Optimal. Leaf size=157 \[ -\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x^2}+\frac {1}{2} c^2 e (a+b) \log (1-c x)+\frac {1}{2} c^2 e (a-b) \log (c x+1)-a c^2 e \log (x)-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac {1}{2} b c^2 e \text {Li}_2(-c x)-\frac {1}{2} b c^2 e \text {Li}_2(c x) \]
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Rubi [A] time = 0.14, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5916, 325, 206, 6085, 801, 5912} \[ \frac {1}{2} b c^2 e \text {PolyLog}(2,-c x)-\frac {1}{2} b c^2 e \text {PolyLog}(2,c x)-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x^2}+\frac {1}{2} c^2 e (a+b) \log (1-c x)+\frac {1}{2} c^2 e (a-b) \log (c x+1)-a c^2 e \log (x)-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 325
Rule 801
Rule 5912
Rule 5916
Rule 6085
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-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 {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x^2}+\left (2 c^2 e\right ) \int \left (\frac {a+b c x}{2 x \left (-1+c^2 x^2\right )}-\frac {b \tanh ^{-1}(c x)}{2 x}\right ) \, dx\\ &=-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x^2}+\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 {\tanh ^{-1}(c x)}{x} \, dx\\ &=-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{2 x}+\frac {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {\left (a+b \tanh ^{-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 e \text {Li}_2(-c x)-\frac {1}{2} b c^2 e \text {Li}_2(c x)+\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\\ &=-a c^2 e \log (x)+\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 {1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {\left (a+b \tanh ^{-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 e \text {Li}_2(-c x)-\frac {1}{2} b c^2 e \text {Li}_2(c x)\\ \end {align*}
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Mathematica [A] time = 0.17, size = 152, normalized size = 0.97 \[ \frac {1}{2} \left (-\frac {e \log \left (1-c^2 x^2\right ) \left (a+\left (b-b c^2 x^2\right ) \tanh ^{-1}(c x)+b c x\right )}{x^2}+c^2 e (a+b) \log (1-c x)+c^2 e (a-b) \log (c x+1)-2 a c^2 e \log (x)-\frac {a d}{x^2}+b c^2 e (\text {Li}_2(-c x)-\text {Li}_2(c x))-\frac {b d \left (c x (c x \log (1-c x)-c x \log (c x+1)+2)+2 \tanh ^{-1}(c x)\right )}{2 x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \operatorname {artanh}\left (c x\right ) + a d + {\left (b e \operatorname {artanh}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\right )}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 26.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctanh \left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b d + \frac {1}{2} \, {\left (c^{2} {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} - \frac {\log \left (-c^{2} x^{2} + 1\right )}{x^{2}}\right )} a e + \frac {1}{4} \, b e {\left (\frac {\log \left (-c x + 1\right )^{2}}{x^{2}} - 2 \, \int -\frac {{\left (c x - 1\right )} \log \left (c x + 1\right )^{2} - c x \log \left (-c x + 1\right )}{c x^{4} - x^{3}}\,{d x}\right )} - \frac {a d}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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