Integrand size = 24, antiderivative size = 163 \[ \int \frac {\left (a+b \arctan \left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}+\frac {i b d \operatorname {PolyLog}\left (2,-i c x^n\right )}{2 n}+\frac {i b e \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,-i c x^n\right )}{2 n}-\frac {i b d \operatorname {PolyLog}\left (2,i c x^n\right )}{2 n}-\frac {i b e \log \left (f x^m\right ) \operatorname {PolyLog}\left (2,i c x^n\right )}{2 n}-\frac {i b e m \operatorname {PolyLog}\left (3,-i c x^n\right )}{2 n^2}+\frac {i b e m \operatorname {PolyLog}\left (3,i c x^n\right )}{2 n^2} \]
a*d*ln(x)+1/2*a*e*ln(f*x^m)^2/m+1/2*I*b*d*polylog(2,-I*c*x^n)/n+1/2*I*b*e* ln(f*x^m)*polylog(2,-I*c*x^n)/n-1/2*I*b*d*polylog(2,I*c*x^n)/n-1/2*I*b*e*l n(f*x^m)*polylog(2,I*c*x^n)/n-1/2*I*b*e*m*polylog(3,-I*c*x^n)/n^2+1/2*I*b* e*m*polylog(3,I*c*x^n)/n^2
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a+b \arctan \left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=-\frac {b c e m x^n \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2},\frac {3}{2};-c^2 x^{2 n}\right )}{n^2}+\frac {b c x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-c^2 x^{2 n}\right ) \left (d+e \log \left (f x^m\right )\right )}{n}+\frac {1}{2} a \log (x) \left (2 d-e m \log (x)+2 e \log \left (f x^m\right )\right ) \]
-((b*c*e*m*x^n*HypergeometricPFQ[{1/2, 1/2, 1/2, 1}, {3/2, 3/2, 3/2}, -(c^ 2*x^(2*n))])/n^2) + (b*c*x^n*HypergeometricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, -(c^2*x^(2*n))]*(d + e*Log[f*x^m]))/n + (a*Log[x]*(2*d - e*m*Log[x] + 2*e* Log[f*x^m]))/2
Time = 0.80 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {7293, 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.
| \(\displaystyle \int \frac {\left (a+b \arctan \left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {d \left (a+b \arctan \left (c x^n\right )\right )}{x}+\frac {e \log \left (f x^m\right ) \left (a+b \arctan \left (c x^n\right )\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}+\frac {i b d \operatorname {PolyLog}\left (2,-i c x^n\right )}{2 n}-\frac {i b d \operatorname {PolyLog}\left (2,i c x^n\right )}{2 n}+\frac {i b e \operatorname {PolyLog}\left (2,-i c x^n\right ) \log \left (f x^m\right )}{2 n}-\frac {i b e \operatorname {PolyLog}\left (2,i c x^n\right ) \log \left (f x^m\right )}{2 n}-\frac {i b e m \operatorname {PolyLog}\left (3,-i c x^n\right )}{2 n^2}+\frac {i b e m \operatorname {PolyLog}\left (3,i c x^n\right )}{2 n^2}\) |
a*d*Log[x] + (a*e*Log[f*x^m]^2)/(2*m) + ((I/2)*b*d*PolyLog[2, (-I)*c*x^n]) /n + ((I/2)*b*e*Log[f*x^m]*PolyLog[2, (-I)*c*x^n])/n - ((I/2)*b*d*PolyLog[ 2, I*c*x^n])/n - ((I/2)*b*e*Log[f*x^m]*PolyLog[2, I*c*x^n])/n - ((I/2)*b*e *m*PolyLog[3, (-I)*c*x^n])/n^2 + ((I/2)*b*e*m*PolyLog[3, I*c*x^n])/n^2
3.2.53.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 212.62 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.36
| method | result | size |
