Integrand size = 18, antiderivative size = 501 \[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\frac {\left (a+b \arctan \left (c x^2\right )\right ) \log (d+e x)}{e}+\frac {b c \log \left (\frac {e \left (1-\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \log \left (-\frac {e \left (1+\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (\frac {e \left (1-\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d+e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}-\frac {b c \log \left (-\frac {e \left (1+\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d-e}\right ) \log (d+e x)}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d+e}\right )}{2 \sqrt {-c^2} e} \] Output:
(a+b*arctan(c*x^2))*ln(e*x+d)/e+1/2*b*c*ln(e*(1-(-c^2)^(1/4)*x)/((-c^2)^(1 /4)*d+e))*ln(e*x+d)/(-c^2)^(1/2)/e+1/2*b*c*ln(-e*(1+(-c^2)^(1/4)*x)/((-c^2 )^(1/4)*d-e))*ln(e*x+d)/(-c^2)^(1/2)/e-1/2*b*c*ln(e*(1-(-(-c^2)^(1/2))^(1/ 2)*x)/((-(-c^2)^(1/2))^(1/2)*d+e))*ln(e*x+d)/(-c^2)^(1/2)/e-1/2*b*c*ln(-e* (1+(-(-c^2)^(1/2))^(1/2)*x)/((-(-c^2)^(1/2))^(1/2)*d-e))*ln(e*x+d)/(-c^2)^ (1/2)/e+1/2*b*c*polylog(2,(-c^2)^(1/4)*(e*x+d)/((-c^2)^(1/4)*d-e))/(-c^2)^ (1/2)/e-1/2*b*c*polylog(2,(-(-c^2)^(1/2))^(1/2)*(e*x+d)/((-(-c^2)^(1/2))^( 1/2)*d-e))/(-c^2)^(1/2)/e+1/2*b*c*polylog(2,(-c^2)^(1/4)*(e*x+d)/((-c^2)^( 1/4)*d+e))/(-c^2)^(1/2)/e-1/2*b*c*polylog(2,(-(-c^2)^(1/2))^(1/2)*(e*x+d)/ ((-(-c^2)^(1/2))^(1/2)*d+e))/(-c^2)^(1/2)/e
Result contains complex when optimal does not.
Time = 14.62 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.65 \[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\frac {a \log (d+e x)}{e}+\frac {b \left (2 \arctan \left (c x^2\right ) \log (d+e x)+i \left (\log (d+e x) \log \left (1-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt [4]{-1} e}\right )+\log (d+e x) \log \left (1-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt [4]{-1} e}\right )-\log (d+e x) \log \left (1-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-(-1)^{3/4} e}\right )-\log (d+e x) \log \left (1-\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+(-1)^{3/4} e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt [4]{-1} e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt [4]{-1} e}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-(-1)^{3/4} e}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+(-1)^{3/4} e}\right )\right )\right )}{2 e} \] Input:
Integrate[(a + b*ArcTan[c*x^2])/(d + e*x),x]
Output:
(a*Log[d + e*x])/e + (b*(2*ArcTan[c*x^2]*Log[d + e*x] + I*(Log[d + e*x]*Lo g[1 - (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - (-1)^(1/4)*e)] + Log[d + e*x]*Log[1 - (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + (-1)^(1/4)*e)] - Log[d + e*x]*Log[1 - (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - (-1)^(3/4)*e)] - Log[d + e*x]*Log[1 - (Sq rt[c]*(d + e*x))/(Sqrt[c]*d + (-1)^(3/4)*e)] + PolyLog[2, (Sqrt[c]*(d + e* x))/(Sqrt[c]*d - (-1)^(1/4)*e)] + PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]* d + (-1)^(1/4)*e)] - PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - (-1)^(3/4 )*e)] - PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + (-1)^(3/4)*e)])))/(2*e )
Time = 1.12 (sec) , antiderivative size = 469, normalized size of antiderivative = 0.94, 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.
