Integrand size = 16, antiderivative size = 863 \[ \int \frac {\arctan (a+b x)}{c+d x^3} \, dx=-\frac {i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {i \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(i+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [6]{-1} \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [6]{-1} \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (i+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{5/6} \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(-1)^{2/3} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{5/6} \log (1-i a-i b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [6]{-1} (1-i a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {i \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d} (i-a-b x)}{b \sqrt [3]{c}+(i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{5/6} \operatorname {PolyLog}\left (2,-\frac {\sqrt [6]{-1} \sqrt [3]{d} (i-a-b x)}{i b \sqrt [3]{c}-\sqrt [6]{-1} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [6]{-1} \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \sqrt [3]{d} (i-a-b x)}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{d} (i+a+b x)}{b \sqrt [3]{c}-(i+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [6]{-1} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (i+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (i+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{5/6} \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{d} (i+a+b x)}{b \sqrt [3]{c}-(-1)^{2/3} (i+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]
-1/6*I*ln(1+I*a+I*b*x)*ln(b*(c^(1/3)+d^(1/3)*x)/(b*c^(1/3)+(I-a)*d^(1/3))) /c^(2/3)/d^(1/3)+1/6*I*ln(1-I*a-I*b*x)*ln(b*(c^(1/3)+d^(1/3)*x)/(b*c^(1/3) -(I+a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*(-1)^(1/6)*ln(1+I*a+I*b*x)*ln(b*(c^(1 /3)-(-1)^(1/3)*d^(1/3)*x)/(b*c^(1/3)-(-1)^(1/3)*(I-a)*d^(1/3)))/c^(2/3)/d^ (1/3)-1/6*(-1)^(1/6)*ln(1-I*a-I*b*x)*ln(b*(c^(1/3)-(-1)^(1/3)*d^(1/3)*x)/( b*c^(1/3)+(-1)^(1/3)*(I+a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*(-1)^(5/6)*ln(1+I *a+I*b*x)*ln(b*(c^(1/3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(2/3)*(I-a)* d^(1/3)))/c^(2/3)/d^(1/3)-1/6*(-1)^(5/6)*ln(1-I*a-I*b*x)*ln(b*(c^(1/3)+(-1 )^(2/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(1/6)*(1-I*a)*d^(1/3)))/c^(2/3)/d^(1/3) -1/6*I*polylog(2,d^(1/3)*(I-a-b*x)/(b*c^(1/3)+(I-a)*d^(1/3)))/c^(2/3)/d^(1 /3)+1/6*(-1)^(5/6)*polylog(2,-(-1)^(1/6)*d^(1/3)*(I-a-b*x)/(I*b*c^(1/3)-(- 1)^(1/6)*(I-a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*(-1)^(1/6)*polylog(2,-(-1)^(1 /3)*d^(1/3)*(I-a-b*x)/(b*c^(1/3)-(-1)^(1/3)*(I-a)*d^(1/3)))/c^(2/3)/d^(1/3 )+1/6*I*polylog(2,-d^(1/3)*(I+a+b*x)/(b*c^(1/3)-(I+a)*d^(1/3)))/c^(2/3)/d^ (1/3)-1/6*(-1)^(1/6)*polylog(2,(-1)^(1/3)*d^(1/3)*(I+a+b*x)/(b*c^(1/3)+(-1 )^(1/3)*(I+a)*d^(1/3)))/c^(2/3)/d^(1/3)-1/6*(-1)^(5/6)*polylog(2,-(-1)^(2/ 3)*d^(1/3)*(I+a+b*x)/(b*c^(1/3)-(-1)^(2/3)*(I+a)*d^(1/3)))/c^(2/3)/d^(1/3)