| risch | \(\frac {\left (-\frac {i e \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )}{4}+\frac {i e \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{4}+\frac {i e \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{4}-\frac {i e \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}}{4}+\frac {e \ln \left (f \right )}{2}+\frac {d}{2}\right ) \left (-i b \operatorname {dilog}\left (1-i c \,x^{n}\right )+2 \ln \left (x^{n}\right ) a +i b \operatorname {dilog}\left (1+i c \,x^{n}\right )\right )}{n}+\frac {i e b \ln \left (x \right )^{2} \ln \left (-i \left (c \,x^{n}+i\right )\right ) m}{2}+\frac {i b e m \operatorname {polylog}\left (3, i c \,x^{n}\right )}{2 n^{2}}-\frac {i e b \operatorname {dilog}\left (-i \left (c \,x^{n}+i\right )\right ) \ln \left (x^{m}\right )}{2 n}-\frac {i e b \ln \left (x \right )^{2} \ln \left (-i \left (-c \,x^{n}+i\right )\right ) m}{2}+\frac {i e b \ln \left (x \right ) \ln \left (1-i c \,x^{n}\right ) \ln \left (x^{m}\right )}{2}+\frac {i e b m \ln \left (x \right ) \operatorname {polylog}\left (2, -i c \,x^{n}\right )}{2 n}-\frac {i e b \operatorname {dilog}\left (-i c \,x^{n}\right ) \ln \left (x^{m}\right )}{2 n}-\frac {i e b \ln \left (-i \left (-c \,x^{n}+i\right )\right ) \ln \left (-i c \,x^{n}\right ) \ln \left (x^{m}\right )}{2 n}+\frac {e a \ln \left (x^{m}\right )^{2}}{2 m}-\frac {i e b \ln \left (x \right ) \ln \left (x^{m}\right ) \ln \left (1+i c \,x^{n}\right )}{2}-\frac {i e b \ln \left (x \right )^{2} \ln \left (1-i c \,x^{n}\right ) m}{2}+\frac {i e b \ln \left (x \right )^{2} \ln \left (1+i c \,x^{n}\right ) m}{2}+\frac {i e b \ln \left (-i \left (-c \,x^{n}+i\right )\right ) \ln \left (-i c \,x^{n}\right ) m \ln \left (x \right )}{2 n}-\frac {i e b \ln \left (x \right ) \ln \left (-i \left (c \,x^{n}+i\right )\right ) \ln \left (x^{m}\right )}{2}+\frac {i e b \operatorname {dilog}\left (-i \left (c \,x^{n}+i\right )\right ) m \ln \left (x \right )}{2 n}-\frac {i b e m \operatorname {polylog}\left (3, -i c \,x^{n}\right )}{2 n^{2}}-\frac {i e b m \ln \left (x \right ) \operatorname {polylog}\left (2, i c \,x^{n}\right )}{2 n}+\frac {i e b \operatorname {dilog}\left (-i c \,x^{n}\right ) m \ln \left (x \right )}{2 n}+\frac {i e b \ln \left (x \right ) \ln \left (x^{m}\right ) \ln \left (-i \left (-c \,x^{n}+i\right )\right )}{2}\) | \(547\) |
(-1/4*I*e*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/4*I*e*Pi*csgn(I*f)*csgn (I*f*x^m)^2+1/4*I*e*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/4*I*e*Pi*csgn(I*f*x^m )^3+1/2*e*ln(f)+1/2*d)/n*(-I*b*dilog(1-I*c*x^n)+2*ln(x^n)*a+I*b*dilog(1+I* c*x^n))+1/2*I*e*b*ln(x)^2*ln(-I*(c*x^n+I))*m+1/2*I*b*e*m*polylog(3,I*c*x^n )/n^2-1/2*I*e*b/n*dilog(-I*(c*x^n+I))*ln(x^m)-1/2*I*e*b*ln(x)^2*ln(-I*(-c* x^n+I))*m+1/2*I*e*b*ln(x)*ln(1-I*c*x^n)*ln(x^m)+1/2*I*e*b*m/n*ln(x)*polylo g(2,-I*c*x^n)-1/2*I*e*b/n*dilog(-I*c*x^n)*ln(x^m)-1/2*I*e*b/n*ln(-I*(-c*x^ n+I))*ln(-I*c*x^n)*ln(x^m)+1/2*e*a/m*ln(x^m)^2-1/2*I*e*b*ln(x)*ln(x^m)*ln( 1+I*c*x^n)-1/2*I*e*b*ln(x)^2*ln(1-I*c*x^n)*m+1/2*I*e*b*ln(x)^2*ln(1+I*c*x^ n)*m+1/2*I*e*b/n*ln(-I*(-c*x^n+I))*ln(-I*c*x^n)*m*ln(x)-1/2*I*e*b*ln(x)*ln (-I*(c*x^n+I))*ln(x^m)+1/2*I*e*b/n*dilog(-I*(c*x^n+I))*m*ln(x)-1/2*I*b*e*m *polylog(3,-I*c*x^n)/n^2-1/2*I*e*b*m/n*ln(x)*polylog(2,I*c*x^n)+1/2*I*e*b/ n*dilog(-I*c*x^n)*m*ln(x)+1/2*I*e*b*ln(x)*ln(x^m)*ln(-I*(-c*x^n+I))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (121) = 242\).
Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.53 \[ \int \frac {\left (a+b \arctan \left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\frac {2 \, a e m n^{2} \log \left (x\right )^{2} + 2 i \, b e m {\rm polylog}\left (3, i \, c x^{n}\right ) - 2 i \, b e m {\rm polylog}\left (3, -i \, c x^{n}\right ) + 2 \, {\left (b e m n^{2} \log \left (x\right )^{2} + 2 \, {\left (b e n^{2} \log \left (f\right ) + b d n^{2}\right )} \log \left (x\right )\right )} \arctan \left (c x^{n}\right ) - 2 \, {\left (i \, b e m n \log \left (x\right ) + i \, b e n \log \left (f\right ) + i \, b d n\right )} {\rm Li}_2\left (i \, c x^{n}\right ) - 2 \, {\left (-i \, b e m n \log \left (x\right ) - i \, b e n \log \left (f\right ) - i \, b d n\right )} {\rm Li}_2\left (-i \, c x^{n}\right ) + {\left (i \, b e m n^{2} \log \left (x\right )^{2} - 2 \, {\left (-i \, b e n^{2} \log \left (f\right ) - i \, b d n^{2}\right )} \log \left (x\right )\right )} \log \left (i \, c x^{n} + 1\right ) + {\left (-i \, b e m n^{2} \log \left (x\right )^{2} - 2 \, {\left (i \, b e n^{2} \log \left (f\right ) + i \, b d n^{2}\right )} \log \left (x\right )\right )} \log \left (-i \, c x^{n} + 1\right ) + 4 \, {\left (a e n^{2} \log \left (f\right ) + a d n^{2}\right )} \log \left (x\right )}{4 \, n^{2}} \]
1/4*(2*a*e*m*n^2*log(x)^2 + 2*I*b*e*m*polylog(3, I*c*x^n) - 2*I*b*e*m*poly log(3, -I*c*x^n) + 2*(b*e*m*n^2*log(x)^2 + 2*(b*e*n^2*log(f) + b*d*n^2)*lo g(x))*arctan(c*x^n) - 2*(I*b*e*m*n*log(x) + I*b*e*n*log(f) + I*b*d*n)*dilo g(I*c*x^n) - 2*(-I*b*e*m*n*log(x) - I*b*e*n*log(f) - I*b*d*n)*dilog(-I*c*x ^n) + (I*b*e*m*n^2*log(x)^2 - 2*(-I*b*e*n^2*log(f) - I*b*d*n^2)*log(x))*lo g(I*c*x^n + 1) + (-I*b*e*m*n^2*log(x)^2 - 2*(I*b*e*n^2*log(f) + I*b*d*n^2) *log(x))*log(-I*c*x^n + 1) + 4*(a*e*n^2*log(f) + a*d*n^2)*log(x))/n^2
Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \arctan \left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x^{n}\right ) + a\right )} {\left (e \log \left (f x^{m}\right ) + d\right )}}{x} \,d x } \]
1/2*a*e*log(f*x^m)^2/m + a*d*log(x) - 1/2*(b*e*m*log(x)^2 - 2*b*e*log(x)*l og(x^m) - 2*(b*e*log(f) + b*d)*log(x))*arctan(c*x^n) - integrate(-1/2*(b*c *e*m*n*x^n*log(x)^2 - 2*b*c*e*n*x^n*log(x)*log(x^m) - 2*(b*c*e*log(f) + b* c*d)*n*x^n*log(x))/(c^2*x*x^(2*n) + x), x)
\[ \int \frac {\left (a+b \arctan \left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \arctan \left (c x^{n}\right ) + a\right )} {\left (e \log \left (f x^{m}\right ) + d\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x^n\right )\right )\,\left (d+e\,\ln \left (f\,x^m\right )\right )}{x} \,d x \]