Integrand size = 18, antiderivative size = 739 \[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\frac {\left (a+b \arctan \left (c x^3\right )\right ) \log (d+e x)}{e}+\frac {b c \log \left (\frac {e \left (1-\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (-\frac {e \left (1+\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \log \left (-\frac {e \left (\sqrt [3]{-1}+\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (-\frac {e \left ((-1)^{2/3}+\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \log \left (\frac {(-1)^{2/3} e \left (1+\sqrt [3]{-1} \sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (\frac {\sqrt [3]{-1} e \left (1+(-1)^{2/3} \sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e} \] Output:
(a+b*arctan(c*x^3))*ln(e*x+d)/e+1/2*b*c*ln(e*(1-(-c^2)^(1/6)*x)/((-c^2)^(1 /6)*d+e))*ln(e*x+d)/(-c^2)^(1/2)/e-1/2*b*c*ln(-e*(1+(-c^2)^(1/6)*x)/((-c^2 )^(1/6)*d-e))*ln(e*x+d)/(-c^2)^(1/2)/e+1/2*b*c*ln(-e*((-1)^(1/3)+(-c^2)^(1 /6)*x)/((-c^2)^(1/6)*d-(-1)^(1/3)*e))*ln(e*x+d)/(-c^2)^(1/2)/e-1/2*b*c*ln( -e*((-1)^(2/3)+(-c^2)^(1/6)*x)/((-c^2)^(1/6)*d-(-1)^(2/3)*e))*ln(e*x+d)/(- c^2)^(1/2)/e+1/2*b*c*ln((-1)^(2/3)*e*(1+(-1)^(1/3)*(-c^2)^(1/6)*x)/((-c^2) ^(1/6)*d+(-1)^(2/3)*e))*ln(e*x+d)/(-c^2)^(1/2)/e-1/2*b*c*ln((-1)^(1/3)*e*( 1+(-1)^(2/3)*(-c^2)^(1/6)*x)/((-c^2)^(1/6)*d+(-1)^(1/3)*e))*ln(e*x+d)/(-c^ 2)^(1/2)/e-1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/((-c^2)^(1/6)*d-e))/(-c^ 2)^(1/2)/e+1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/((-c^2)^(1/6)*d+e))/(-c^ 2)^(1/2)/e+1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/((-c^2)^(1/6)*d-(-1)^(1/ 3)*e))/(-c^2)^(1/2)/e-1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/((-c^2)^(1/6) *d+(-1)^(1/3)*e))/(-c^2)^(1/2)/e-1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/(( -c^2)^(1/6)*d-(-1)^(2/3)*e))/(-c^2)^(1/2)/e+1/2*b*c*polylog(2,(-c^2)^(1/6) *(e*x+d)/((-c^2)^(1/6)*d+(-1)^(2/3)*e))/(-c^2)^(1/2)/e
Result contains complex when optimal does not.
Time = 9.90 (sec) , antiderivative size = 522, normalized size of antiderivative = 0.71 \[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\frac {a \log (d+e x)}{e}+\frac {b \left (2 \arctan \left (c x^3\right ) \log (d+e x)-i \left (\log \left (\frac {e \left (-i+\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 \sqrt [3]{c} d+\left (-i+\sqrt {3}\right ) e}\right ) \log (d+e x)-\log \left (\frac {e \left (i+\sqrt {3}-2 \sqrt [3]{c} x\right )}{2 \sqrt [3]{c} d+\left (i+\sqrt {3}\right ) e}\right ) \log (d+e x)+\log \left (\frac {e \left (i-\sqrt [3]{c} x\right )}{\sqrt [3]{c} d+i e}\right ) \log (d+e x)-\log \left (-\frac {e \left (i+\sqrt [3]{c} x\right )}{\sqrt [3]{c} d-i e}\right ) \log (d+e x)-\log \left (\frac {e \left (-i+\sqrt {3}+2 \sqrt [3]{c} x\right )}{-2 \sqrt [3]{c} d+\left (-i+\sqrt {3}\right ) e}\right ) \log (d+e x)+\log \left (\frac {e \left (i+\sqrt {3}+2 \sqrt [3]{c} x\right )}{-2 \sqrt [3]{c} d+\left (i+\sqrt {3}\right ) e}\right ) \log (d+e x)-\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-i e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+i e}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+i e-\sqrt {3} e}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+\left (-i+\sqrt {3}\right ) e}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-\left (i+\sqrt {3}\right ) e}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+\left (i+\sqrt {3}\right ) e}\right )\right )\right )}{2 e} \] Input:
Integrate[(a + b*ArcTan[c*x^3])/(d + e*x),x]
Output:
