Optimal. Leaf size=256 \[ \frac {7 b c^3 e}{60 x^2}+\frac {2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \tanh ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \tanh ^{-1}(c x)\right )^2}{5 b}-\frac {5}{6} b c^5 e \log (x)+\frac {19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac {b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac {1}{10} b c^5 \left (d+e \log \left (1-c^2 x^2\right )\right ) \log \left (1-\frac {1}{1-c^2 x^2}\right )-\frac {1}{10} b c^5 e \text {PolyLog}\left (2,\frac {1}{1-c^2 x^2}\right ) \]
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Rubi [A]
time = 0.45, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 16, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used =
{6228, 2525, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46, 6129, 6037, 272, 36, 29, 6095}
\begin {gather*} -\frac {c^5 e \left (a+b \tanh ^{-1}(c x)\right )^2}{5 b}+\frac {2 c^4 e \left (a+b \tanh ^{-1}(c x)\right )}{5 x}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{5 x^5}+\frac {2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}-\frac {5}{6} b c^5 e \log (x)+\frac {7 b c^3 e}{60 x^2}-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 x^4}+\frac {1}{10} b c^5 \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}{10} b c^5 e \text {Li}_2\left (\frac {1}{1-c^2 x^2}\right )+\frac {19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac {b c^3 \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 272
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2458
Rule 2525
Rule 6037
Rule 6095
Rule 6129
Rule 6228
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^6} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac {1}{5} (b c) \int \frac {d+e \log \left (1-c^2 x^2\right )}{x^5 \left (1-c^2 x^2\right )} \, dx-\frac {1}{5} \left (2 c^2 e\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^4 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac {1}{10} (b c) \text {Subst}\left (\int \frac {d+e \log \left (1-c^2 x\right )}{x^3 \left (1-c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{5} \left (2 c^2 e\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^4} \, dx-\frac {1}{5} \left (2 c^4 e\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx\\ &=\frac {2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac {b \text {Subst}\left (\int \frac {d+e \log (x)}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )^3} \, dx,x,1-c^2 x^2\right )}{10 c}-\frac {1}{15} \left (2 b c^3 e\right ) \int \frac {1}{x^3 \left (1-c^2 x^2\right )} \, dx-\frac {1}{5} \left (2 c^4 e\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx-\frac {1}{5} \left (2 c^6 e\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=\frac {2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \tanh ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \tanh ^{-1}(c x)\right )^2}{5 b}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac {b \text {Subst}\left (\int \frac {d+e \log (x)}{\left (\frac {1}{c^2}-\frac {x}{c^2}\right )^3} \, dx,x,1-c^2 x^2\right )}{10 c}-\frac {1}{10} (b c) \text {Subst}\left (\int \frac {d+e \log (x)}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )-\frac {1}{15} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{5} \left (2 b c^5 e\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx\\ &=\frac {2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \tanh ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \tanh ^{-1}(c x)\right )^2}{5 b}-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac {1}{10} (b c) \text {Subst}\left (\int \frac {d+e \log (x)}{\left (\frac {1}{c^2}-\frac {x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )-\frac {1}{10} \left (b c^3\right ) \text {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 )+\frac {1}{20} (b c e) \text {Subst}\left (\int \frac {1}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )-\frac {1}{15} \left (b c^3 e\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {c^2}{x}-\frac {c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{5} \left (b c^5 e\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=\frac {b c^3 e}{15 x^2}+\frac {2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \tanh ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \tanh ^{-1}(c x)\right )^2}{5 b}-\frac {2}{15} b c^5 e \log (x)+\frac {1}{15} b c^5 e \log \left (1-c^2 x^2\right )-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac {b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac {1}{10} \left (b c^3\right ) \text {Subst}\left (\int \frac {d+e \log (x)}{\frac {1}{c^2}-\frac {x}{c^2}} \, dx,x,1-c^2 x^2\right )-\frac {1}{10} \left (b c^5\right ) \text {Subst}\left (\int \frac {d+e \log (x)}{x} \, dx,x,1-c^2 x^2\right )+\frac {1}{20} (b c e) \text {Subst}\left (\int \left (\frac {c^4}{(-1+x)^2}-\frac {c^4}{-1+x}+\frac {c^4}{x}\right ) \, dx,x,1-c^2 x^2\right )+\frac {1}{10} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x}{c^2}} \, dx,x,1-c^2 x^2\right )-\frac {1}{5} \left (b c^5 e\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{5} \left (b c^7 e\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac {7 b c^3 e}{60 x^2}+\frac {2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \tanh ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \tanh ^{-1}(c x)\right )^2}{5 b}+\frac {1}{5} b c^5 d \log (x)-\frac {5}{6} b c^5 e \log (x)+\frac {19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac {b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac {b c^5 \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{20 e}-\frac {1}{10} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {\log (x)}{\frac {1}{c^2}-\frac {x}{c^2}} \, dx,x,1-c^2 x^2\right )\\ &=\frac {7 b c^3 e}{60 x^2}+\frac {2 c^2 e \left (a+b \tanh ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \tanh ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \tanh ^{-1}(c x)\right )^2}{5 b}+\frac {1}{5} b c^5 d \log (x)-\frac {5}{6} b c^5 e \log (x)+\frac {19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac {b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac {b c^5 \left (d+e \log \left (1-c^2 x^2\right )\right )^2}{20 e}-\frac {1}{10} b c^5 e \text {Li}_2\left (c^2 x^2\right )\\ \end {align*}
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Mathematica [F]
time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 24.64, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arctanh \left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{6}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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^{6}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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