Time = 0.70 (sec) , antiderivative size = 701, normalized size of antiderivative = 0.81 \[ \int \frac {\arctan (a+b x)}{c+d x^3} \, dx=\frac {-i \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-i+a) \sqrt [3]{d}}\right )+i \log (-i (i+a+b x)) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(i+a) \sqrt [3]{d}}\right )+\sqrt [6]{-1} \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (-i+a) \sqrt [3]{d}}\right )-\sqrt [6]{-1} \log (-i (i+a+b x)) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (i+a) \sqrt [3]{d}}\right )-(-1)^{5/6} \log (-i (i+a+b x)) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [6]{-1} (1-i a) \sqrt [3]{d}}\right )+(-1)^{5/6} \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (-i+a) \sqrt [3]{d}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d} (-i+a+b x)}{-b \sqrt [3]{c}+(-i+a) \sqrt [3]{d}}\right )+(-1)^{5/6} \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-1} \sqrt [3]{d} (-i+a+b x)}{i b \sqrt [3]{c}+\sqrt [6]{-1} (-i+a) \sqrt [3]{d}}\right )+\sqrt [6]{-1} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (-i+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (-i+a) \sqrt [3]{d}}\right )+i \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d} (i+a+b x)}{-b \sqrt [3]{c}+(i+a) \sqrt [3]{d}}\right )-\sqrt [6]{-1} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (i+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (i+a) \sqrt [3]{d}}\right )-(-1)^{5/6} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{d} (i+a+b x)}{-b \sqrt [3]{c}+(-1)^{2/3} (i+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]
((-I)*Log[1 + I*a + I*b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (-I + a)*d^(1/3))] + I*Log[(-I)*(I + a + b*x)]*Log[(b*(c^(1/3) + d^(1/3)*x))/( b*c^(1/3) - (I + a)*d^(1/3))] + (-1)^(1/6)*Log[1 + I*a + I*b*x]*Log[(b*(c^ (1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(-I + a)*d^(1/3))] - (-1)^(1/6)*Log[(-I)*(I + a + b*x)]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)* x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3))] - (-1)^(5/6)*Log[(-I)*(I + a + b*x)]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/6)* (1 - I*a)*d^(1/3))] + (-1)^(5/6)*Log[1 + I*a + I*b*x]*Log[(b*(c^(1/3) + (- 1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(2/3)*(-I + a)*d^(1/3))] - I*PolyLo g[2, (d^(1/3)*(-I + a + b*x))/(-(b*c^(1/3)) + (-I + a)*d^(1/3))] + (-1)^(5 /6)*PolyLog[2, ((-1)^(1/6)*d^(1/3)*(-I + a + b*x))/(I*b*c^(1/3) + (-1)^(1/ 6)*(-I + a)*d^(1/3))] + (-1)^(1/6)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(-I + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(-I + a)*d^(1/3))] + I*PolyLog[2, (d^(1/3) *(I + a + b*x))/(-(b*c^(1/3)) + (I + a)*d^(1/3))] - (-1)^(1/6)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(I + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3) )] - (-1)^(5/6)*PolyLog[2, ((-1)^(2/3)*d^(1/3)*(I + a + b*x))/(-(b*c^(1/3) ) + (-1)^(2/3)*(I + a)*d^(1/3))])/(6*c^(2/3)*d^(1/3))
Time = 1.57 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5574, 2856, 2009}
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.
| \(\displaystyle \int \frac {\arctan (a+b x)}{c+d x^3} \, dx\) |
| \(\Big \downarrow \) 5574 |
| \(\displaystyle \frac {1}{2} i \int \frac {\log (-i a-i b x+1)}{d x^3+c}dx-\frac {1}{2} i \int \frac {\log (i a+i b x+1)}{d x^3+c}dx\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \frac {1}{2} i \int \left (-\frac {\log (-i a-i b x+1)}{3 c^{2/3} \left (-\sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (-i a-i b x+1)}{3 c^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (-i a-i b x+1)}{3 c^{2/3} \left (-(-1)^{2/3} \sqrt [3]{d} x-\sqrt [3]{c}\right )}\right )dx-\frac {1}{2} i \int \left (-\frac {\log (i a+i b x+1)}{3 c^{2/3} \left (-\sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (i a+i b x+1)}{3 c^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (i