Integrand size = 20, antiderivative size = 171 \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^2} \, dx=-\frac {b c}{2 d^2 (1+c x)}+\frac {b c \text {arctanh}(c x)}{2 d^2}-\frac {a+b \text {arctanh}(c x)}{d^2 x}-\frac {c (a+b \text {arctanh}(c x))}{d^2 (1+c x)}-\frac {2 a c \log (x)}{d^2}+\frac {b c \log (x)}{d^2}-\frac {2 c (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d^2}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {b c \operatorname {PolyLog}(2,-c x)}{d^2}-\frac {b c \operatorname {PolyLog}(2,c x)}{d^2}+\frac {b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{d^2} \] Output:
-1/2*b*c/d^2/(c*x+1)+1/2*b*c*arctanh(c*x)/d^2-(a+b*arctanh(c*x))/d^2/x-c*( a+b*arctanh(c*x))/d^2/(c*x+1)-2*a*c*ln(x)/d^2+b*c*ln(x)/d^2-2*c*(a+b*arcta nh(c*x))*ln(2/(c*x+1))/d^2-1/2*b*c*ln(-c^2*x^2+1)/d^2+b*c*polylog(2,-c*x)/ d^2-b*c*polylog(2,c*x)/d^2+b*c*polylog(2,1-2/(c*x+1))/d^2
Time = 0.57 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^2} \, dx=\frac {-\frac {4 a}{x}-\frac {4 a c}{1+c x}-8 a c \log (x)+8 a c \log (1+c x)+b c \left (-\cosh (2 \text {arctanh}(c x))+4 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+4 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+\sinh (2 \text {arctanh}(c x))+\text {arctanh}(c x) \left (-\frac {4}{c x}-2 \cosh (2 \text {arctanh}(c x))-8 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+2 \sinh (2 \text {arctanh}(c x))\right )\right )}{4 d^2} \] Input:
Integrate[(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)^2),x]
Output:
((-4*a)/x - (4*a*c)/(1 + c*x) - 8*a*c*Log[x] + 8*a*c*Log[1 + c*x] + b*c*(- Cosh[2*ArcTanh[c*x]] + 4*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 4*PolyLog[2, E^(-2 *ArcTanh[c*x])] + Sinh[2*ArcTanh[c*x]] + ArcTanh[c*x]*(-4/(c*x) - 2*Cosh[2 *ArcTanh[c*x]] - 8*Log[1 - E^(-2*ArcTanh[c*x])] + 2*Sinh[2*ArcTanh[c*x]])) )/(4*d^2)
Time = 0.56 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x^2 (c d x+d)^2} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {2 c^2 (a+b \text {arctanh}(c x))}{d^2 (c x+1)}+\frac {c^2 (a+b \text {arctanh}(c x))}{d^2 (c x+1)^2}+\frac {a+b \text {arctanh}(c x)}{d^2 x^2}-\frac {2 c (a+b \text {arctanh}(c x))}{d^2 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c (a+b \text {arctanh}(c x))}{d^2 (c x+1)}-\frac {a+b \text {arctanh}(c x)}{d^2 x}-\frac {2 c \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^2}-\frac {2 a c \log (x)}{d^2}+\frac {b c \text {arctanh}(c x)}{2 d^2}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d^2}+\frac {b c \operatorname {PolyLog}(2,-c x)}{d^2}-\frac {b c \operatorname {PolyLog}(2,c x)}{d^2}+\frac {b c \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{d^2}-\frac {b c}{2 d^2 (c x+1)}+\frac {b c \log (x)}{d^2}\) |
Input:
Int[(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)^2),x]
Output:
-1/2*(b*c)/(d^2*(1 + c*x)) + (b*c*ArcTanh[c*x])/(2*d^2) - (a + b*ArcTanh[c *x])/(d^2*x) - (c*(a + b*ArcTanh[c*x]))/(d^2*(1 + c*x)) - (2*a*c*Log[x])/d ^2 + (b*c*Log[x])/d^2 - (2*c*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/d^2 - (b*c*Log[1 - c^2*x^2])/(2*d^2) + (b*c*PolyLog[2, -(c*x)])/d^2 - (b*c*PolyL og[2, c*x])/d^2 + (b*c*PolyLog[2, 1 - 2/(1 + c*x)])/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 = 184, normalized size of antiderivative = 1.08
| method | result | size |
