Integrand size = 19, antiderivative size = 261 \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+e x)} \, dx=-\frac {b c}{2 d x}+\frac {b c^2 \text {arctanh}(c x)}{2 d}-\frac {a+b \text {arctanh}(c x)}{2 d x^2}+\frac {e (a+b \text {arctanh}(c x))}{d^2 x}-\frac {b c e \log (x)}{d^2}+\frac {a e^2 \log (x)}{d^3}+\frac {e^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d^3}-\frac {e^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{d^3}+\frac {b c e \log \left (1-c^2 x^2\right )}{2 d^2}-\frac {b e^2 \operatorname {PolyLog}(2,-c x)}{2 d^3}+\frac {b e^2 \operatorname {PolyLog}(2,c x)}{2 d^3}-\frac {b e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d^3}+\frac {b e^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 d^3} \] Output:
-1/2*b*c/d/x+1/2*b*c^2*arctanh(c*x)/d-1/2*(a+b*arctanh(c*x))/d/x^2+e*(a+b* arctanh(c*x))/d^2/x-b*c*e*ln(x)/d^2+a*e^2*ln(x)/d^3+e^2*(a+b*arctanh(c*x)) *ln(2/(c*x+1))/d^3-e^2*(a+b*arctanh(c*x))*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/ d^3+1/2*b*c*e*ln(-c^2*x^2+1)/d^2-1/2*b*e^2*polylog(2,-c*x)/d^3+1/2*b*e^2*p olylog(2,c*x)/d^3-1/2*b*e^2*polylog(2,1-2/(c*x+1))/d^3+1/2*b*e^2*polylog(2 ,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/d^3
Result contains complex when optimal does not.
Time = 4.06 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.67 \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+e x)} \, dx=-\frac {a}{2 d x^2}+\frac {a e}{d^2 x}+\frac {a e^2 \log (x)}{d^3}-\frac {a e^2 \log (d+e x)}{d^3}-\frac {b \left (\frac {c^2 d^3}{x}+i c d e^2 \pi \text {arctanh}(c x)-\frac {2 c d^2 e \text {arctanh}(c x)}{x}+\frac {c d^3 \left (1-c^2 x^2\right ) \text {arctanh}(c x)}{x^2}+2 c d e^2 \text {arctanh}\left (\frac {c d}{e}\right ) \text {arctanh}(c x)-c d e^2 \text {arctanh}(c x)^2+e^3 \text {arctanh}(c x)^2-\sqrt {1-\frac {c^2 d^2}{e^2}} e^3 e^{-\text {arctanh}\left (\frac {c d}{e}\right )} \text {arctanh}(c x)^2-2 c d e^2 \text {arctanh}(c x) \log \left (1-e^{-2 \text {arctanh}(c x)}\right )-i c d e^2 \pi \log \left (1+e^{2 \text {arctanh}(c x)}\right )+2 c d e^2 \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )+2 c d e^2 \text {arctanh}(c x) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )+i c d e^2 \pi \log \left (\frac {1}{\sqrt {1-c^2 x^2}}\right )+2 c^2 d^2 e \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-2 c d e^2 \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )+c d e^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )-c d e^2 \operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )\right )}{2 c d^4} \] Input:
Integrate[(a + b*ArcTanh[c*x])/(x^3*(d + e*x)),x]
Output:
-1/2*a/(d*x^2) + (a*e)/(d^2*x) + (a*e^2*Log[x])/d^3 - (a*e^2*Log[d + e*x]) /d^3 - (b*((c^2*d^3)/x + I*c*d*e^2*Pi*ArcTanh[c*x] - (2*c*d^2*e*ArcTanh[c* x])/x + (c*d^3*(1 - c^2*x^2)*ArcTanh[c*x])/x^2 + 2*c*d*e^2*ArcTanh[(c*d)/e ]*ArcTanh[c*x] - c*d*e^2*ArcTanh[c*x]^2 + e^3*ArcTanh[c*x]^2 - (Sqrt[1 - ( c^2*d^2)/e^2]*e^3*ArcTanh[c*x]^2)/E^ArcTanh[(c*d)/e] - 2*c*d*e^2*ArcTanh[c *x]*Log[1 - E^(-2*ArcTanh[c*x])] - I*c*d*e^2*Pi*Log[1 + E^(2*ArcTanh[c*x]) ] + 2*c*d*e^2*ArcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c *x]))] + 2*c*d*e^2*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[ c*x]))] + I*c*d*e^2*Pi*Log[1/Sqrt[1 - c^2*x^2]] + 2*c^2*d^2*e*Log[(c*x)/Sq rt[1 - c^2*x^2]] - 2*c*d*e^2*ArcTanh[(c*d)/e]*Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] + c*d*e^2*PolyLog[2, E^(-2*ArcTanh[c*x])] - c*d*e^2*PolyL og[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))]))/(2*c*d^4)
