Optimal. Leaf size=244 \[ \frac {1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac {1}{12} c^4 e (3 a-4 b) \log (c x+1)-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x^4}-\frac {1}{2} a c^4 e \log (x)+\frac {a c^2 e}{4 x^2}+\frac {1}{4} b c^4 e \text {Li}_2(-c x)-\frac {1}{4} b c^4 e \text {Li}_2(c x)-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)+\frac {5 b c^3 e}{12 x}-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{12 x^3}+\frac {b c^2 e \tanh ^{-1}(c x)}{4 x^2}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x} \]
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Rubi [A] time = 0.26, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5916, 325, 206, 6085, 1802, 6044, 5912} \[ \frac {1}{4} b c^4 e \text {PolyLog}(2,-c x)-\frac {1}{4} b c^4 e \text {PolyLog}(2,c x)-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x^4}+\frac {1}{12} c^4 e (3 a+4 b) \log (1-c x)+\frac {1}{12} c^4 e (3 a-4 b) \log (c x+1)+\frac {a c^2 e}{4 x^2}-\frac {1}{2} a c^4 e \log (x)-\frac {b c^3 \left (e \log \left (1-c^2 x^2\right )+d\right )}{4 x}-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{12 x^3}+\frac {1}{4} b c^4 \tanh ^{-1}(c x) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac {b c^2 e \tanh ^{-1}(c x)}{4 x^2}+\frac {5 b c^3 e}{12 x}-\frac {1}{4} b c^4 e \tanh ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 206
Rule 325
Rule 1802
Rule 5912
Rule 5916
Rule 6044
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^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 {1}{4} b c^4 \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 )}{4 x^4}+\left (2 c^2 e\right ) \int \left (\frac {3 a+b c x+3 b c^3 x^3}{12 x^3 \left (-1+c^2 x^2\right )}-\frac {b \left (1+c^2 x^2\right ) \tanh ^{-1}(c x)}{4 x^3}\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 {1}{4} b c^4 \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 )}{4 x^4}+\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 {\left (1+c^2 x^2\right ) \tanh ^{-1}(c x)}{x^3} \, 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 {1}{4} b c^4 \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 )}{4 x^4}+\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 \left (\frac {\tanh ^{-1}(c x)}{x^3}+\frac {c^2 \tanh ^{-1}(c x)}{x}\right ) \, dx\\ &=\frac {a c^2 e}{4 x^2}+\frac {b c^3 e}{6 x}-\frac {1}{2} a c^4 e \log (x)+\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 {1}{4} b c^4 \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 )}{4 x^4}-\frac {1}{2} \left (b c^2 e\right ) \int \frac {\tanh ^{-1}(c x)}{x^3} \, dx-\frac {1}{2} \left (b c^4 e\right ) \int \frac {\tanh ^{-1}(c x)}{x} \, dx\\ &=\frac {a c^2 e}{4 x^2}+\frac {b c^3 e}{6 x}+\frac {b c^2 e \tanh ^{-1}(c x)}{4 x^2}-\frac {1}{2} a c^4 e \log (x)+\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 {1}{4} b c^4 \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 )}{4 x^4}+\frac {1}{4} b c^4 e \text {Li}_2(-c x)-\frac {1}{4} b c^4 e \text {Li}_2(c x)-\frac {1}{4} \left (b c^3 e\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=\frac {a c^2 e}{4 x^2}+\frac {5 b c^3 e}{12 x}+\frac {b c^2 e \tanh ^{-1}(c x)}{4 x^2}-\frac {1}{2} a c^4 e \log (x)+\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 {1}{4} b c^4 \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 )}{4 x^4}+\frac {1}{4} b c^4 e \text {Li}_2(-c x)-\frac {1}{4} b c^4 e \text {Li}_2(c x)-\frac {1}{4} \left (b c^5 e\right ) \int \frac {1}{1-c^2 x^2} \, dx\\ &=\frac {a c^2 e}{4 x^2}+\frac {5 b c^3 e}{12 x}-\frac {1}{4} b c^4 e \tanh ^{-1}(c x)+\frac {b c^2 e \tanh ^{-1}(c x)}{4 x^2}-\frac {1}{2} a c^4 e \log (x)+\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 {1}{4} b c^4 \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 )}{4 x^4}+\frac {1}{4} b c^4 e \text {Li}_2(-c x)-\frac {1}{4} b c^4 e \text {Li}_2(c x)\\ \end {align*}
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Mathematica [A] time = 0.16, size = 299, normalized size = 1.23 \[ \frac {1}{12} \log (1-c x) \left (3 a c^4 e+4 b c^4 e\right )+\frac {1}{12} \log (c x+1) \left (3 a c^4 e-4 b c^4 e\right )+\frac {e \log \left (1-c^2 x^2\right ) \left (-3 a+3 b c^4 x^4 \tanh ^{-1}(c x)-3 b c^3 x^3-b c x-3 b \tanh ^{-1}(c x)\right )}{12 x^4}-\frac {1}{2} a c^4 e \log (x)+\frac {a c^2 e}{4 x^2}-\frac {a d}{4 x^4}-\frac {1}{4} b c^4 e (\text {Li}_2(c x)-\text {Li}_2(-c x))+\frac {b c^3 e}{6 x}+b c^4 d \left (\frac {1}{4} \left (-\frac {1}{3 c^3 x^3}-\frac {1}{c x}-\frac {1}{2} \log (1-c x)+\frac {1}{2} \log (c x+1)\right )-\frac {\tanh ^{-1}(c x)}{4 c^4 x^4}\right )-\frac {1}{2} b c^4 e \left (\frac {1}{2} \left (-\frac {1}{c x}-\frac {1}{2} \log (1-c x)+\frac {1}{2} \log (c x+1)\right )-\frac {\tanh ^{-1}(c x)}{2 c^2 x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b d \operatorname {artanh}\left (c x\right ) + a d + {\left (b e \operatorname {artanh}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\right )}{x^{5}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{5}}\,{d x} \]
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 \[ \int \frac {\left (a +b \arctanh \left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{5}}\, dx \]
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 \[ \frac {1}{24} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} b d + \frac {1}{4} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c^{2} - \frac {\log \left (-c^{2} x^{2} + 1\right )}{x^{4}}\right )} a e + \frac {1}{8} \, b e {\left (\frac {\log \left (-c x + 1\right )^{2}}{x^{4}} - 4 \, \int -\frac {2 \, {\left (c x - 1\right )} \log \left (c x + 1\right )^{2} - c x \log \left (-c x + 1\right )}{2 \, {\left (c x^{6} - x^{5}\right )}}\,{d x}\right )} - \frac {a d}{4 \, x^{4}} \]
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 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^5} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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