| parts | \(\frac {a \left (-\frac {c}{c x +1}+2 c \ln \left (c x +1\right )-\frac {1}{x}-2 c \ln \left (x \right )\right )}{d^{2}}+\frac {b c \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {3 \ln \left (c x -1\right )}{4}+\ln \left (c x \right )-\frac {1}{2 \left (c x +1\right )}-\frac {\ln \left (c x +1\right )}{4}+\operatorname {dilog}\left (c x \right )+\operatorname {dilog}\left (c x +1\right )+\ln \left (c x \right ) \ln \left (c x +1\right )-\frac {\ln \left (c x +1\right )^{2}}{2}+\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )-\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )\right )}{d^{2}}\) | \(184\) |
| derivativedivides | \(c \left (\frac {a \left (-\frac {1}{c x}-2 \ln \left (c x \right )-\frac {1}{c x +1}+2 \ln \left (c x +1\right )\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {3 \ln \left (c x -1\right )}{4}+\ln \left (c x \right )-\frac {1}{2 \left (c x +1\right )}-\frac {\ln \left (c x +1\right )}{4}+\operatorname {dilog}\left (c x \right )+\operatorname {dilog}\left (c x +1\right )+\ln \left (c x \right ) \ln \left (c x +1\right )-\frac {\ln \left (c x +1\right )^{2}}{2}+\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )-\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )\right )}{d^{2}}\right )\) | \(187\) |
| default | \(c \left (\frac {a \left (-\frac {1}{c x}-2 \ln \left (c x \right )-\frac {1}{c x +1}+2 \ln \left (c x +1\right )\right )}{d^{2}}+\frac {b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+2 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {3 \ln \left (c x -1\right )}{4}+\ln \left (c x \right )-\frac {1}{2 \left (c x +1\right )}-\frac {\ln \left (c x +1\right )}{4}+\operatorname {dilog}\left (c x \right )+\operatorname {dilog}\left (c x +1\right )+\ln \left (c x \right ) \ln \left (c x +1\right )-\frac {\ln \left (c x +1\right )^{2}}{2}+\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )-\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )\right )}{d^{2}}\right )\) | \(187\) |
| risch | \(-\frac {a}{d^{2} x}-\frac {2 c a \ln \left (-c x \right )}{d^{2}}+\frac {c a}{d^{2} \left (-c x -1\right )}+\frac {2 c a \ln \left (-c x -1\right )}{d^{2}}+\frac {b c \ln \left (-c x -1\right )}{4 d^{2}}+\frac {c^{2} b \ln \left (-c x +1\right ) x}{4 d^{2} \left (-c x -1\right )}-\frac {c b \ln \left (-c x +1\right )}{4 d^{2} \left (-c x -1\right )}-\frac {c b \operatorname {dilog}\left (-c x +1\right )}{d^{2}}+\frac {c b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{d^{2}}-\frac {c b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{d^{2}}+\frac {c b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{d^{2}}+\frac {c b \ln \left (-c x \right )}{2 d^{2}}-\frac {c b \ln \left (-c x +1\right )}{2 d^{2}}+\frac {b \ln \left (-c x +1\right )}{2 d^{2} x}+\frac {b c \ln \left (c x +1\right )^{2}}{2 d^{2}}+\frac {b c \operatorname {dilog}\left (c x +1\right )}{d^{2}}-\frac {b c \ln \left (c x +1\right )}{2 d^{2} \left (c x +1\right )}-\frac {b c}{2 d^{2} \left (c x +1\right )}+\frac {b c \ln \left (c x \right )}{2 d^{2}}-\frac {b c \ln \left (c x +1\right )}{2 d^{2}}-\frac {b \ln \left (c x +1\right )}{2 d^{2} x}\) | \(322\) |
Input:
int((a+b*arctanh(c*x))/x^2/(c*d*x+d)^2,x,method=_RETURNVERBOSE)
Output:
