Integrand size = 18, antiderivative size = 70 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx=-\frac {d (a+b \text {arctanh}(c x))}{x}+a c d \log (x)+b c d \log (x)-\frac {1}{2} b c d \log \left (1-c^2 x^2\right )-\frac {1}{2} b c d \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b c d \operatorname {PolyLog}(2,c x) \]
-d*(a+b*arctanh(c*x))/x+a*c*d*ln(x)+b*c*d*ln(x)-1/2*b*c*d*ln(-c^2*x^2+1)-1 /2*b*c*d*polylog(2,-c*x)+1/2*b*c*d*polylog(2,c*x)
Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx=-\frac {a d}{x}+a c d \log (x)+b c d \left (-\frac {\text {arctanh}(c x)}{c x}+\log (c x)-\frac {1}{2} \log \left (1-c^2 x^2\right )\right )+\frac {1}{2} b c d (-\operatorname {PolyLog}(2,-c x)+\operatorname {PolyLog}(2,c x)) \]
-((a*d)/x) + a*c*d*Log[x] + b*c*d*(-(ArcTanh[c*x]/(c*x)) + Log[c*x] - Log[ 1 - c^2*x^2]/2) + (b*c*d*(-PolyLog[2, -(c*x)] + PolyLog[2, c*x]))/2
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
\(\displaystyle \int \frac {(c d x+d) (a+b \text {arctanh}(c x))}{x^2} \, dx\) |
\(\Big \downarrow \) 6502 |
\(\displaystyle \int \left (\frac {d (a+b \text {arctanh}(c x))}{x^2}+\frac {c d (a+b \text {arctanh}(c x))}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d (a+b \text {arctanh}(c x))}{x}+a c d \log (x)-\frac {1}{2} b c d \log \left (1-c^2 x^2\right )-\frac {1}{2} b c d \operatorname {PolyLog}(2,-c x)+\frac {1}{2} b c d \operatorname {PolyLog}(2,c x)+b c d \log (x)\) |
-((d*(a + b*ArcTanh[c*x]))/x) + a*c*d*Log[x] + b*c*d*Log[x] - (b*c*d*Log[1 - c^2*x^2])/2 - (b*c*d*PolyLog[2, -(c*x)])/2 + (b*c*d*PolyLog[2, c*x])/2
3.1.6.3.1 Defintions of rubi rules used
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 = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24
method | result | size |
parts | \(a d \left (-\frac {1}{x}+c \ln \left (x \right )\right )+b d c \left (\ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )\) | \(87\) |
derivativedivides | \(c \left (a d \left (\ln \left (c x \right )-\frac {1}{c x}\right )+b d \left (\ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )\right )\) | \(91\) |
default | \(c \left (a d \left (\ln \left (c x \right )-\frac {1}{c x}\right )+b d \left (\ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\ln \left (c x +1\right )}{2}-\frac {\ln \left (c x -1\right )}{2}+\ln \left (c x \right )-\frac {\operatorname {dilog}\left (c x +1\right )}{2}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {\operatorname {dilog}\left (c x \right )}{2}\right )\right )\) | \(91\) |
risch | \(\frac {c d b \ln \left (-c x \right )}{2}-\frac {\ln \left (-c x +1\right ) b c d}{2}+\frac {d b \ln \left (-c x +1\right )}{2 x}+\frac {c d \operatorname {dilog}\left (-c x +1\right ) b}{2}-\frac {a d}{x}+c d \ln \left (-c x \right ) a +\frac {b c d \ln \left (c x \right )}{2}-\frac {\ln \left (c x +1\right ) b c d}{2}-\frac {b d \ln \left (c x +1\right )}{2 x}-\frac {b c d \operatorname {dilog}\left (c x +1\right )}{2}\) | \(110\) |
a*d*(-1/x+c*ln(x))+b*d*c*(ln(c*x)*arctanh(c*x)-1/c/x*arctanh(c*x)-1/2*ln(c *x+1)-1/2*ln(c*x-1)+ln(c*x)-1/2*dilog(c*x+1)-1/2*ln(c*x)*ln(c*x+1)-1/2*dil og(c*x))
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx=d \left (\int \frac {a}{x^{2}}\, dx + \int \frac {a c}{x}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {b c \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \]
d*(Integral(a/x**2, x) + Integral(a*c/x, x) + Integral(b*atanh(c*x)/x**2, x) + Integral(b*c*atanh(c*x)/x, x))
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
1/2*b*c*d*integrate((log(c*x + 1) - log(-c*x + 1))/x, x) + a*c*d*log(x) - 1/2*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*b*d - a*d/x
\[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx=\int { \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {(d+c d x) (a+b \text {arctanh}(c x))}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+c\,d\,x\right )}{x^2} \,d x \]