Integrand size = 19, antiderivative size = 200 \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=-\frac {a+b \text {arctanh}(c x)}{d x}+\frac {b c \log (x)}{d}-\frac {a e \log (x)}{d^2}-\frac {e (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d^2}+\frac {e (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^2}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d}+\frac {b e \operatorname {PolyLog}(2,-c x)}{2 d^2}-\frac {b e \operatorname {PolyLog}(2,c x)}{2 d^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d^2}-\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^2} \] Output:
-(a+b*arctanh(c*x))/d/x+b*c*ln(x)/d-a*e*ln(x)/d^2-e*(a+b*arctanh(c*x))*ln( 2/(c*x+1))/d^2+e*(a+b*arctanh(c*x))*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/d^2-1/ 2*b*c*ln(-c^2*x^2+1)/d+1/2*b*e*polylog(2,-c*x)/d^2-1/2*b*e*polylog(2,c*x)/ d^2+1/2*b*e*polylog(2,1-2/(c*x+1))/d^2-1/2*b*e*polylog(2,1-2*c*(e*x+d)/(c* d+e)/(c*x+1))/d^2
Result contains complex when optimal does not.
Time = 2.37 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.80 \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=-\frac {\frac {2 a d^2}{x}-i b d e \pi \text {arctanh}(c x)+\frac {2 b d^2 \text {arctanh}(c x)}{x}-2 b d e \text {arctanh}\left (\frac {c d}{e}\right ) \text {arctanh}(c x)+b d e \text {arctanh}(c x)^2-\frac {b e^2 \text {arctanh}(c x)^2}{c}+\frac {b \sqrt {1-\frac {c^2 d^2}{e^2}} e^2 e^{-\text {arctanh}\left (\frac {c d}{e}\right )} \text {arctanh}(c x)^2}{c}+2 b d e \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+i b d e \pi \log \left (1+e^{2 \text {arctanh}(c x)}\right )-2 b d e \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-2 b d e \text {arctanh}(c x) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )+2 a d e \log (x)-2 a d e \log (d+e x)-2 b c d^2 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+\frac {1}{2} i b d e \pi \log \left (1-c^2 x^2\right )+2 b d e \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )-b d e \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+b d e \operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )}{2 d^3} \] Input:
Integrate[(a + b*ArcTanh[c*x])/(x^2*(d + e*x)),x]
Output:
-1/2*((2*a*d^2)/x - I*b*d*e*Pi*ArcTanh[c*x] + (2*b*d^2*ArcTanh[c*x])/x - 2 *b*d*e*ArcTanh[(c*d)/e]*ArcTanh[c*x] + b*d*e*ArcTanh[c*x]^2 - (b*e^2*ArcTa nh[c*x]^2)/c + (b*Sqrt[1 - (c^2*d^2)/e^2]*e^2*ArcTanh[c*x]^2)/(c*E^ArcTanh [(c*d)/e]) + 2*b*d*e*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + I*b*d*e*P i*Log[1 + E^(2*ArcTanh[c*x])] - 2*b*d*e*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(Ar cTanh[(c*d)/e] + ArcTanh[c*x]))] - 2*b*d*e*ArcTanh[c*x]*Log[1 - E^(-2*(Arc Tanh[(c*d)/e] + ArcTanh[c*x]))] + 2*a*d*e*Log[x] - 2*a*d*e*Log[d + e*x] - 2*b*c*d^2*Log[(c*x)/Sqrt[1 - c^2*x^2]] + (I/2)*b*d*e*Pi*Log[1 - c^2*x^2] + 2*b*d*e*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - b *d*e*PolyLog[2, E^(-2*ArcTanh[c*x])] + b*d*e*PolyLog[2, E^(-2*(ArcTanh[(c* d)/e] + ArcTanh[c*x]))])/d^3
Time = 0.49 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6502, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {e^2 (a+b \text {arctanh}(c x))}{d^2 (d+e x)}-\frac {e (a+b \text {arctanh}(c x))}{d^2 x}+\frac {a+b \text {arctanh}(c x)}{d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {e \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^2}+\frac {e (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d^2}-\frac {a+b \text {arctanh}(c x)}{d x}-\frac {a e \log (x)}{d^2}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d}+\frac {b e \operatorname {PolyLog}(2,-c x)}{2 d^2}-\frac {b e \operatorname {PolyLog}(2,c x)}{2 d^2}+\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 d^2}-\frac {b e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 d^2}+\frac {b c \log (x)}{d}\) |
Input:
Int[(a + b*ArcTanh[c*x])/(x^2*(d + e*x)),x]
Output:
-((a + b*ArcTanh[c*x])/(d*x)) + (b*c*Log[x])/d - (a*e*Log[x])/d^2 - (e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^2 + (e*(a + b*ArcTanh[c*x])*Log[(2*c *(d + e*x))/((c*d + e)*(1 + c*x))])/d^2 - (b*c*Log[1 - c^2*x^2])/(2*d) + ( b*e*PolyLog[2, -(c*x)])/(2*d^2) - (b*e*PolyLog[2, c*x])/(2*d^2) + (b*e*Pol yLog[2, 1 - 2/(1 + c*x)])/(2*d^2) - (b*e*PolyLog[2, 1 - (2*c*(d + e*x))/(( c*d + e)*(1 + c*x))])/(2*d^2)
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Time = 0.36 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.34
| method | result | size |