(a*Log[d + e*x])/e + (b*(2*ArcTan[c*x^3]*Log[d + e*x] - I*(Log[(e*(-I + Sq rt[3] - 2*c^(1/3)*x))/(2*c^(1/3)*d + (-I + Sqrt[3])*e)]*Log[d + e*x] - Log [(e*(I + Sqrt[3] - 2*c^(1/3)*x))/(2*c^(1/3)*d + (I + Sqrt[3])*e)]*Log[d + e*x] + Log[(e*(I - c^(1/3)*x))/(c^(1/3)*d + I*e)]*Log[d + e*x] - Log[-((e* (I + c^(1/3)*x))/(c^(1/3)*d - I*e))]*Log[d + e*x] - Log[(e*(-I + Sqrt[3] + 2*c^(1/3)*x))/(-2*c^(1/3)*d + (-I + Sqrt[3])*e)]*Log[d + e*x] + Log[(e*(I + Sqrt[3] + 2*c^(1/3)*x))/(-2*c^(1/3)*d + (I + Sqrt[3])*e)]*Log[d + e*x] - PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d - I*e)] + PolyLog[2, (c^(1/3)* (d + e*x))/(c^(1/3)*d + I*e)] - PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3 )*d + I*e - Sqrt[3]*e)] + PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d + (-I + Sqrt[3])*e)] + PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d - (I + Sqrt[3])*e)] - PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d + (I + Sqrt[3 ])*e)])))/(2*e)
Time = 1.49 (sec) , antiderivative size = 687, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5391, 2863, 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.
| \(\displaystyle \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 5391 |
| \(\displaystyle \frac {\log (d+e x) \left (a+b \arctan \left (c x^3\right )\right )}{e}-\frac {3 b c \int \frac {x^2 \log (d+e x)}{c^2 x^6+1}dx}{e}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {\log (d+e x) \left (a+b \arctan \left (c x^3\right )\right )}{e}-\frac {3 b c \int \left (-\frac {c^2 \log (d+e x) x^2}{2 \sqrt {-c^2} \left (\sqrt {-c^2}-c^2 x^3\right )}-\frac {c^2 \log (d+e x) x^2}{2 \sqrt {-c^2} \left (c^2 x^3+\sqrt {-c^2}\right )}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (d+e x) \left (a+b \arctan \left (c x^3\right )\right )}{e}-\frac {3 b c \left (\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-e}\right )}{6 \sqrt {-c^2}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+e}\right )}{6 \sqrt {-c^2}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{6 \sqrt {-c^2}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{6 \sqrt {-c^2}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{6 \sqrt {-c^2}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{6 \sqrt {-c^2}}-\frac {\log (d+e x) \log \left (\frac {e \left (1-\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+e}\right )}{6 \sqrt {-c^2}}+\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d-e}\right )}{6 \sqrt {-c^2}}-\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt [6]{-c^2} x+\sqrt [3]{-1}\right )}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{6 \sqrt {-c^2}}+\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt [6]{-c^2} x+(-1)^{2/3}\right )}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{6 \sqrt {-c^2}}-\frac {\log (d+e x) \log \left (\frac {(-1)^{2/3} e \left (\sqrt [3]{-1} \sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{6 \sqrt {-c^2}}+\frac {\log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{6 \sqrt {-c^2}}\right )}{e}\) |
Input:
Int[(a + b*ArcTan[c*x^3])/(d + e*x),x]
Output:
((a + b*ArcTan[c*x^3])*Log[d + e*x])/e - (3*b*c*(-1/6*(Log[(e*(1 - (-c^2)^ (1/6)*x))/((-c^2)^(1/6)*d + e)]*Log[d + e*x])/Sqrt[-c^2] + (Log[-((e*(1 + (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d - e))]*Log[d + e*x])/(6*Sqrt[-c^2]) - (Lo g[-((e*((-1)^(1/3) + (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d - (-1)^(1/3)*e))]*Lo g[d + e*x])/(6*Sqrt[-c^2]) + (Log[-((e*((-1)^(2/3) + (-c^2)^(1/6)*x))/((-c ^2)^(1/6)*d - (-1)^(2/3)*e))]*Log[d + e*x])/(6*Sqrt[-c^2]) - (Log[((-1)^(2 /3)*e*(1 + (-1)^(1/3)*(-c^2)^(1/6)*x))/((-c^2)^(1/6)*d + (-1)^(2/3)*e)]*Lo g[d + e*x])/(6*Sqrt[-c^2]) + (Log[((-1)^(1/3)*e*(1 + (-1)^(2/3)*(-c^2)^(1/ 6)*x))/((-c^2)^(1/6)*d + (-1)^(1/3)*e)]*Log[d + e*x])/(6*Sqrt[-c^2]) + Pol yLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d - e)]/(6*Sqrt[-c^2]) - Po lyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d + e)]/(6*Sqrt[-c^2]) - P olyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d - (-1)^(1/3)*e)]/(6*Sqr t[-c^2]) + PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d + (-1)^(1/3 )*e)]/(6*Sqrt[-c^2]) + PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d - (-1)^(2/3)*e)]/(6*Sqrt[-c^2]) - PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((- c^2)^(1/6)*d + (-1)^(2/3)*e)]/(6*Sqrt[-c^2])))/e
