Optimal. Leaf size=64 \[ -\frac {a+b \tan ^{-1}(c x)}{5 x^5}+\frac {1}{5} b c^5 \log (x)+\frac {b c^3}{10 x^2}-\frac {1}{10} b c^5 \log \left (c^2 x^2+1\right )-\frac {b c}{20 x^4} \]
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Rubi [A] time = 0.04, antiderivative size = 64, 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 = {4852, 266, 44} \[ -\frac {a+b \tan ^{-1}(c x)}{5 x^5}+\frac {b c^3}{10 x^2}-\frac {1}{10} b c^5 \log \left (c^2 x^2+1\right )+\frac {1}{5} b c^5 \log (x)-\frac {b c}{20 x^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 4852
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^6} \, dx &=-\frac {a+b \tan ^{-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 \tan ^{-1}(c x)}{5 x^5}+\frac {1}{10} (b c) \operatorname {Subst}\left (\int \frac {1}{x^3 \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {a+b \tan ^{-1}(c x)}{5 x^5}+\frac {1}{10} (b c) \operatorname {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 \tan ^{-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 = 0.02, size = 64, normalized size = 1.00 \[ -\frac {a}{5 x^5}+\frac {1}{10} b c \left (2 c^4 \log (x)+\frac {c^2}{x^2}-c^4 \log \left (c^2 x^2+1\right )-\frac {1}{2 x^4}\right )-\frac {b \tan ^{-1}(c x)}{5 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 59, normalized size = 0.92 \[ -\frac {2 \, b c^{5} x^{5} \log \left (c^{2} x^{2} + 1\right ) - 4 \, b c^{5} x^{5} \log \relax (x) - 2 \, b c^{3} x^{3} + b c x + 4 \, b \arctan \left (c x\right ) + 4 \, a}{20 \, x^{5}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.94 \[ -\frac {a}{5 x^{5}}-\frac {b \arctan \left (c x \right )}{5 x^{5}}-\frac {b c}{20 x^{4}}+\frac {c^{5} b \ln \left (c x \right )}{5}+\frac {b \,c^{3}}{10 x^{2}}-\frac {b \,c^{5} \ln \left (c^{2} x^{2}+1\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 62, normalized size = 0.97 \[ -\frac {1}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b - \frac {a}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 56, normalized size = 0.88 \[ \frac {b\,c^5\,\ln \relax (x)}{5}-\frac {b\,\mathrm {atan}\left (c\,x\right )}{5\,x^5}-\frac {b\,c^5\,\ln \left (c^2\,x^2+1\right )}{10}-\frac {-\frac {b\,c^3\,x^3}{2}+\frac {b\,c\,x}{4}+a}{5\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.67, size = 71, normalized size = 1.11 \[ \begin {cases} - \frac {a}{5 x^{5}} + \frac {b c^{5} \log {\relax (x )}}{5} - \frac {b c^{5} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{10} + \frac {b c^{3}}{10 x^{2}} - \frac {b c}{20 x^{4}} - \frac {b \operatorname {atan}{\left (c x \right )}}{5 x^{5}} & \text {for}\: c \neq 0 \\- \frac {a}{5 x^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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