Optimal. Leaf size=70 \[ -\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c d \log (x)-\frac {1}{2} b c d \log \left (1-c^2 x^2\right )-\frac {1}{2} b c d \text {Li}_2(-c x)+\frac {1}{2} b c d \text {Li}_2(c x)+b c d \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5940, 5916, 266, 36, 29, 31, 5912} \[ -\frac {1}{2} b c d \text {PolyLog}(2,-c x)+\frac {1}{2} b c d \text {PolyLog}(2,c x)-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c d \log (x)-\frac {1}{2} b c d \log \left (1-c^2 x^2\right )+b c d \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 266
Rule 5912
Rule 5916
Rule 5940
Rubi steps
\begin {align*} \int \frac {(d+c d x) \left (a+b \tanh ^{-1}(c x)\right )}{x^2} \, dx &=\int \left (\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{x^2}+\frac {c d \left (a+b \tanh ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+(c d) \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c d \log (x)-\frac {1}{2} b c d \text {Li}_2(-c x)+\frac {1}{2} b c d \text {Li}_2(c x)+(b c d) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c d \log (x)-\frac {1}{2} b c d \text {Li}_2(-c x)+\frac {1}{2} b c d \text {Li}_2(c x)+\frac {1}{2} (b c d) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{x}+a c d \log (x)-\frac {1}{2} b c d \text {Li}_2(-c x)+\frac {1}{2} b c d \text {Li}_2(c x)+\frac {1}{2} (b c d) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (b c^3 d\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {d \left (a+b \tanh ^{-1}(c x)\right )}{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 \text {Li}_2(-c x)+\frac {1}{2} b c d \text {Li}_2(c x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 71, normalized size = 1.01 \[ a c d \log (x)-\frac {a d}{x}+b c d \left (-\frac {1}{2} \log \left (1-c^2 x^2\right )+\log (c x)-\frac {\tanh ^{-1}(c x)}{c x}\right )+\frac {1}{2} b c d (\text {Li}_2(c x)-\text {Li}_2(-c x)) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a c d x + a d + {\left (b c d x + b d\right )} \operatorname {artanh}\left (c x\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d x + d\right )} {\left (b \operatorname {artanh}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 105, normalized size = 1.50 \[ c d a \ln \left (c x \right )-\frac {a d}{x}+c d b \arctanh \left (c x \right ) \ln \left (c x \right )-\frac {d b \arctanh \left (c x \right )}{x}+c d b \ln \left (c x \right )-\frac {c d b \ln \left (c x -1\right )}{2}-\frac {c d b \ln \left (c x +1\right )}{2}-\frac {c d b \dilog \left (c x \right )}{2}-\frac {c d b \dilog \left (c x +1\right )}{2}-\frac {c d b \ln \left (c x \right ) \ln \left (c x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, b c d \int \frac {\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{x}\,{d x} + a c d \log \relax (x) - \frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b d - \frac {a d}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+c\,d\,x\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ d \left (\int \frac {a}{x^{2}}\, dx + \int \frac {a c}{x}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {b c \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________