| \(\displaystyle \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 5391 |
| \(\displaystyle \frac {\log (d+e x) \left (a+b \arctan \left (c x^2\right )\right )}{e}-\frac {2 b c \int \frac {x \log (d+e x)}{c^2 x^4+1}dx}{e}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {\log (d+e x) \left (a+b \arctan \left (c x^2\right )\right )}{e}-\frac {2 b c \int \left (-\frac {x \log (d+e x) c^2}{2 \sqrt {-c^2} \left (\sqrt {-c^2}-c^2 x^2\right )}-\frac {x \log (d+e x) c^2}{2 \sqrt {-c^2} \left (c^2 x^2+\sqrt {-c^2}\right )}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (d+e x) \left (a+b \arctan \left (c x^2\right )\right )}{e}-\frac {2 b c \left (-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d-e}\right )}{4 \sqrt {-c^2}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d-e}\right )}{4 \sqrt {-c^2}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{-c^2} (d+e x)}{\sqrt [4]{-c^2} d+e}\right )}{4 \sqrt {-c^2}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {-\sqrt {-c^2}} (d+e x)}{\sqrt {-\sqrt {-c^2}} d+e}\right )}{4 \sqrt {-c^2}}-\frac {\log (d+e x) \log \left (\frac {e \left (1-\sqrt [4]{-c^2} x\right )}{\sqrt [4]{-c^2} d+e}\right )}{4 \sqrt {-c^2}}-\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt [4]{-c^2} x+1\right )}{\sqrt [4]{-c^2} d-e}\right )}{4 \sqrt {-c^2}}+\frac {\log (d+e x) \log \left (\frac {e \left (1-\sqrt {-\sqrt {-c^2}} x\right )}{\sqrt {-\sqrt {-c^2}} d+e}\right )}{4 \sqrt {-c^2}}+\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt {-\sqrt {-c^2}} x+1\right )}{\sqrt {-\sqrt {-c^2}} d-e}\right )}{4 \sqrt {-c^2}}\right )}{e}\) |
Input:
Int[(a + b*ArcTan[c*x^2])/(d + e*x),x]
Output:
((a + b*ArcTan[c*x^2])*Log[d + e*x])/e - (2*b*c*(-1/4*(Log[(e*(1 - (-c^2)^ (1/4)*x))/((-c^2)^(1/4)*d + e)]*Log[d + e*x])/Sqrt[-c^2] - (Log[-((e*(1 + (-c^2)^(1/4)*x))/((-c^2)^(1/4)*d - e))]*Log[d + e*x])/(4*Sqrt[-c^2]) + (Lo g[(e*(1 - Sqrt[-Sqrt[-c^2]]*x))/(Sqrt[-Sqrt[-c^2]]*d + e)]*Log[d + e*x])/( 4*Sqrt[-c^2]) + (Log[-((e*(1 + Sqrt[-Sqrt[-c^2]]*x))/(Sqrt[-Sqrt[-c^2]]*d - e))]*Log[d + e*x])/(4*Sqrt[-c^2]) - PolyLog[2, ((-c^2)^(1/4)*(d + e*x))/ ((-c^2)^(1/4)*d - e)]/(4*Sqrt[-c^2]) + PolyLog[2, (Sqrt[-Sqrt[-c^2]]*(d + e*x))/(Sqrt[-Sqrt[-c^2]]*d - e)]/(4*Sqrt[-c^2]) - PolyLog[2, ((-c^2)^(1/4) *(d + e*x))/((-c^2)^(1/4)*d + e)]/(4*Sqrt[-c^2]) + PolyLog[2, (Sqrt[-Sqrt[ -c^2]]*(d + e*x))/(Sqrt[-Sqrt[-c^2]]*d + e)]/(4*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 = 0.38 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.28
| method | result | size |
| default | \(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctan \left (c \,x^{2}\right )}{e}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{4}-4 c^{2} d \,\textit {\_Z}^{3}+6 c^{2} d^{2} \textit {\_Z}^{2}-4 c^{2} d^{3} \textit {\_Z} +c^{2} d^{4}+e^{4}\right )}{\sum }\frac {\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} d +d^{2}}\right )}{2 c}\) | \(138\) |