a+i b x+1)}{3 c^{2/3} \left (-(-1)^{2/3} \sqrt [3]{d} x-\sqrt [3]{c}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} i \left (\frac {\log (-i a-i b x+1) \log \left (\frac {b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-(a+i) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (-i a-i b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} \sqrt [3]{d} (a+i)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (-i a-i b x+1) \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{\sqrt [6]{-1} \sqrt [3]{d} (1-i a)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{d} (a+b x+i)}{b \sqrt [3]{c}-(a+i) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (a+b x+i)}{\sqrt [3]{-1} \sqrt [3]{d} (a+i)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{d} (a+b x+i)}{b \sqrt [3]{c}-(-1)^{2/3} (a+i) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\frac {1}{2} i \left (\frac {\log (i a+i b x+1) \log \left (\frac {b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\sqrt [3]{d} (i-a)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (i a+i b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (i a+i b x+1) \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{(-1)^{2/3} \sqrt [3]{d} (i-a)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d} (-a-b x+i)}{\sqrt [3]{d} (i-a)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,-\frac {\sqrt [6]{-1} \sqrt [3]{d} (-a-b x+i)}{i b \sqrt [3]{c}-\sqrt [6]{-1} (i-a) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \sqrt [3]{d} (-a-b x+i)}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\) |
(-1/2*I)*((Log[1 + I*a + I*b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) + (I - a)*d^(1/3))])/(3*c^(2/3)*d^(1/3)) + ((-1)^(2/3)*Log[1 + I*a + I*b*x] *Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(1/3)*(I - a)* d^(1/3))])/(3*c^(2/3)*d^(1/3)) - ((-1)^(1/3)*Log[1 + I*a + I*b*x]*Log[(b*( c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(2/3)*(I - a)*d^(1/3))] )/(3*c^(2/3)*d^(1/3)) + PolyLog[2, (d^(1/3)*(I - a - b*x))/(b*c^(1/3) + (I - a)*d^(1/3))]/(3*c^(2/3)*d^(1/3)) - ((-1)^(1/3)*PolyLog[2, -(((-1)^(1/6) *d^(1/3)*(I - a - b*x))/(I*b*c^(1/3) - (-1)^(1/6)*(I - a)*d^(1/3)))])/(3*c ^(2/3)*d^(1/3)) + ((-1)^(2/3)*PolyLog[2, -(((-1)^(1/3)*d^(1/3)*(I - a - b* x))/(b*c^(1/3) - (-1)^(1/3)*(I - a)*d^(1/3)))])/(3*c^(2/3)*d^(1/3))) + (I/ 2)*((Log[1 - I*a - I*b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (I + a)*d^(1/3))])/(3*c^(2/3)*d^(1/3)) + ((-1)^(2/3)*Log[1 - I*a - I*b*x]*Log[( b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3 ))])/(3*c^(2/3)*d^(1/3)) - ((-1)^(1/3)*Log[1 - I*a - I*b*x]*Log[(b*(c^(1/3 ) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/6)*(1 - I*a)*d^(1/3))])/(3 *c^(2/3)*d^(1/3)) + PolyLog[2, -((d^(1/3)*(I + a + b*x))/(b*c^(1/3) - (I + a)*d^(1/3)))]/(3*c^(2/3)*d^(1/3)) + ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*d^ (1/3)*(I + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(I + a)*d^(1/3))])/(3*c^(2/3) *d^(1/3)) - ((-1)^(1/3)*PolyLog[2, -(((-1)^(2/3)*d^(1/3)*(I + a + b*x))/(b *c^(1/3) - (-1)^(2/3)*(I + a)*d^(1/3)))])/(3*c^(2/3)*d^(1/3)))
3.1.52.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. )*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) ^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
