Integrand size = 20, antiderivative size = 113 \[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=-\frac {x}{2 a^3 c}+\frac {\arctan (a x)}{2 a^4 c}+\frac {x^2 \arctan (a x)}{2 a^2 c}+\frac {i \arctan (a x)^2}{2 a^4 c}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c} \]
-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
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=-\frac {x}{2 a^3 c}+\frac {\arctan (a x)}{2 a^4 c}+\frac {x^2 \arctan (a x)}{2 a^2 c}+\frac {i \arctan (a x)^2}{2 a^4 c}+\frac {\arctan (a x) \log \left (\frac {2 i}{i-a x}\right )}{a^4 c}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i+a x}{i-a x}\right )}{2 a^4 c} \]
-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)
Time = 0.55 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5451, 27, 5361, 262, 216, 5455, 5379, 2849, 2752}
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.
\(\displaystyle \int \frac {x^3 \arctan (a x)}{a^2 c x^2+c} \, dx\) |
\(\Big \downarrow \) 5451 |
\(\displaystyle \frac {\int x \arctan (a x)dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)}{c \left (a^2 x^2+1\right )}dx}{a^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int x \arctan (a x)dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2 c}\) |
\(\Big \downarrow \) 5361 |
\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \int \frac {x^2}{a^2 x^2+1}dx}{a^2 c}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2 c}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\int \frac {1}{a^2 x^2+1}dx}{a^2}\right )}{a^2 c}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2 c}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2 c}-\frac {\int \frac {x \arctan (a x)}{a^2 x^2+1}dx}{a^2 c}\) |
\(\Big \downarrow \) 5455 |
\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2 c}-\frac {-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2 c}\) |
\(\Big \downarrow \) 5379 |
\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2 c}-\frac {-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2 c}\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2 c}-\frac {-\frac {\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}}{a}-\frac {i \arctan (a x)^2}{2 a^2}}{a^2 c}\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {\frac {1}{2} x^2 \arctan (a x)-\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )}{a^2 c}-\frac {-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}}{a^2 c}\) |
((x^2*ArcTan[a*x])/2 - (a*(x/a^2 - ArcTan[a*x]/a^3))/2)/(a^2*c) - (((-1/2* I)*ArcTan[a*x]^2)/a^2 - ((ArcTan[a*x]*Log[2/(1 + I*a*x)])/a + ((I/2)*PolyL og[2, 1 - 2/(1 + I*a*x)])/a)/a)/(a^2*c)
3.2.74.3.1 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)^(-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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[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]
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_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[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]
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]
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 | \(\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c \,a^{4}}+\frac {i \ln \left (-i a x +1\right ) x^{2}}{4 c \,a^{2}}+\frac {i \ln \left (-i a x +1\right )}{4 c \,a^{4}}-\frac {x}{2 a^{3} c}-\frac {i \ln \left (i a x +1\right ) x^{2}}{4 c \,a^{2}}-\frac {i \ln \left (i a x +1\right )}{4 c \,a^{4}}-\frac {i \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c \,a^{4}}+\frac {i \ln \left (i a x +1\right )^{2}}{8 c \,a^{4}}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c \,a^{4}}+\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c \,a^{4}}-\frac {i \ln \left (-i a x +1\right )^{2}}{8 c \,a^{4}}\) | \(207\) |
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)))
\[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )}{a^{2} c x^{2} + c} \,d x } \]
\[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
\[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )}{a^{2} c x^{2} + c} \,d x } \]
\[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )}{a^{2} c x^{2} + c} \,d x } \]
Timed out. \[ \int \frac {x^3 \arctan (a x)}{c+a^2 c x^2} \, dx=\int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{c\,a^2\,x^2+c} \,d x \]