Optimal. Leaf size=41 \[ -\frac {a+b \tanh ^{-1}\left (c x^2\right )}{4 x^4}+\frac {1}{4} b c^2 \tanh ^{-1}\left (c x^2\right )-\frac {b c}{4 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6097, 275, 325, 206} \[ -\frac {a+b \tanh ^{-1}\left (c x^2\right )}{4 x^4}+\frac {1}{4} b c^2 \tanh ^{-1}\left (c x^2\right )-\frac {b c}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 275
Rule 325
Rule 6097
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^2\right )}{x^5} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{4 x^4}+\frac {1}{2} (b c) \int \frac {1}{x^3 \left (1-c^2 x^4\right )} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{4 x^4}+\frac {1}{4} (b c) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {b c}{4 x^2}-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{4 x^4}+\frac {1}{4} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,x^2\right )\\ &=-\frac {b c}{4 x^2}+\frac {1}{4} b c^2 \tanh ^{-1}\left (c x^2\right )-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 65, normalized size = 1.59 \[ -\frac {a}{4 x^4}-\frac {1}{8} b c^2 \log \left (1-c x^2\right )+\frac {1}{8} b c^2 \log \left (c x^2+1\right )-\frac {b c}{4 x^2}-\frac {b \tanh ^{-1}\left (c x^2\right )}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 49, normalized size = 1.20 \[ -\frac {2 \, b c x^{2} - {\left (b c^{2} x^{4} - b\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a}{8 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 67, normalized size = 1.63 \[ \frac {1}{8} \, b c^{2} \log \left (c x^{2} + 1\right ) - \frac {1}{8} \, b c^{2} \log \left (c x^{2} - 1\right ) - \frac {b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )}{8 \, x^{4}} - \frac {b c x^{2} + a}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 55, normalized size = 1.34 \[ -\frac {a}{4 x^{4}}-\frac {b \arctanh \left (c \,x^{2}\right )}{4 x^{4}}-\frac {b c}{4 x^{2}}+\frac {b \,c^{2} \ln \left (c \,x^{2}+1\right )}{8}-\frac {b \,c^{2} \ln \left (c \,x^{2}-1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 51, normalized size = 1.24 \[ \frac {1}{8} \, {\left ({\left (c \log \left (c x^{2} + 1\right ) - c \log \left (c x^{2} - 1\right ) - \frac {2}{x^{2}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x^{2}\right )}{x^{4}}\right )} b - \frac {a}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 52, normalized size = 1.27 \[ \frac {b\,c^2\,\mathrm {atanh}\left (c\,x^2\right )}{4}-\frac {\frac {a}{4}+\frac {b\,\ln \left (c\,x^2+1\right )}{8}-\frac {b\,\ln \left (1-c\,x^2\right )}{8}+\frac {b\,c\,x^2}{4}}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.07, size = 41, normalized size = 1.00 \[ - \frac {a}{4 x^{4}} + \frac {b c^{2} \operatorname {atanh}{\left (c x^{2} \right )}}{4} - \frac {b c}{4 x^{2}} - \frac {b \operatorname {atanh}{\left (c x^{2} \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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