Int[ArcTan[(a_) + (b_.)*(x_)]/((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[ I/2 Int[Log[1 - I*a - I*b*x]/(c + d*x^n), x], x] - Simp[I/2 Int[Log[1 + I*a + I*b*x]/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.88 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(-\frac {i b^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 d \right ) \textit {\_Z}^{2}+\left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 a^{2} d +3 d \right ) \textit {\_Z} -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a^{3} d +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) b^{3} c +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d +3 a^{2} d -d \right )}{\sum }\frac {\ln \left (-b x i-i a +1\right ) \ln \left (\frac {b x i+i a +\textit {\_R1} -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {b x i+i a +\textit {\_R1} -1}{\textit {\_R1}}\right )}{2 \textit {\_R1} a i+\textit {\_R1}^{2}-a^{2}-2 i a -2 \textit {\_R1} +1}\right )}{6 d}+\frac {i b^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 d \right ) \textit {\_Z}^{2}+\left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 a^{2} d +3 d \right ) \textit {\_Z} +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a^{3} d -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) b^{3} c -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d +3 a^{2} d -d \right )}{\sum }\frac {\ln \left (b x i+i a +1\right ) \ln \left (\frac {-b x i-i a +\textit {\_R1} -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-b x i-i a +\textit {\_R1} -1}{\textit {\_R1}}\right )}{-2 \textit {\_R1} a i+\textit {\_R1}^{2}-a^{2}+2 i a -2 \textit {\_R1} +1}\right )}{6 d}\) | \(380\) |
| derivativedivides | \(\frac {-\frac {b^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 a^{2} d \textit {\_Z} -d \,a^{3}+b^{3} c \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right ) \arctan \left (b x +a \right )}{3 d}+\frac {b^{3} \left (\arctan \left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 a^{2} d \textit {\_Z} -d \,a^{3}+b^{3} c \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right )-3 d \left (-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (d \,a^{3}+3 i a^{2} d -b^{3} c -3 a d -i d \right ) \textit {\_Z}^{6}+\left (3 d \,a^{3}+3 i a^{2} d -3 b^{3} c +3 a d +3 i d \right ) \textit {\_Z}^{4}+\left (3 d \,a^{3}-3 i a^{2} d -3 b^{3} c +3 a d -3 i d \right ) \textit {\_Z}^{2}-3 i a^{2} d +d \,a^{3}-b^{3} c +i d -3 a d \right )}{\sum }\frac {i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{a^{3} d \,\textit {\_R1}^{4}+3 i a^{2} d \,\textit {\_R1}^{4}-b^{3} c \,\textit {\_R1}^{4}-3 a d \,\textit {\_R1}^{4}-i d \,\textit {\_R1}^{4}+2 a^{3} d \,\textit {\_R1}^{2}+2 i a^{2} d \,\textit {\_R1}^{2}-2 b^{3} c \,\textit {\_R1}^{2}+2 a d \,\textit {\_R1}^{2}+2 i d \,\textit {\_R1}^{2}+d \,a^{3}-i a^{2} d -b^{3} c +a d -i d}\right )}{3}-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (d \,a^{3}+3 i a^{2} d -b^{3} c -3 a d -i d \right ) \textit {\_Z}^{6}+\left (3 d \,a^{3}+3 i a^{2} d -3 b^{3} c +3 a d +3 i d \right ) \textit {\_Z}^{4}+\left (3 d \,a^{3}-3 i a^{2} d -3 b^{3} c +3 a d -3 i d \right ) \textit {\_Z}^{2}-3 i a^{2} d +d \,a^{3}-b^{3} c +i d -3 a d \right )}{\sum }\frac {\textit {\_R1}^{2} \left (i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{a^{3} d \,\textit {\_R1}^{4}+3 i a^{2} d \,\textit {\_R1}^{4}-b^{3} c \,\textit {\_R1}^{4}-3 a d \,\textit {\_R1}^{4}-i d \,\textit {\_R1}^{4}+2 a^{3} d \,\textit {\_R1}^{2}+2 i a^{2} d \,\textit {\_R1}^{2}-2 b^{3} c \,\textit {\_R1}^{2}+2 a d \,\textit {\_R1}^{2}+2 i d \,\textit {\_R1}^{2}+d \,a^{3}-i a^{2} d -b^{3} c +a d -i d}\right )}{3}\right )\right )}{3 d}}{b}\) | \(787\) |