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[d + e*x]*((a + b*ArcTan[c*x^n])/e), x] - Simp[b*c*(n/e) Int[x ^(n - 1)*(Log[d + e*x]/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.46 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.23
| method | result | size |
| default | \(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctan \left (c \,x^{3}\right )}{e}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c^{2}-6 c^{2} d \,\textit {\_Z}^{5}+15 c^{2} d^{2} \textit {\_Z}^{4}-20 c^{2} d^{3} \textit {\_Z}^{3}+15 c^{2} d^{4} \textit {\_Z}^{2}-6 c^{2} d^{5} \textit {\_Z} +c^{2} d^{6}+e^{6}\right )}{\sum }\frac {\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{3}-3 \textit {\_R1}^{2} d +3 \textit {\_R1} \,d^{2}-d^{3}}\right )}{2 c}\) | \(172\) |
| parts | \(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctan \left (c \,x^{3}\right )}{e}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c^{2}-6 c^{2} d \,\textit {\_Z}^{5}+15 c^{2} d^{2} \textit {\_Z}^{4}-20 c^{2} d^{3} \textit {\_Z}^{3}+15 c^{2} d^{4} \textit {\_Z}^{2}-6 c^{2} d^{5} \textit {\_Z} +c^{2} d^{6}+e^{6}\right )}{\sum }\frac {\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{3}-3 \textit {\_R1}^{2} d +3 \textit {\_R1} \,d^{2}-d^{3}}\right )}{2 c}\) | \(172\) |
| risch | \(\frac {i b \ln \left (e x +d \right ) \ln \left (-i c \,x^{3}+1\right )}{2 e}-\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}+e^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right )\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}+\frac {a \ln \left (e x +d \right )}{e}-\frac {i b \ln \left (e x +d \right ) \ln \left (i c \,x^{3}+1\right )}{2 e}+\frac {i b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}-e^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right )\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}\) | \(232\) |
Input:
int((a+b*arctan(c*x^3))/(e*x+d),x,method=_RETURNVERBOSE)
Output:
a*ln(e*x+d)/e+b*ln(e*x+d)/e*arctan(c*x^3)-1/2*b*e^2/c*sum(1/(_R1^3-3*_R1^2 *d+3*_R1*d^2-d^3)*(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog((-e*x+_R1-d)/_R1)) ,_R1=RootOf(_Z^6*c^2-6*_Z^5*c^2*d+15*_Z^4*c^2*d^2-20*_Z^3*c^2*d^3+15*_Z^2* c^2*d^4-6*_Z*c^2*d^5+c^2*d^6+e^6))
\[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{3}\right ) + a}{e x + d} \,d x } \] Input:
integrate((a+b*arctan(c*x^3))/(e*x+d),x, algorithm="fricas")
Output:
integral((b*arctan(c*x^3) + a)/(e*x + d), x)
Timed out. \[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\text {Timed out} \] Input:
integrate((a+b*atan(c*x**3))/(e*x+d),x)
Output:
Timed out
\[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{3}\right ) + a}{e x + d} \,d x } \] Input:
integrate((a+b*arctan(c*x^3))/(e*x+d),x, algorithm="maxima")
Output:
2*b*integrate(1/2*arctan(c*x^3)/(e*x + d), x) + a*log(e*x + d)/e
\[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{3}\right ) + a}{e x + d} \,d x } \] Input:
integrate((a+b*arctan(c*x^3))/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctan(c*x^3) + a)/(e*x + d), x)
Timed out. \[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x^3\right )}{d+e\,x} \,d x \] Input:
int((a + b*atan(c*x^3))/(d + e*x),x)
Output:
int((a + b*atan(c*x^3))/(d + e*x), x)
\[ \int \frac {a+b \arctan \left (c x^3\right )}{d+e x} \, dx=\frac {\left (\int \frac {\mathit {atan} \left (c \,x^{3}\right )}{e x +d}d x \right ) b e +\mathrm {log}\left (e x +d \right ) a}{e} \] Input:
int((a+b*atan(c*x^3))/(e*x+d),x)
Output:
(int(atan(c*x**3)/(d + e*x),x)*b*e + log(d + e*x)*a)/e