a/d^2*(-c/(c*x+1)+2*c*ln(c*x+1)-1/x-2*c*ln(x))+b/d^2*c*(-arctanh(c*x)/c/x- 2*arctanh(c*x)*ln(c*x)-1/(c*x+1)*arctanh(c*x)+2*arctanh(c*x)*ln(c*x+1)-3/4 *ln(c*x-1)+ln(c*x)-1/2/(c*x+1)-1/4*ln(c*x+1)+dilog(c*x)+dilog(c*x+1)+ln(c* x)*ln(c*x+1)-1/2*ln(c*x+1)^2+(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)- dilog(1/2*c*x+1/2))
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^2} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{2} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^2/(c*d*x+d)^2,x, algorithm="fricas")
Output:
integral((b*arctanh(c*x) + a)/(c^2*d^2*x^4 + 2*c*d^2*x^3 + d^2*x^2), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^2} \, dx=\frac {\int \frac {a}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{4} + 2 c x^{3} + x^{2}}\, dx}{d^{2}} \] Input:
integrate((a+b*atanh(c*x))/x**2/(c*d*x+d)**2,x)
Output:
(Integral(a/(c**2*x**4 + 2*c*x**3 + x**2), x) + Integral(b*atanh(c*x)/(c** 2*x**4 + 2*c*x**3 + x**2), x))/d**2
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^2} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{2} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^2/(c*d*x+d)^2,x, algorithm="maxima")
Output:
-a*((2*c*x + 1)/(c*d^2*x^2 + d^2*x) - 2*c*log(c*x + 1)/d^2 + 2*c*log(x)/d^ 2) + 1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/(c^2*d^2*x^4 + 2*c*d^2 *x^3 + d^2*x^2), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^2} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{2} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^2/(c*d*x+d)^2,x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)/((c*d*x + d)^2*x^2), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^2} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^2\,{\left (d+c\,d\,x\right )}^2} \,d x \] Input:
int((a + b*atanh(c*x))/(x^2*(d + c*d*x)^2),x)
Output:
int((a + b*atanh(c*x))/(x^2*(d + c*d*x)^2), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^2} \, dx=\frac {-\mathit {atanh} \left (c x \right )^{2} b \,c^{2} x^{2}-\mathit {atanh} \left (c x \right )^{2} b c x -2 \mathit {atanh} \left (c x \right ) b \,c^{2} x^{2}+2 \mathit {atanh} \left (c x \right ) b -4 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{3} x^{5}+c^{2} x^{4}-c \,x^{3}-x^{2}}d x \right ) b c \,x^{2}-4 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{3} x^{5}+c^{2} x^{4}-c \,x^{3}-x^{2}}d x \right ) b x +4 \,\mathrm {log}\left (c x +1\right ) a \,c^{2} x^{2}+4 \,\mathrm {log}\left (c x +1\right ) a c x +2 \,\mathrm {log}\left (c x +1\right ) b \,c^{2} x^{2}+2 \,\mathrm {log}\left (c x +1\right ) b c x -4 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}-4 \,\mathrm {log}\left (x \right ) a c x -2 \,\mathrm {log}\left (x \right ) b \,c^{2} x^{2}-2 \,\mathrm {log}\left (x \right ) b c x +4 a \,c^{2} x^{2}-2 a}{2 d^{2} x \left (c x +1\right )} \] Input:
int((a+b*atanh(c*x))/x^2/(c*d*x+d)^2,x)
Output:
( - atanh(c*x)**2*b*c**2*x**2 - atanh(c*x)**2*b*c*x - 2*atanh(c*x)*b*c**2* x**2 + 2*atanh(c*x)*b - 4*int(atanh(c*x)/(c**3*x**5 + c**2*x**4 - c*x**3 - x**2),x)*b*c*x**2 - 4*int(atanh(c*x)/(c**3*x**5 + c**2*x**4 - c*x**3 - x* *2),x)*b*x + 4*log(c*x + 1)*a*c**2*x**2 + 4*log(c*x + 1)*a*c*x + 2*log(c*x + 1)*b*c**2*x**2 + 2*log(c*x + 1)*b*c*x - 4*log(x)*a*c**2*x**2 - 4*log(x) *a*c*x - 2*log(x)*b*c**2*x**2 - 2*log(x)*b*c*x + 4*a*c**2*x**2 - 2*a)/(2*d **2*x*(c*x + 1))