| default | \(\frac {-\frac {b^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 a^{2} d \textit {\_Z} -d \,a^{3}+b^{3} c \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right ) \arctan \left (b x +a \right )}{3 d}+\frac {b^{3} \left (\arctan \left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 a^{2} d \textit {\_Z} -d \,a^{3}+b^{3} c \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right )-3 d \left (-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (d \,a^{3}+3 i a^{2} d -b^{3} c -3 a d -i d \right ) \textit {\_Z}^{6}+\left (3 d \,a^{3}+3 i a^{2} d -3 b^{3} c +3 a d +3 i d \right ) \textit {\_Z}^{4}+\left (3 d \,a^{3}-3 i a^{2} d -3 b^{3} c +3 a d -3 i d \right ) \textit {\_Z}^{2}-3 i a^{2} d +d \,a^{3}-b^{3} c +i d -3 a d \right )}{\sum }\frac {i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{a^{3} d \,\textit {\_R1}^{4}+3 i a^{2} d \,\textit {\_R1}^{4}-b^{3} c \,\textit {\_R1}^{4}-3 a d \,\textit {\_R1}^{4}-i d \,\textit {\_R1}^{4}+2 a^{3} d \,\textit {\_R1}^{2}+2 i a^{2} d \,\textit {\_R1}^{2}-2 b^{3} c \,\textit {\_R1}^{2}+2 a d \,\textit {\_R1}^{2}+2 i d \,\textit {\_R1}^{2}+d \,a^{3}-i a^{2} d -b^{3} c +a d -i d}\right )}{3}-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (d \,a^{3}+3 i a^{2} d -b^{3} c -3 a d -i d \right ) \textit {\_Z}^{6}+\left (3 d \,a^{3}+3 i a^{2} d -3 b^{3} c +3 a d +3 i d \right ) \textit {\_Z}^{4}+\left (3 d \,a^{3}-3 i a^{2} d -3 b^{3} c +3 a d -3 i d \right ) \textit {\_Z}^{2}-3 i a^{2} d +d \,a^{3}-b^{3} c +i d -3 a d \right )}{\sum }\frac {\textit {\_R1}^{2} \left (i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{a^{3} d \,\textit {\_R1}^{4}+3 i a^{2} d \,\textit {\_R1}^{4}-b^{3} c \,\textit {\_R1}^{4}-3 a d \,\textit {\_R1}^{4}-i d \,\textit {\_R1}^{4}+2 a^{3} d \,\textit {\_R1}^{2}+2 i a^{2} d \,\textit {\_R1}^{2}-2 b^{3} c \,\textit {\_R1}^{2}+2 a d \,\textit {\_R1}^{2}+2 i d \,\textit {\_R1}^{2}+d \,a^{3}-i a^{2} d -b^{3} c +a d -i d}\right )}{3}\right )\right )}{3 d}}{b}\) | \(787\) |
-1/6*I*b^2/d*sum(1/(1+2*I*a*_R1-2*I*a+_R1^2-a^2-2*_R1)*(ln(1-I*a-I*b*x)*ln ((_R1+I*b*x+I*a-1)/_R1)+dilog((_R1+I*b*x+I*a-1)/_R1)),_R1=RootOf(d*_Z^3+(3 *RootOf(_Z^2+1,index=1)*a*d-3*d)*_Z^2+(-6*RootOf(_Z^2+1,index=1)*a*d-3*a^2 *d+3*d)*_Z-RootOf(_Z^2+1,index=1)*a^3*d+RootOf(_Z^2+1,index=1)*b^3*c+3*Roo tOf(_Z^2+1,index=1)*a*d+3*a^2*d-d))+1/6*I*b^2/d*sum(1/(1-2*I*a*_R1+2*I*a+_ R1^2-a^2-2*_R1)*(ln(1+I*a+I*b*x)*ln((_R1-I*b*x-I*a-1)/_R1)+dilog((_R1-I*b* x-I*a-1)/_R1)),_R1=RootOf(d*_Z^3+(-3*RootOf(_Z^2+1,index=1)*a*d-3*d)*_Z^2+ (6*RootOf(_Z^2+1,index=1)*a*d-3*a^2*d+3*d)*_Z+RootOf(_Z^2+1,index=1)*a^3*d -RootOf(_Z^2+1,index=1)*b^3*c-3*RootOf(_Z^2+1,index=1)*a*d+3*a^2*d-d))
\[ \int \frac {\arctan (a+b x)}{c+d x^3} \, dx=\int { \frac {\arctan \left (b x + a\right )}{d x^{3} + c} \,d x } \]
Timed out. \[ \int \frac {\arctan (a+b x)}{c+d x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\arctan (a+b x)}{c+d x^3} \, dx=\int { \frac {\arctan \left (b x + a\right )}{d x^{3} + c} \,d x } \]
\[ \int \frac {\arctan (a+b x)}{c+d x^3} \, dx=\int { \frac {\arctan \left (b x + a\right )}{d x^{3} + c} \,d x } \]
Timed out. \[ \int \frac {\arctan (a+b x)}{c+d x^3} \, dx=\int \frac {\mathrm {atan}\left (a+b\,x\right )}{d\,x^3+c} \,d x \]