Optimal. Leaf size=339 \[ \frac {1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac {1}{12} c^4 e (3 a-4 b) \log (c x+1)-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x^4}-\frac {1}{2} a c^4 e \log (x)+\frac {a c^2 e}{4 x^2}+\frac {1}{4} b c^4 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )+\frac {1}{4} b c^4 e \text {Li}_2\left (\frac {2}{c x+1}-1\right )-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)^2-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2+\frac {1}{2} b c^4 e \log \left (\frac {2}{1-c x}\right ) \tanh ^{-1}(c x)-\frac {1}{2} b c^4 e \log \left (2-\frac {2}{c x+1}\right ) \coth ^{-1}(c x)+\frac {5 b c^3 e}{12 x}-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{12 x^3}+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x} \]
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Rubi [A] time = 0.72, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.518, Rules used = {5917, 325, 206, 6086, 6725, 1802, 5983, 5989, 5933, 2447, 5984, 5918, 2402, 2315} \[ \frac {1}{4} b c^4 e \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+\frac {1}{4} b c^4 e \text {PolyLog}\left (2,\frac {2}{c x+1}-1\right )-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x^4}+\frac {1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac {1}{12} c^4 e (3 a-4 b) \log (c x+1)+\frac {a c^2 e}{4 x^2}-\frac {1}{2} a c^4 e \log (x)-\frac {b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x}-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{12 x^3}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}+\frac {5 b c^3 e}{12 x}-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)^2-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2+\frac {1}{2} b c^4 e \log \left (\frac {2}{1-c x}\right ) \tanh ^{-1}(c x)-\frac {1}{2} b c^4 e \log \left (2-\frac {2}{c x+1}\right ) \coth ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 206
Rule 325
Rule 1802
Rule 2315
Rule 2402
Rule 2447
Rule 5917
Rule 5918
Rule 5933
Rule 5983
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^5} \, dx &=-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac {1}{4} b c^4 \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 {3 a+b c x+3 b c^3 x^3+3 b \coth ^{-1}(c x)}{12 x^3 \left (-1+c^2 x^2\right )}-\frac {b c^4 x \tanh ^{-1}(c x)}{4 \left (-1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{6} \left (c^2 e\right ) \int \frac {3 a+b c x+3 b c^3 x^3+3 b \coth ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (b c^6 e\right ) \int \frac {x \tanh ^{-1}(c x)}{-1+c^2 x^2} \, dx\\ &=-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)^2-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{6} \left (c^2 e\right ) \int \left (\frac {3 a+b c x+3 b c^3 x^3}{x^3 \left (-1+c^2 x^2\right )}+\frac {3 b \coth ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )}\right ) \, dx+\frac {1}{2} \left (b c^5 e\right ) \int \frac {\tanh ^{-1}(c x)}{1-c x} \, dx\\ &=-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)^2+\frac {1}{2} b c^4 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 )}{12 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{6} \left (c^2 e\right ) \int \frac {3 a+b c x+3 b c^3 x^3}{x^3 \left (-1+c^2 x^2\right )} \, dx+\frac {1}{2} \left (b c^2 e\right ) \int \frac {\coth ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )} \, dx-\frac {1}{2} \left (b c^5 e\right ) \int \frac {\log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)^2+\frac {1}{2} b c^4 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 )}{12 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{6} \left (c^2 e\right ) \int \left (-\frac {3 a}{x^3}-\frac {b c}{x^2}-\frac {3 a c^2}{x}+\frac {(3 a+4 b) c^3}{2 (-1+c x)}+\frac {(3 a-4 b) c^3}{2 (1+c x)}\right ) \, dx-\frac {1}{2} \left (b c^2 e\right ) \int \frac {\coth ^{-1}(c x)}{x^3} \, dx+\frac {1}{2} \left (b c^4 e\right ) \int \frac {\coth ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx+\frac {1}{2} \left (b c^4 e\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c x}\right )\\ &=\frac {a c^2 e}{4 