| parts | \(a \left (-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}+\frac {e \ln \left (e x +d \right )}{d^{2}}\right )+b c \left (-\frac {\operatorname {arctanh}\left (c x \right )}{d c x}-\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c x \right )}{c \,d^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c e x +c d \right )}{c \,d^{2}}-c \left (\frac {\frac {\ln \left (c x -1\right )}{2}-\ln \left (c x \right )+\frac {\ln \left (c x +1\right )}{2}}{d c}+\frac {e \left (-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d^{2} c^{2}}-\frac {\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}}{d^{2} c^{2}}\right )\right )\) | \(267\) |
| derivativedivides | \(c \left (\frac {a e \ln \left (c e x +c d \right )}{c \,d^{2}}-\frac {a}{d c x}-\frac {a e \ln \left (c x \right )}{c \,d^{2}}+b c \left (\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c e x +c d \right )}{d^{2} c^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{d \,c^{2} x}-\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{2}}-\frac {\frac {\ln \left (c x -1\right )}{2}-\ln \left (c x \right )+\frac {\ln \left (c x +1\right )}{2}}{d c}-\frac {e \left (-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d^{2} c^{2}}+\frac {\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}}{d^{2} c^{2}}\right )\right )\) | \(280\) |
| default | \(c \left (\frac {a e \ln \left (c e x +c d \right )}{c \,d^{2}}-\frac {a}{d c x}-\frac {a e \ln \left (c x \right )}{c \,d^{2}}+b c \left (\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c e x +c d \right )}{d^{2} c^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{d \,c^{2} x}-\frac {\operatorname {arctanh}\left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{2}}-\frac {\frac {\ln \left (c x -1\right )}{2}-\ln \left (c x \right )+\frac {\ln \left (c x +1\right )}{2}}{d c}-\frac {e \left (-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d^{2} c^{2}}+\frac {\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}}{d^{2} c^{2}}\right )\right )\) | \(280\) |
| risch | \(-\frac {b e \operatorname {dilog}\left (-c x +1\right )}{2 d^{2}}-\frac {b e \operatorname {dilog}\left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 d^{2}}-\frac {b e \ln \left (-c x +1\right ) \ln \left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 d^{2}}+\frac {c b \ln \left (-c x \right )}{2 d}-\frac {c b \ln \left (-c x +1\right )}{2 d}+\frac {b \ln \left (-c x +1\right )}{2 d x}-\frac {a}{x d}-\frac {a e \ln \left (-c x \right )}{d^{2}}+\frac {a e \ln \left (\left (-c x +1\right ) e -c d -e \right )}{d^{2}}+\frac {b e \operatorname {dilog}\left (c x +1\right )}{2 d^{2}}+\frac {b e \operatorname {dilog}\left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 d^{2}}+\frac {b e \ln \left (c x +1\right ) \ln \left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 d^{2}}+\frac {b c \ln \left (c x \right )}{2 d}-\frac {b c \ln \left (c x +1\right )}{2 d}-\frac {b \ln \left (c x +1\right )}{2 d x}\) | \(301\) |
Input:
int((a+b*arctanh(c*x))/x^2/(e*x+d),x,method=_RETURNVERBOSE)
Output:
a*(-1/d/x-e/d^2*ln(x)+e/d^2*ln(e*x+d))+b*c*(-arctanh(c*x)/d/c/x-1/c*arctan h(c*x)*e/d^2*ln(c*x)+1/c*arctanh(c*x)*e/d^2*ln(c*e*x+c*d)-c*(1/d/c*(1/2*ln (c*x-1)-ln(c*x)+1/2*ln(c*x+1))+1/d^2/c^2*e*(-1/2*dilog(c*x)-1/2*dilog(c*x+ 1)-1/2*ln(c*x)*ln(c*x+1))-1/d^2/c^2*(1/2*e*(dilog((c*e*x-e)/(-c*d-e))+ln(c *e*x+c*d)*ln((c*e*x-e)/(-c*d-e)))-1/2*e*(dilog((c*e*x+e)/(-c*d+e))+ln(c*e* x+c*d)*ln((c*e*x+e)/(-c*d+e))))))
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^2/(e*x+d),x, algorithm="fricas")
Output:
integral((b*arctanh(c*x) + a)/(e*x^3 + d*x^2), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{x^{2} \left (d + e x\right )}\, dx \] Input:
integrate((a+b*atanh(c*x))/x**2/(e*x+d),x)
Output:
Integral((a + b*atanh(c*x))/(x**2*(d + e*x)), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^2/(e*x+d),x, algorithm="maxima")
Output:
a*(e*log(e*x + d)/d^2 - e*log(x)/d^2 - 1/(d*x)) + 1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/(e*x^3 + d*x^2), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^2/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)/((e*x + d)*x^2), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^2\,\left (d+e\,x\right )} \,d x \] Input:
int((a + b*atanh(c*x))/(x^2*(d + e*x)),x)
Output:
int((a + b*atanh(c*x))/(x^2*(d + e*x)), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+e x)} \, dx=\frac {-\left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} e \,x^{5}+c^{2} d \,x^{4}-e \,x^{3}-d \,x^{2}}d x \right ) b \,d^{2} x +\left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} e \,x^{3}+c^{2} d \,x^{2}-e x -d}d x \right ) b \,c^{2} d^{2} x +\mathrm {log}\left (e x +d \right ) a e x -\mathrm {log}\left (x \right ) a e x -a d}{d^{2} x} \] Input:
int((a+b*atanh(c*x))/x^2/(e*x+d),x)
Output:
( - int(atanh(c*x)/(c**2*d*x**4 + c**2*e*x**5 - d*x**2 - e*x**3),x)*b*d**2 *x + int(atanh(c*x)/(c**2*d*x**2 + c**2*e*x**3 - d - e*x),x)*b*c**2*d**2*x + log(d + e*x)*a*e*x - log(x)*a*e*x - a*d)/(d**2*x)