Time = 0.54 (sec) , antiderivative size = 261, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (-\frac {e^3 (a+b \text {arctanh}(c x))}{d^3 (d+e x)}+\frac {e^2 (a+b \text {arctanh}(c x))}{d^3 x}-\frac {e (a+b \text {arctanh}(c x))}{d^2 x^2}+\frac {a+b \text {arctanh}(c x)}{d x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {e^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{d^3}+\frac {e (a+b \text {arctanh}(c x))}{d^2 x}-\frac {a+b \text {arctanh}(c x)}{2 d x^2}+\frac {a e^2 \log (x)}{d^3}+\frac {b c^2 \text {arctanh}(c x)}{2 d}+\frac {b c e \log \left (1-c^2 x^2\right )}{2 d^2}-\frac {b e^2 \operatorname {PolyLog}(2,-c x)}{2 d^3}+\frac {b e^2 \operatorname {PolyLog}(2,c x)}{2 d^3}-\frac {b e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 d^3}+\frac {b e^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 d^3}-\frac {b c e \log (x)}{d^2}-\frac {b c}{2 d x}\) |
Input:
Int[(a + b*ArcTanh[c*x])/(x^3*(d + e*x)),x]
Output:
-1/2*(b*c)/(d*x) + (b*c^2*ArcTanh[c*x])/(2*d) - (a + b*ArcTanh[c*x])/(2*d* x^2) + (e*(a + b*ArcTanh[c*x]))/(d^2*x) - (b*c*e*Log[x])/d^2 + (a*e^2*Log[ x])/d^3 + (e^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^3 - (e^2*(a + b*Ar cTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/d^3 + (b*c*e*Log[1 - c^2*x^2])/(2*d^2) - (b*e^2*PolyLog[2, -(c*x)])/(2*d^3) + (b*e^2*PolyLog [2, c*x])/(2*d^3) - (b*e^2*PolyLog[2, 1 - 2/(1 + c*x)])/(2*d^3) + (b*e^2*P olyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*d^3)
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 = 0.46 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.23
| method | result | size |
| parts | \(a \left (-\frac {1}{2 d \,x^{2}}+\frac {e^{2} \ln \left (x \right )}{d^{3}}+\frac {e}{d^{2} x}-\frac {e^{2} \ln \left (e x +d \right )}{d^{3}}\right )+b \,c^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 d \,c^{2} x^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) e^{2} \ln \left (c x \right )}{c^{2} d^{3}}+\frac {\operatorname {arctanh}\left (c x \right ) e}{c^{2} d^{2} x}-\frac {\operatorname {arctanh}\left (c x \right ) e^{2} \ln \left (c e x +c d \right )}{c^{2} d^{3}}-\frac {c \left (-\frac {2 e^{2} \left (-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d^{3} c^{3}}+\frac {2 e \left (\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}\right )}{d^{3} c^{3}}-\frac {-\frac {d}{x}-2 e \ln \left (c x \right )+\left (-\frac {c d}{2}+e \right ) \ln \left (c x -1\right )+\left (\frac {c d}{2}+e \right ) \ln \left (c x +1\right )}{c^{2} d^{2}}\right )}{2}\right )\) | \(322\) |
