Optimal. Leaf size=93 \[ -\frac {a+b \tanh ^{-1}(c x)}{d x}-\frac {c \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d}+\frac {b c \text {Li}_2\left (\frac {2}{c x+1}-1\right )}{2 d}+\frac {b c \log (x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5934, 5916, 266, 36, 29, 31, 5932, 2447} \[ \frac {b c \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )}{2 d}-\frac {a+b \tanh ^{-1}(c x)}{d x}-\frac {c \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d}+\frac {b c \log (x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 266
Rule 2447
Rule 5916
Rule 5932
Rule 5934
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{x^2 (d+c d x)} \, dx &=-\left (c \int \frac {a+b \tanh ^{-1}(c x)}{x (d+c d x)} \, dx\right )+\frac {\int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx}{d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{d x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {(b c) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx}{d}+\frac {\left (b c^2\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{d x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b c \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{d x}-\frac {c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b c \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}+\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {a+b \tanh ^{-1}(c x)}{d x}+\frac {b c \log (x)}{d}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d}-\frac {c \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{d}+\frac {b c \text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 93, normalized size = 1.00 \[ \frac {b c x \text {Li}_2\left (e^{-2 \tanh ^{-1}(c x)}\right )-2 \left (a c x \log (x)-a c x \log (c x+1)+a-b c x \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+b \tanh ^{-1}(c x) \left (c x \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )+1\right )\right )}{2 d x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x\right ) + a}{c d x^{3} + d x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 225, normalized size = 2.42 \[ -\frac {a}{d x}-\frac {c a \ln \left (c x \right )}{d}+\frac {c a \ln \left (c x +1\right )}{d}-\frac {b \arctanh \left (c x \right )}{d x}-\frac {c b \arctanh \left (c x \right ) \ln \left (c x \right )}{d}+\frac {c b \arctanh \left (c x \right ) \ln \left (c x +1\right )}{d}+\frac {c b \ln \left (c x \right )}{d}-\frac {c b \ln \left (c x -1\right )}{2 d}-\frac {c b \ln \left (c x +1\right )}{2 d}+\frac {c b \dilog \left (c x \right )}{2 d}+\frac {c b \dilog \left (c x +1\right )}{2 d}+\frac {c b \ln \left (c x \right ) \ln \left (c x +1\right )}{2 d}-\frac {c b \ln \left (c x +1\right )^{2}}{4 d}+\frac {c b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{2 d}-\frac {c b \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 d}-\frac {c b \dilog \left (\frac {1}{2}+\frac {c x}{2}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a {\left (\frac {c \log \left (c x + 1\right )}{d} - \frac {c \log \relax (x)}{d} - \frac {1}{d x}\right )} + \frac {1}{2} \, b \int \frac {\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{c d x^{3} + d x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^2\,\left (d+c\,d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a}{c x^{3} + x^{2}}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c x^{3} + x^{2}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________