Optimal. Leaf size=65 \[ -\frac {b c}{20 x^4}-\frac {b c^3}{10 x^2}-\frac {a+b \tanh ^{-1}(c x)}{5 x^5}+\frac {1}{5} b c^5 \log (x)-\frac {1}{10} b c^5 \log \left (1-c^2 x^2\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6037, 272, 46}
\begin {gather*} -\frac {a+b \tanh ^{-1}(c x)}{5 x^5}+\frac {1}{5} b c^5 \log (x)-\frac {b c^3}{10 x^2}-\frac {1}{10} b c^5 \log \left (1-c^2 x^2\right )-\frac {b c}{20 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 6037
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{x^6} \, dx &=-\frac {a+b \tanh ^{-1}(c x)}{5 x^5}+\frac {1}{5} (b c) \int \frac {1}{x^5 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {a+b \tanh ^{-1}(c x)}{5 x^5}+\frac {1}{10} (b c) \text {Subst}\left (\int \frac {1}{x^3 \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a+b \tanh ^{-1}(c x)}{5 x^5}+\frac {1}{10} (b c) \text {Subst}\left (\int \left (\frac {1}{x^3}+\frac {c^2}{x^2}+\frac {c^4}{x}-\frac {c^6}{-1+c^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b c}{20 x^4}-\frac {b c^3}{10 x^2}-\frac {a+b \tanh ^{-1}(c x)}{5 x^5}+\frac {1}{5} b c^5 \log (x)-\frac {1}{10} b c^5 \log \left (1-c^2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 70, normalized size = 1.08 \begin {gather*} -\frac {a}{5 x^5}-\frac {b c}{20 x^4}-\frac {b c^3}{10 x^2}-\frac {b \tanh ^{-1}(c x)}{5 x^5}+\frac {1}{5} b c^5 \log (x)-\frac {1}{10} b c^5 \log \left (1-c^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 71, normalized size = 1.09
method | result | size |
derivativedivides | \(c^{5} \left (-\frac {a}{5 c^{5} x^{5}}-\frac {b \arctanh \left (c x \right )}{5 c^{5} x^{5}}-\frac {b \ln \left (c x -1\right )}{10}-\frac {b}{20 c^{4} x^{4}}-\frac {b}{10 c^{2} x^{2}}+\frac {b \ln \left (c x \right )}{5}-\frac {b \ln \left (c x +1\right )}{10}\right )\) | \(71\) |
default | \(c^{5} \left (-\frac {a}{5 c^{5} x^{5}}-\frac {b \arctanh \left (c x \right )}{5 c^{5} x^{5}}-\frac {b \ln \left (c x -1\right )}{10}-\frac {b}{20 c^{4} x^{4}}-\frac {b}{10 c^{2} x^{2}}+\frac {b \ln \left (c x \right )}{5}-\frac {b \ln \left (c x +1\right )}{10}\right )\) | \(71\) |
risch | \(-\frac {b \ln \left (c x +1\right )}{10 x^{5}}+\frac {4 b \,c^{5} \ln \left (x \right ) x^{5}-2 b \,c^{5} \ln \left (c^{2} x^{2}-1\right ) x^{5}-2 b \,c^{3} x^{3}-b c x +2 b \ln \left (-c x +1\right )-4 a}{20 x^{5}}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 61, normalized size = 0.94 \begin {gather*} -\frac {1}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac {2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x\right )}{x^{5}}\right )} b - \frac {a}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 70, normalized size = 1.08 \begin {gather*} -\frac {2 \, b c^{5} x^{5} \log \left (c^{2} x^{2} - 1\right ) - 4 \, b c^{5} x^{5} \log \left (x\right ) + 2 \, b c^{3} x^{3} + b c x + 2 \, b \log \left (-\frac {c x + 1}{c x - 1}\right ) + 4 \, a}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.57, size = 80, normalized size = 1.23 \begin {gather*} \begin {cases} - \frac {a}{5 x^{5}} + \frac {b c^{5} \log {\left (x \right )}}{5} - \frac {b c^{5} \log {\left (x - \frac {1}{c} \right )}}{5} - \frac {b c^{5} \operatorname {atanh}{\left (c x \right )}}{5} - \frac {b c^{3}}{10 x^{2}} - \frac {b c}{20 x^{4}} - \frac {b \operatorname {atanh}{\left (c x \right )}}{5 x^{5}} & \text {for}\: c \neq 0 \\- \frac {a}{5 x^{5}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 397 vs.
\(2 (55) = 110\).
time = 0.41, size = 397, normalized size = 6.11 \begin {gather*} \frac {1}{5} \, {\left (b c^{4} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - b c^{4} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {{\left (\frac {5 \, {\left (c x + 1\right )}^{4} b c^{4}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{2} b c^{4}}{{\left (c x - 1\right )}^{2}} + b c^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {5 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {10 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {2 \, {\left (\frac {5 \, {\left (c x + 1\right )}^{4} a c^{4}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{2} a c^{4}}{{\left (c x - 1\right )}^{2}} + a c^{4} + \frac {2 \, {\left (c x + 1\right )}^{4} b c^{4}}{{\left (c x - 1\right )}^{4}} + \frac {4 \, {\left (c x + 1\right )}^{3} b c^{4}}{{\left (c x - 1\right )}^{3}} + \frac {4 \, {\left (c x + 1\right )}^{2} b c^{4}}{{\left (c x - 1\right )}^{2}} + \frac {2 \, {\left (c x + 1\right )} b c^{4}}{c x - 1}\right )}}{\frac {{\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {5 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {10 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {10 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {5 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.91, size = 71, normalized size = 1.09 \begin {gather*} \frac {b\,c^5\,\ln \left (x\right )}{5}-\frac {b\,c^5\,\ln \left (c^2\,x^2-1\right )}{10}-\frac {\frac {b\,c^3\,x^3}{2}+\frac {b\,c\,x}{4}+a}{5\,x^5}-\frac {b\,\ln \left (c\,x+1\right )}{10\,x^5}+\frac {b\,\ln \left (1-c\,x\right )}{10\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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