Optimal. Leaf size=45 \[ -\frac {b}{4 c x^2}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{4 x^4}+\frac {b \tanh ^{-1}\left (\frac {x^2}{c}\right )}{4 c^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6037, 269, 281,
331, 213} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{4 x^4}+\frac {b \tanh ^{-1}\left (\frac {x^2}{c}\right )}{4 c^2}-\frac {b}{4 c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 269
Rule 281
Rule 331
Rule 6037
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x^5} \, dx &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{4 x^4}-\frac {1}{2} (b c) \int \frac {1}{\left (1-\frac {c^2}{x^4}\right ) x^7} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{4 x^4}-\frac {1}{2} (b c) \int \frac {1}{x^3 \left (-c^2+x^4\right )} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{4 x^4}-\frac {1}{4} (b c) \text {Subst}\left (\int \frac {1}{x^2 \left (-c^2+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {b}{4 c x^2}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{4 x^4}-\frac {b \text {Subst}\left (\int \frac {1}{-c^2+x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {b}{4 c x^2}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{4 x^4}+\frac {b \tanh ^{-1}\left (\frac {x^2}{c}\right )}{4 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 64, normalized size = 1.42 \begin {gather*} -\frac {a}{4 x^4}-\frac {b}{4 c x^2}-\frac {b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{4 x^4}-\frac {b \log \left (-c+x^2\right )}{8 c^2}+\frac {b \log \left (c+x^2\right )}{8 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 57, normalized size = 1.27
method | result | size |
derivativedivides | \(-\frac {a}{4 x^{4}}-\frac {b \arctanh \left (\frac {c}{x^{2}}\right )}{4 x^{4}}-\frac {b}{4 c \,x^{2}}-\frac {b \ln \left (\frac {c}{x^{2}}-1\right )}{8 c^{2}}+\frac {b \ln \left (1+\frac {c}{x^{2}}\right )}{8 c^{2}}\) | \(57\) |
default | \(-\frac {a}{4 x^{4}}-\frac {b \arctanh \left (\frac {c}{x^{2}}\right )}{4 x^{4}}-\frac {b}{4 c \,x^{2}}-\frac {b \ln \left (\frac {c}{x^{2}}-1\right )}{8 c^{2}}+\frac {b \ln \left (1+\frac {c}{x^{2}}\right )}{8 c^{2}}\) | \(57\) |
risch | \(-\frac {b \ln \left (x^{2}+c \right )}{8 x^{4}}-\frac {2 i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}+i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-2 i \pi b \,c^{2}-i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}-i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )-i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{3}+i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}+i \pi b \,c^{2} \mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}-i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{3}-i \pi b \,c^{2} \mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}-2 b \ln \left (-x^{2}-c \right ) x^{4}+2 b \ln \left (-x^{2}+c \right ) x^{4}-2 b \ln \left (-x^{2}+c \right ) c^{2}+4 b c \,x^{2}+4 a \,c^{2}}{16 c^{2} x^{4}}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 56, normalized size = 1.24 \begin {gather*} \frac {1}{8} \, {\left (c {\left (\frac {\log \left (x^{2} + c\right )}{c^{3}} - \frac {\log \left (x^{2} - c\right )}{c^{3}} - \frac {2}{c^{2} x^{2}}\right )} - \frac {2 \, \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x^{4}}\right )} b - \frac {a}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 52, normalized size = 1.16 \begin {gather*} -\frac {2 \, b c x^{2} + 2 \, a c^{2} - {\left (b x^{4} - b c^{2}\right )} \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{8 \, c^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.82, size = 49, normalized size = 1.09 \begin {gather*} \begin {cases} - \frac {a}{4 x^{4}} - \frac {b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{4 x^{4}} - \frac {b}{4 c x^{2}} + \frac {b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\- \frac {a}{4 x^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 66, normalized size = 1.47 \begin {gather*} \frac {b \log \left (x^{2} + c\right )}{8 \, c^{2}} - \frac {b \log \left (-x^{2} + c\right )}{8 \, c^{2}} - \frac {b \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{8 \, x^{4}} - \frac {b x^{2} + a c}{4 \, c x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.00, size = 59, normalized size = 1.31 \begin {gather*} \frac {\frac {b\,x^4\,\mathrm {atanh}\left (\frac {x^2}{c}\right )}{4}-\frac {b\,c\,x^2}{4}}{c^2\,x^4}-\frac {\frac {a}{4}-\frac {b\,\ln \left (x^2-c\right )}{8}+\frac {b\,\ln \left (x^2+c\right )}{8}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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