Optimal. Leaf size=105 \[ -\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{x}-\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}+\frac {1}{2} b c \log \left (1-\frac {1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {1}{2} b c e \text {Li}_2\left (\frac {1}{1-c^2 x^2}\right ) \]
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Rubi [A] time = 0.27, antiderivative size = 94, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6082, 2475, 2411, 2344, 2301, 2316, 2315, 5949} \[ -\frac {1}{2} b c e \text {PolyLog}\left (2,c^2 x^2\right )-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{x}-\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )^2}{4 e}+b c d \log (x) \]
Antiderivative was successfully verified.
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Rule 2301
Rule 2315
Rule 2316
Rule 2344
Rule 2411
Rule 2475
Rule 5949
Rule 6082
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^2} \, dx &=-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}+(b c) \int \frac {d+e \log \left (1-c^2 x^2\right )}{x \left (1-c^2 x^2\right )} \, dx-\left (2 c^2 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=-\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}+\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {d+e \log \left (1-c^2 x\right )}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}-\frac {b \operatorname {Subst}\left (\int \frac {d+e \log (x)}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )} \, dx,x,1-c^2 x^2\right )}{2 c}\\ &=-\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}-\frac {b \operatorname {Subst}\left (\int \frac {d+e \log (x)}{\frac {1}{c^2}-\frac {x}{c^2}} \, dx,x,1-c^2 x^2\right )}{2 c}-\frac {1}{2} (b c) \operatorname {Subst}\left (\int \frac {d+e \log (x)}{x} \, dx,x,1-c^2 x^2\right )\\ &=-\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}+b c d \log (x)-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{4 e}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {1}{c^2}-\frac {x}{c^2}} \, dx,x,1-c^2 x^2\right )}{2 c}\\ &=-\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}+b c d \log (x)-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{4 e}-\frac {1}{2} b c e \text {Li}_2\left (c^2 x^2\right )\\ \end {align*}
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Mathematica [B] time = 0.21, size = 332, normalized size = 3.16 \[ -\frac {4 a e \log \left (1-c^2 x^2\right )+8 a c e x \tanh ^{-1}(c x)+4 a d+2 b c d x \log \left (1-c^2 x^2\right )-4 b c e x \log (x) \log \left (1-c^2 x^2\right )+2 b c e x \log \left (x-\frac {1}{c}\right ) \log \left (1-c^2 x^2\right )+2 b c e x \log \left (\frac {1}{c}+x\right ) \log \left (1-c^2 x^2\right )+4 b e \log \left (1-c^2 x^2\right ) \coth ^{-1}(c x)-4 b c d x \log (x)+4 b d \coth ^{-1}(c x)+4 b c e x \text {Li}_2(-c x)+4 b c e x \text {Li}_2(c x)-2 b c e x \text {Li}_2\left (\frac {1}{2}-\frac {c x}{2}\right )-2 b c e x \text {Li}_2\left (\frac {1}{2} (c x+1)\right )-b c e x \log ^2\left (x-\frac {1}{c}\right )-b c e x \log ^2\left (\frac {1}{c}+x\right )-2 b c e x \log \left (\frac {1}{c}+x\right ) \log \left (\frac {1}{2} (1-c x)\right )+4 b c e x \log (x) \log (1-c x)-2 b c e x \log \left (x-\frac {1}{c}\right ) \log \left (\frac {1}{2} (c x+1)\right )+4 b c e x \log (x) \log (c x+1)+4 b c e x \coth ^{-1}(c x)^2}{4 x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \operatorname {arcoth}\left (c x\right ) + a d + {\left (b e \operatorname {arcoth}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 5.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{2}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {arcoth}\left (c x\right )}{x}\right )} b d - {\left (c^{2} {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} + \frac {\log \left (-c^{2} x^{2} + 1\right )}{x}\right )} a e - \frac {1}{2} \, b e {\left (\frac {\log \left (c x + 1\right )^{2}}{x} - \int -\frac {{\left (c x + 1\right )} \log \left (c x - 1\right )^{2} - {\left (i \, \pi + {\left (i \, \pi c + 2 \, c\right )} x\right )} \log \left (c x + 1\right ) - {\left (-i \, \pi - i \, \pi c x\right )} \log \left (c x - 1\right )}{c x^{3} + x^{2}}\,{d x}\right )} - \frac {a d}{x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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