x^2}+\frac {b c^3 e}{6 x}+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)^2-\frac {1}{2} a c^4 e \log (x)+\frac {1}{2} b c^4 e \tanh ^{-1}(c x) \log \left (\frac {2}{1-c x}\right )+\frac {1}{12} (3 a+4 b) c^4 e \log (1-c x)+\frac {1}{12} (3 a-4 b) c^4 e \log (1+c x)-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {1}{4} b c^4 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )-\frac {1}{4} \left (b c^3 e\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx-\frac {1}{2} \left (b c^4 e\right ) \int \frac {\coth ^{-1}(c x)}{x (1+c x)} \, dx\\ &=\frac {a c^2 e}{4 x^2}+\frac {5 b c^3 e}{12 x}+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)^2-\frac {1}{2} a c^4 e \log (x)+\frac {1}{2} b c^4 e \tanh ^{-1}(c x) \log \left (\frac {2}{1-c x}\right )+\frac {1}{12} (3 a+4 b) c^4 e \log (1-c x)+\frac {1}{12} (3 a-4 b) c^4 e \log (1+c x)-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {1}{2} b c^4 e \coth ^{-1}(c x) \log \left (2-\frac {2}{1+c x}\right )+\frac {1}{4} b c^4 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )-\frac {1}{4} \left (b c^5 e\right ) \int \frac {1}{1-c^2 x^2} \, dx+\frac {1}{2} \left (b c^5 e\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=\frac {a c^2 e}{4 x^2}+\frac {5 b c^3 e}{12 x}+\frac {b c^2 e \coth ^{-1}(c x)}{4 x^2}-\frac {1}{4} b c^4 e \coth ^{-1}(c x)^2-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)^2-\frac {1}{2} a c^4 e \log (x)+\frac {1}{2} b c^4 e \tanh ^{-1}(c x) \log \left (\frac {2}{1-c x}\right )+\frac {1}{12} (3 a+4 b) c^4 e \log (1-c x)+\frac {1}{12} (3 a-4 b) c^4 e \log (1+c x)-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{12 x^3}-\frac {b c^3 \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{4 x^4}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (d+e \log \left (1-c^2 x^2\right )\right )-\frac {1}{2} b c^4 e \coth ^{-1}(c x) \log \left (2-\frac {2}{1+c x}\right )+\frac {1}{4} b c^4 e \text {Li}_2\left (1-\frac {2}{1-c x}\right )+\frac {1}{4} b c^4 e \text {Li}_2\left (-1+\frac {2}{1+c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 307, normalized size = 0.91 \[ \frac {1}{12} \log (1-c x) \left (3 a c^4 e+4 b c^4 e\right )+\frac {1}{12} \log (c x+1) \left (3 a c^4 e-4 b c^4 e\right )+\frac {e \log \left (1-c^2 x^2\right ) \left (-3 a+3 b c^4 x^4 \coth ^{-1}(c x)-3 b c^3 x^3-b c x-3 b \coth ^{-1}(c x)\right )}{12 x^4}-\frac {1}{2} a c^4 e \log (x)+\frac {a c^2 e}{4 x^2}-\frac {a d}{4 x^4}-\frac {1}{4} b c^4 e \left (\text {Li}_2\left (-\frac {1}{c x}\right )-\text {Li}_2\left (\frac {1}{c x}\right )\right )+\frac {b c^3 e}{6 x}+b c^4 d \left (\frac {1}{4} \left (-\frac {1}{3 c^3 x^3}-\frac {1}{c x}-\frac {1}{2} \log (1-c x)+\frac {1}{2} \log (c x+1)\right )-\frac {\coth ^{-1}(c x)}{4 c^4 x^4}\right )-\frac {1}{2} b c^4 e \left (\frac {1}{2} \left (-\frac {1}{c x}-\frac {1}{2} \log (1-c x)+\frac {1}{2} \log (c x+1)\right )-\frac {\coth ^{-1}(c x)}{2 c^2 x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \operatorname {arcoth}\left (c x\right ) + a d + {\left (b e \operatorname {arcoth}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\right )}{x^{5}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{5}}\, dx \]
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 \[ \frac {1}{24} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {arcoth}\left (c x\right )}{x^{4}}\right )} b d + \frac {1}{4} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c^{2} - \frac {\log \left (-c^{2} x^{2} + 1\right )}{x^{4}}\right )} a e - \frac {1}{8} \, b e {\left (\frac {\log \left (c x + 1\right )^{2}}{x^{4}} - 4 \, \int -\frac {2 \, {\left (c x + 1\right )} \log \left (c x - 1\right )^{2} - {\left (2 i \, \pi + {\left (2 i \, \pi c + c\right )} x\right )} \log \left (c x + 1\right ) - {\left (-2 i \, \pi - 2 i \, \pi c x\right )} \log \left (c x - 1\right )}{2 \, {\left (c x^{6} + x^{5}\right )}}\,{d x}\right )} - \frac {a d}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^5} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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