| derivativedivides | \(c^{2} \left (-\frac {a \,e^{2} \ln \left (c e x +c d \right )}{c^{2} d^{3}}-\frac {a}{2 d \,c^{2} x^{2}}+\frac {a \,e^{2} \ln \left (c x \right )}{c^{2} d^{3}}+\frac {a e}{c^{2} d^{2} x}+b c \left (-\frac {\operatorname {arctanh}\left (c x \right ) e^{2} \ln \left (c e x +c d \right )}{d^{3} c^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 d \,c^{3} x^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) e^{2} \ln \left (c x \right )}{d^{3} c^{3}}+\frac {\operatorname {arctanh}\left (c x \right ) e}{d^{2} c^{3} x}-\frac {\left (\frac {c d}{2}-e \right ) \ln \left (c x -1\right )+2 e \ln \left (c x \right )+\frac {d}{x}+\left (-\frac {c d}{2}-e \right ) \ln \left (c x +1\right )}{2 d^{2} c^{2}}+\frac {e^{2} \left (-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d^{3} c^{3}}-\frac {e \left (\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}\right )}{d^{3} c^{3}}\right )\right )\) | \(340\) |
| default | \(c^{2} \left (-\frac {a \,e^{2} \ln \left (c e x +c d \right )}{c^{2} d^{3}}-\frac {a}{2 d \,c^{2} x^{2}}+\frac {a \,e^{2} \ln \left (c x \right )}{c^{2} d^{3}}+\frac {a e}{c^{2} d^{2} x}+b c \left (-\frac {\operatorname {arctanh}\left (c x \right ) e^{2} \ln \left (c e x +c d \right )}{d^{3} c^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{2 d \,c^{3} x^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) e^{2} \ln \left (c x \right )}{d^{3} c^{3}}+\frac {\operatorname {arctanh}\left (c x \right ) e}{d^{2} c^{3} x}-\frac {\left (\frac {c d}{2}-e \right ) \ln \left (c x -1\right )+2 e \ln \left (c x \right )+\frac {d}{x}+\left (-\frac {c d}{2}-e \right ) \ln \left (c x +1\right )}{2 d^{2} c^{2}}+\frac {e^{2} \left (-\frac {\operatorname {dilog}\left (c x \right )}{2}-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}\right )}{d^{3} c^{3}}-\frac {e \left (\frac {e \left (\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )\right )}{2}-\frac {e \left (\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )\right )}{2}\right )}{d^{3} c^{3}}\right )\right )\) | \(340\) |
| risch | \(\frac {c^{2} b \ln \left (-c x \right )}{4 d}-\frac {b c}{2 d x}-\frac {c^{2} b \ln \left (-c x +1\right )}{4 d}+\frac {b \ln \left (-c x +1\right )}{4 d \,x^{2}}+\frac {b \,e^{2} \operatorname {dilog}\left (-c x +1\right )}{2 d^{3}}+\frac {b \,e^{2} \operatorname {dilog}\left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 d^{3}}+\frac {b \,e^{2} \ln \left (-c x +1\right ) \ln \left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 d^{3}}-\frac {c b e \ln \left (-c x \right )}{2 d^{2}}+\frac {c b e \ln \left (-c x +1\right )}{2 d^{2}}-\frac {b e \ln \left (-c x +1\right )}{2 d^{2} x}-\frac {a}{2 d \,x^{2}}+\frac {a e}{d^{2} x}+\frac {a \,e^{2} \ln \left (-c x \right )}{d^{3}}-\frac {a \,e^{2} \ln \left (\left (-c x +1\right ) e -c d -e \right )}{d^{3}}-\frac {b \,c^{2} \ln \left (c x \right )}{4 d}+\frac {b \,c^{2} \ln \left (c x +1\right )}{4 d}-\frac {b \ln \left (c x +1\right )}{4 d \,x^{2}}-\frac {b \,e^{2} \operatorname {dilog}\left (c x +1\right )}{2 d^{3}}-\frac {b \,e^{2} \operatorname {dilog}\left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 d^{3}}-\frac {b \,e^{2} \ln \left (c x +1\right ) \ln \left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 d^{3}}-\frac {b c e \ln \left (c x \right )}{2 d^{2}}+\frac {b c e \ln \left (c x +1\right )}{2 d^{2}}+\frac {b e \ln \left (c x +1\right )}{2 d^{2} x}\) | \(431\) |
