Optimal. Leaf size=247 \[ -\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x^2}+\frac {1}{2} c^2 e (a+b) \log (1-c x)+\frac {1}{2} c^2 e (a-b) \log (c x+1)-a c^2 e \log (x)-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac {1}{2} b c^2 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )+\frac {1}{2} b c^2 e \text {Li}_2\left (\frac {2}{c x+1}-1\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x)^2-\frac {1}{2} b c^2 e \coth ^{-1}(c x)^2+b c^2 e \log \left (\frac {2}{1-c x}\right ) \tanh ^{-1}(c x)-b c^2 e \log \left (2-\frac {2}{c x+1}\right ) \coth ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.49, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {5917, 325, 206, 6086, 6725, 801, 5989, 5933, 2447, 5984, 5918, 2402, 2315} \[ \frac {1}{2} b c^2 e \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{2} b c^2 e \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x^2}+\frac {1}{2} c^2 e (a+b) \log (1-c x)+\frac {1}{2} c^2 e (a-b) \log (c x+1)-a c^2 e \log (x)-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {1}{2} b c^2 e \tanh ^{-1}(c x)^2-\frac {1}{2} b c^2 e \coth ^{-1}(c x)^2+b c^2 e \log \left (\frac {2}{1-c x}\right ) \tanh ^{-1}(c x)-b c^2 e \log \left (2-\frac {2}{c x+1}\right ) \coth ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 325
Rule 801
Rule 2315
Rule 2402
Rule 2447
Rule 5917
Rule 5918
Rule 5933
Rule 5984
Rule 5989
Rule 6086
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^3} \, dx &=-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\left (2 c^2 e\right ) \int \left (\frac {a+b c x+b \coth ^{-1}(c x)}{2 x \left (-1+c^2 x^2\right )}-\frac {b c^2 x \tanh ^{-1}(c x)}{2 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\left (c^2 e\right ) \int \frac {a+b c x+b \coth ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx-\left (b c^4 e\right ) \int \frac {x \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx\\ &=-\frac {1}{2} b c^2 e \tanh ^{-1}(c x)^2-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\left (c^2 e\right ) \int \left (\frac {a+b c x}{x \left (-1+c^2 x^2\right )}+\frac {b \coth ^{-1}(c x)}{x \left (-1+c^2 x^2\right )}\right ) \, dx+\left (b c^3 e\right ) \int \frac {\tanh ^{-1}(c x)}{1-c x} \, dx\\ &=-\frac {1}{2} b c^2 e \tanh ^{-1}(c x)^2+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1-c x}\right )-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\left (c^2 e\right ) \int \frac {a+b c x}{x \left (-1+c^2 x^2\right )} \, dx+\left (b c^2 e\right ) \int \frac {\coth ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx-\left (b c^3 e\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {1}{2} b c^2 e \coth ^{-1}(c x)^2-\frac {1}{2} b c^2 e \tanh ^{-1}(c x)^2+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1-c x}\right )-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\left (c^2 e\right ) \int \left (-\frac {a}{x}+\frac {(a+b) c}{2 (-1+c x)}+\frac {(a-b) c}{2 (1+c x)}\right ) \, dx-\left (b c^2 e\right ) \int \frac {\coth ^{-1}(c x)}{x (1+c x)} \, dx+\left (b c^2 e\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )\\ &=-\frac {1}{2} b c^2 e \coth ^{-1}(c x)^2-\frac {1}{2} b c^2 e \tanh ^{-1}(c x)^2-a c^2 e \log (x)+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1-c x}\right )+\frac {1}{2} (a+b) c^2 e \log (1-c x)+\frac {1}{2} (a-b) c^2 e \log (1+c x)-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-b c^2 e \coth ^{-1}(c x) \log \left (2-\frac {2}{1+c x}\right )+\frac {1}{2} b c^2 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )+\left (b c^3 e\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {1}{2} b c^2 e \coth ^{-1}(c x)^2-\frac {1}{2} b c^2 e \tanh ^{-1}(c x)^2-a c^2 e \log (x)+b c^2 e \tanh ^{-1}(c x) \log \left (\frac {2}{1-c x}\right )+\frac {1}{2} (a+b) c^2 e \log (1-c x)+\frac {1}{2} (a-b) c^2 e \log (1+c x)-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-b c^2 e \coth ^{-1}(c x) \log \left (2-\frac {2}{1+c x}\right )+\frac {1}{2} b c^2 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )+\frac {1}{2} b c^2 e \text {Li}_2\left (-1+\frac {2}{1+c x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 161, normalized size = 0.65 \[ \frac {1}{2} \left (-\frac {e \log \left (1-c^2 x^2\right ) \left (a+\left (b-b c^2 x^2\right ) \coth ^{-1}(c x)+b c x\right )}{x^2}+c^2 e (a+b) \log (1-c x)+c^2 e (a-b) \log (c x+1)-2 a c^2 e \log (x)-\frac {a d}{x^2}-b c^2 e \left (\text {Li}_2\left (-\frac {1}{c x}\right )-\text {Li}_2\left (\frac {1}{c x}\right )\right )-\frac {b d \left (c x (c x \log (1-c x)-c x \log (c x+1)+2)+2 \coth ^{-1}(c x)\right )}{2 x^2}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \operatorname {arcoth}\left (c x\right ) + a d + {\left (b e \operatorname {arcoth}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 11.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {arcoth}\left (c x\right )}{x^{2}}\right )} b d + \frac {1}{2} \, {\left (c^{2} {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} - \frac {\log \left (-c^{2} x^{2} + 1\right )}{x^{2}}\right )} a e - \frac {1}{4} \, b e {\left (\frac {\log \left (c x + 1\right )^{2}}{x^{2}} - 2 \, \int -\frac {{\left (c x + 1\right )} \log \left (c x - 1\right )^{2} - {\left (i \, \pi + {\left (i \, \pi c + c\right )} x\right )} \log \left (c x + 1\right ) - {\left (-i \, \pi - i \, \pi c x\right )} \log \left (c x - 1\right )}{c x^{4} + x^{3}}\,{d x}\right )} - \frac {a d}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________