\frac {x}{2 a^{4} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2 a^{5}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+1\right )}{\sum }\frac {2 \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (a^{2} x^{2}+1\right )-a^{2} \left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{a^{2} \underline {\hspace {1.25 ex}}\alpha }+2 \underline {\hspace {1.25 ex}}\alpha \ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )+2 \underline {\hspace {1.25 ex}}\alpha \operatorname {dilog}\left (\frac {x +\underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha }\right )\right )}{\underline {\hspace {1.25 ex}}\alpha }}{4 a^{6}}\right )}{2 c^{2}}\) | \(172\) |
| derivativedivides | \(\frac {\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{2}}+\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {\frac {a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c^{2}}}{a^{4}}\) | \(200\) |
| default | \(\frac {\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{2}}+\frac {\arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {\frac {a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )}{2}-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{2}}{2 c^{2}}}{a^{4}}\) | \(200\) |
| risch | \(-\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c^{2} a^{4}}-\frac {i \ln \left (-i a x +1\right )}{16 c^{2} a^{4} \left (-i a x -1\right )}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c^{2} a^{4}}+\frac {i}{8 c^{2} a^{4} \left (-i a x +1\right )}-\frac {\arctan \left (a x \right )}{8 a^{4} c^{2}}-\frac {\ln \left (-i a x +1\right ) x}{16 c^{2} a^{3} \left (-i a x -1\right )}+\frac {i \ln \left (-i a x +1\right )^{2}}{8 c^{2} a^{4}}+\frac {i \ln \left (-i a x +1\right )}{8 c^{2} a^{4} \left (-i a x +1\right )}+\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c^{2} a^{4}}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c^{2} a^{4}}-\frac {i \ln \left (i a x +1\right )^{2}}{8 c^{2} a^{4}}-\frac {i}{8 c^{2} a^{4} \left (i a x +1\right )}-\frac {i \ln \left (i a x +1\right )}{8 c^{2} a^{4} \left (i a x +1\right )}+\frac {i \ln \left (i a x +1\right )}{16 c^{2} a^{4} \left (i a x -1\right )}-\frac {\ln \left (i a x +1\right ) x}{16 c^{2} a^{3} \left (i a x -1\right )}\) | \(328\) |
1/2/c^2*arctan(a*x)/a^4*ln(a^2*x^2+1)+1/2*arctan(a*x)/a^4/c^2/(a^2*x^2+1)- 1/2/c^2*a*(1/2/a^4*x/(a^2*x^2+1)+1/2/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)))
\[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
\[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
\[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
\[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^3 \arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]