Input:
int((a+b*arctanh(c*x))/x^3/(e*x+d),x,method=_RETURNVERBOSE)
Output:
a*(-1/2/d/x^2+e^2/d^3*ln(x)+e/d^2/x-e^2/d^3*ln(e*x+d))+b*c^2*(-1/2*arctanh (c*x)/d/c^2/x^2+1/c^2*arctanh(c*x)*e^2/d^3*ln(c*x)+1/c^2*arctanh(c*x)*e/d^ 2/x-1/c^2*arctanh(c*x)*e^2/d^3*ln(c*e*x+c*d)-1/2*c*(-2/d^3/c^3*e^2*(-1/2*d ilog(c*x)-1/2*dilog(c*x+1)-1/2*ln(c*x)*ln(c*x+1))+2/d^3/c^3*e*(1/2*e*(dilo g((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))))-1/c^2/d^2*(-d/x- 2*e*ln(c*x)+(-1/2*c*d+e)*ln(c*x-1)+(1/2*c*d+e)*ln(c*x+1))))
\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{3}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^3/(e*x+d),x, algorithm="fricas")
Output:
integral((b*arctanh(c*x) + a)/(e*x^4 + d*x^3), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+e x)} \, dx=\int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{x^{3} \left (d + e x\right )}\, dx \] Input:
integrate((a+b*atanh(c*x))/x**3/(e*x+d),x)
Output:
Integral((a + b*atanh(c*x))/(x**3*(d + e*x)), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{3}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^3/(e*x+d),x, algorithm="maxima")
Output:
-1/2*a*(2*e^2*log(e*x + d)/d^3 - 2*e^2*log(x)/d^3 - (2*e*x - d)/(d^2*x^2)) + 1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/(e*x^4 + d*x^3), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+e x)} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (e x + d\right )} x^{3}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^3/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)/((e*x + d)*x^3), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+e x)} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^3\,\left (d+e\,x\right )} \,d x \] Input:
int((a + b*atanh(c*x))/(x^3*(d + e*x)),x)
Output:
int((a + b*atanh(c*x))/(x^3*(d + e*x)), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^3 (d+e x)} \, dx=\frac {\mathit {atanh} \left (c x \right )^{2} b c d e \,x^{2}+\mathit {atanh} \left (c x \right ) b \,c^{2} d^{2} x^{2}-\mathit {atanh} \left (c x \right ) b \,d^{2}+2 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{2} e \,x^{5}+c^{2} d \,x^{4}-e \,x^{3}-d \,x^{2}}d x \right ) b \,d^{2} e \,x^{2}+2 \left (\int \frac {\mathit {atanh} \left (c x \right ) x}{c^{2} e \,x^{3}+c^{2} d \,x^{2}-e x -d}d x \right ) b \,c^{2} d \,e^{2} x^{2}-2 \,\mathrm {log}\left (e x +d \right ) a \,e^{2} x^{2}+2 \,\mathrm {log}\left (x \right ) a \,e^{2} x^{2}-a \,d^{2}+2 a d e x -b c \,d^{2} x}{2 d^{3} x^{2}} \] Input:
int((a+b*atanh(c*x))/x^3/(e*x+d),x)
Output:
(atanh(c*x)**2*b*c*d*e*x**2 + atanh(c*x)*b*c**2*d**2*x**2 - atanh(c*x)*b*d **2 + 2*int(atanh(c*x)/(c**2*d*x**4 + c**2*e*x**5 - d*x**2 - e*x**3),x)*b* d**2*e*x**2 + 2*int((atanh(c*x)*x)/(c**2*d*x**2 + c**2*e*x**3 - d - e*x),x )*b*c**2*d*e**2*x**2 - 2*log(d + e*x)*a*e**2*x**2 + 2*log(x)*a*e**2*x**2 - a*d**2 + 2*a*d*e*x - b*c*d**2*x)/(2*d**3*x**2)