Integrand size = 19, antiderivative size = 148 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\frac {a \log (x)}{d}+\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d}-\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d}-\frac {b \operatorname {PolyLog}(2,-c x)}{2 d}+\frac {b \operatorname {PolyLog}(2,c x)}{2 d}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d} \] Output:
a*ln(x)/d+(a+b*arctanh(c*x))*ln(2/(c*x+1))/d-(a+b*arctanh(c*x))*ln(2*c*(e* x+d)/(c*d+e)/(c*x+1))/d-1/2*b*polylog(2,-c*x)/d+1/2*b*polylog(2,c*x)/d-1/2 *b*polylog(2,1-2/(c*x+1))/d+1/2*b*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1)) /d
Result contains complex when optimal does not.
Time = 1.19 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.99 \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\frac {2 a d \log (x)-2 a d \log (d+e x)+\frac {b \left (-i c d \pi \text {arctanh}(c x)-2 c d \text {arctanh}\left (\frac {c d}{e}\right ) \text {arctanh}(c x)+c d \text {arctanh}(c x)^2-e \text {arctanh}(c x)^2+\sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\text {arctanh}\left (\frac {c d}{e}\right )} \text {arctanh}(c x)^2+2 c d \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+i c d \pi \log \left (1+e^{2 \text {arctanh}(c x)}\right )-2 c d \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-2 c d \text {arctanh}(c x) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )+\frac {1}{2} i c d \pi \log \left (1-c^2 x^2\right )+2 c d \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )-c d \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+c d \operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )\right )}{c}}{2 d^2} \] Input:
Integrate[(a + b*ArcTanh[c*x])/(x*(d + e*x)),x]
Output:
(2*a*d*Log[x] - 2*a*d*Log[d + e*x] + (b*((-I)*c*d*Pi*ArcTanh[c*x] - 2*c*d* ArcTanh[(c*d)/e]*ArcTanh[c*x] + c*d*ArcTanh[c*x]^2 - e*ArcTanh[c*x]^2 + (S qrt[1 - (c^2*d^2)/e^2]*e*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] + 2*c*d*ArcTan h[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] + I*c*d*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 2*c*d*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x])) ] - 2*c*d*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] + (I/2)*c*d*Pi*Log[1 - c^2*x^2] + 2*c*d*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh [(c*d)/e] + ArcTanh[c*x]]] - c*d*PolyLog[2, E^(-2*ArcTanh[c*x])] + c*d*Pol yLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/c)/(2*d^2)
Time = 0.41 (sec) , antiderivative size = 148, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {a+b \text {arctanh}(c x)}{d x}-\frac {e (a+b \text {arctanh}(c x))}{d (d+e x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d}+\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d}+\frac {a \log (x)}{d}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 d}-\frac {b \operatorname {PolyLog}(2,-c x)}{2 d}+\frac {b \operatorname {PolyLog}(2,c x)}{2 d}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 d}\) |
Input:
Int[(a + b*ArcTanh[c*x])/(x*(d + e*x)),x]
Output:
(a*Log[x])/d + ((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d - ((a + b*ArcTanh [c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d - (b*PolyLog[2, -(c*x )])/(2*d) + (b*PolyLog[2, c*x])/(2*d) - (b*PolyLog[2, 1 - 2/(1 + c*x)])/(2 *d) + (b*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*d)
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 = 0.30 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(\frac {b \operatorname {dilog}\left (-c x +1\right )}{2 d}+\frac {b \operatorname {dilog}\left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 d}+\frac {b \ln \left (-c x +1\right ) \ln \left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 d}+\frac {a \ln \left (-c x \right )}{d}-\frac {a \ln \left (\left (-c x +1\right ) e -c d -e \right )}{d}-\frac {b \operatorname {dilog}\left (c x +1\right )}{2 d}-\frac {b \operatorname {dilog}\left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 d}-\frac {b \ln \left (c x +1\right ) \ln \left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 d}\) | \(203\) |
| parts | \(\frac {a \ln \left (x \right )}{d}-\frac {a \ln \left (e x +d \right )}{d}+b \left (\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{d}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c e x +c d \right )}{d}-c \left (\frac {\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}}{d c e}-\frac {-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}}{d c}\right )\right )\) | \(204\) |
| derivativedivides | \(-\frac {a \ln \left (c e x +c d \right )}{d}+\frac {a \ln \left (c x \right )}{d}+b c \left (-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c e x +c d \right )}{d c}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{d c}-\frac {\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}}{d c e}+\frac {-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}}{d c}\right )\) | \(212\) |
| default | \(-\frac {a \ln \left (c e x +c d \right )}{d}+\frac {a \ln \left (c x \right )}{d}+b c \left (-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c e x +c d \right )}{d c}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )}{d c}-\frac {\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}}{d c e}+\frac {-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}}{d c}\right )\) | \(212\) |
Input:
int((a+b*arctanh(c*x))/x/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/2*b/d*dilog(-c*x+1)+1/2*b/d*dilog(((-c*x+1)*e-c*d-e)/(-c*d-e))+1/2*b/d*l n(-c*x+1)*ln(((-c*x+1)*e-c*d-e)/(-c*d-e))+a/d*ln(-c*x)-a/d*ln((-c*x+1)*e-c *d-e)-1/2*b/d*dilog(c*x+1)-1/2*b/d*dilog(((c*x+1)*e+c*d-e)/(c*d-e))-1/2*b/ d*ln(c*x+1)*ln(((c*x+1)*e+c*d-e)/(c*d-e))
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x/(e*x+d),x, algorithm="fricas")
Output:
integral((b*arctanh(c*x) + a)/(e*x^2 + d*x), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{x \left (d + e x\right )}\, dx \] Input:
integrate((a+b*atanh(c*x))/x/(e*x+d),x)
Output:
Integral((a + b*atanh(c*x))/(x*(d + e*x)), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x/(e*x+d),x, algorithm="maxima")
Output:
-a*(log(e*x + d)/d - log(x)/d) + 1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/(e*x^2 + d*x), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)/((e*x + d)*x), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x\,\left (d+e\,x\right )} \,d x \] Input:
int((a + b*atanh(c*x))/(x*(d + e*x)),x)
Output:
int((a + b*atanh(c*x))/(x*(d + e*x)), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x (d+e x)} \, dx=\frac {-\mathit {atanh} \left (c x \right )^{2} b c d -2 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} e \,x^{4}+c^{2} d \,x^{3}-e \,x^{2}-d x}d x \right ) b d e -2 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} e \,x^{3}+c^{2} d \,x^{2}-e x -d}d x \right ) b \,c^{2} d^{2}-2 \,\mathrm {log}\left (e x +d \right ) a e +2 \,\mathrm {log}\left (x \right ) a e}{2 d e} \] Input:
int((a+b*atanh(c*x))/x/(e*x+d),x)
Output:
( - atanh(c*x)**2*b*c*d - 2*int(atanh(c*x)/(c**2*d*x**3 + c**2*e*x**4 - d* x - e*x**2),x)*b*d*e - 2*int(atanh(c*x)/(c**2*d*x**2 + c**2*e*x**3 - d - e *x),x)*b*c**2*d**2 - 2*log(d + e*x)*a*e + 2*log(x)*a*e)/(2*d*e)