| parts | \(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctan \left (c \,x^{2}\right )}{e}-\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{4}-4 c^{2} d \,\textit {\_Z}^{3}+6 c^{2} d^{2} \textit {\_Z}^{2}-4 c^{2} d^{3} \textit {\_Z} +c^{2} d^{4}+e^{4}\right )}{\sum }\frac {\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} d +d^{2}}\right )}{2 c}\) | \(138\) |
| risch | \(\frac {i b \ln \left (e x +d \right ) \ln \left (-i c \,x^{2}+1\right )}{2 e}-\frac {i b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-i c}-\left (e x +d \right ) c +c d}{e \sqrt {-i c}+c d}\right )}{2 e}-\frac {i b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-i c}+\left (e x +d \right ) c -c d}{e \sqrt {-i c}-c d}\right )}{2 e}-\frac {i b \operatorname {dilog}\left (\frac {e \sqrt {-i c}-\left (e x +d \right ) c +c d}{e \sqrt {-i c}+c d}\right )}{2 e}-\frac {i b \operatorname {dilog}\left (\frac {e \sqrt {-i c}+\left (e x +d \right ) c -c d}{e \sqrt {-i c}-c d}\right )}{2 e}+\frac {a \ln \left (e x +d \right )}{e}-\frac {i b \ln \left (e x +d \right ) \ln \left (i c \,x^{2}+1\right )}{2 e}+\frac {i b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {i c}-\left (e x +d \right ) c +c d}{e \sqrt {i c}+c d}\right )}{2 e}+\frac {i b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {i c}+\left (e x +d \right ) c -c d}{e \sqrt {i c}-c d}\right )}{2 e}+\frac {i b \operatorname {dilog}\left (\frac {e \sqrt {i c}-\left (e x +d \right ) c +c d}{e \sqrt {i c}+c d}\right )}{2 e}+\frac {i b \operatorname {dilog}\left (\frac {e \sqrt {i c}+\left (e x +d \right ) c -c d}{e \sqrt {i c}-c d}\right )}{2 e}\) | \(431\) |
Input:
int((a+b*arctan(c*x^2))/(e*x+d),x,method=_RETURNVERBOSE)
Output:
a*ln(e*x+d)/e+b*ln(e*x+d)/e*arctan(c*x^2)-1/2*b*e/c*sum(1/(_R1^2-2*_R1*d+d ^2)*(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog((-e*x+_R1-d)/_R1)),_R1=RootOf(_Z ^4*c^2-4*_Z^3*c^2*d+6*_Z^2*c^2*d^2-4*_Z*c^2*d^3+c^2*d^4+e^4))
\[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{2}\right ) + a}{e x + d} \,d x } \] Input:
integrate((a+b*arctan(c*x^2))/(e*x+d),x, algorithm="fricas")
Output:
integral((b*arctan(c*x^2) + a)/(e*x + d), x)
Timed out. \[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\text {Timed out} \] Input:
integrate((a+b*atan(c*x**2))/(e*x+d),x)
Output:
Timed out
\[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{2}\right ) + a}{e x + d} \,d x } \] Input:
integrate((a+b*arctan(c*x^2))/(e*x+d),x, algorithm="maxima")
Output:
2*b*integrate(1/2*arctan(c*x^2)/(e*x + d), x) + a*log(e*x + d)/e
\[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\int { \frac {b \arctan \left (c x^{2}\right ) + a}{e x + d} \,d x } \] Input:
integrate((a+b*arctan(c*x^2))/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctan(c*x^2) + a)/(e*x + d), x)
Timed out. \[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x^2\right )}{d+e\,x} \,d x \] Input:
int((a + b*atan(c*x^2))/(d + e*x),x)
Output:
int((a + b*atan(c*x^2))/(d + e*x), x)
\[ \int \frac {a+b \arctan \left (c x^2\right )}{d+e x} \, dx=\frac {\left (\int \frac {\mathit {atan} \left (c \,x^{2}\right )}{e x +d}d x \right ) b e +\mathrm {log}\left (e x +d \right ) a}{e} \] Input:
int((a+b*atan(c*x^2))/(e*x+d),x)
Output:
(int(atan(c*x**2)/(d + e*x),x)*b*e + log(d + e*x)*a)/e