Optimal. Leaf size=57 \[ -\frac {b}{6 c x^2}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x}\right )}{3 x^3}+\frac {b \log (x)}{3 c^3}-\frac {b \log \left (c^2-x^2\right )}{6 c^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6037, 269, 272,
46} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (\frac {c}{x}\right )}{3 x^3}+\frac {b \log (x)}{3 c^3}-\frac {b \log \left (c^2-x^2\right )}{6 c^3}-\frac {b}{6 c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 269
Rule 272
Rule 6037
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (\frac {c}{x}\right )}{x^4} \, dx &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x}\right )}{3 x^3}-\frac {1}{3} (b c) \int \frac {1}{\left (1-\frac {c^2}{x^2}\right ) x^5} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x}\right )}{3 x^3}-\frac {1}{3} (b c) \int \frac {1}{x^3 \left (-c^2+x^2\right )} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x}\right )}{3 x^3}-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {1}{x^2 \left (-c^2+x\right )} \, dx,x,x^2\right )\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x}\right )}{3 x^3}-\frac {1}{6} (b c) \text {Subst}\left (\int \left (-\frac {1}{c^4 \left (c^2-x\right )}-\frac {1}{c^2 x^2}-\frac {1}{c^4 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b}{6 c x^2}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x}\right )}{3 x^3}+\frac {b \log (x)}{3 c^3}-\frac {b \log \left (c^2-x^2\right )}{6 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 62, normalized size = 1.09 \begin {gather*} -\frac {a}{3 x^3}-\frac {b}{6 c x^2}-\frac {b \tanh ^{-1}\left (\frac {c}{x}\right )}{3 x^3}+\frac {b \log (x)}{3 c^3}-\frac {b \log \left (-c^2+x^2\right )}{6 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 62, normalized size = 1.09
method | result | size |
derivativedivides | \(-\frac {\frac {a \,c^{3}}{3 x^{3}}+\frac {b \,c^{3} \arctanh \left (\frac {c}{x}\right )}{3 x^{3}}+\frac {b \,c^{2}}{6 x^{2}}+\frac {b \ln \left (\frac {c}{x}-1\right )}{6}+\frac {b \ln \left (1+\frac {c}{x}\right )}{6}}{c^{3}}\) | \(62\) |
default | \(-\frac {\frac {a \,c^{3}}{3 x^{3}}+\frac {b \,c^{3} \arctanh \left (\frac {c}{x}\right )}{3 x^{3}}+\frac {b \,c^{2}}{6 x^{2}}+\frac {b \ln \left (\frac {c}{x}-1\right )}{6}+\frac {b \ln \left (1+\frac {c}{x}\right )}{6}}{c^{3}}\) | \(62\) |
risch | \(-\frac {b \ln \left (x +c \right )}{6 x^{3}}-\frac {-i \pi b \,c^{3} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}-i \pi b \,c^{3} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (x +c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x +c \right )}{x}\right )+i \pi b \,c^{3} \mathrm {csgn}\left (i \left (x +c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{2}+i \pi b \,c^{3} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{2}-2 i \pi b \,c^{3}-i \pi b \,c^{3} \mathrm {csgn}\left (i \left (c -x \right )\right ) \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}+2 i \pi b \,c^{3} \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}-i \pi b \,c^{3} \mathrm {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{3}-i \pi b \,c^{3} \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{3}+i \pi b \,c^{3} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (c -x \right )\right ) \mathrm {csgn}\left (\frac {i \left (c -x \right )}{x}\right )-4 b \ln \left (x \right ) x^{3}+2 b \ln \left (c^{2}-x^{2}\right ) x^{3}-2 b \ln \left (c -x \right ) c^{3}+4 a \,c^{3}+2 b \,c^{2} x}{12 c^{3} x^{3}}\) | \(324\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 55, normalized size = 0.96 \begin {gather*} -\frac {1}{6} \, {\left (c {\left (\frac {\log \left (-c^{2} + x^{2}\right )}{c^{4}} - \frac {\log \left (x^{2}\right )}{c^{4}} + \frac {1}{c^{2} x^{2}}\right )} + \frac {2 \, \operatorname {artanh}\left (\frac {c}{x}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 62, normalized size = 1.09 \begin {gather*} -\frac {b x^{3} \log \left (-c^{2} + x^{2}\right ) - 2 \, b x^{3} \log \left (x\right ) + b c^{3} \log \left (-\frac {c + x}{c - x}\right ) + 2 \, a c^{3} + b c^{2} x}{6 \, c^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.43, size = 68, normalized size = 1.19 \begin {gather*} \begin {cases} - \frac {a}{3 x^{3}} - \frac {b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{3 x^{3}} - \frac {b}{6 c x^{2}} + \frac {b \log {\left (x \right )}}{3 c^{3}} - \frac {b \log {\left (- c + x \right )}}{3 c^{3}} - \frac {b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{3 c^{3}} & \text {for}\: c \neq 0 \\- \frac {a}{3 x^{3}} & \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. 234 vs.
\(2 (49) = 98\).
time = 0.42, size = 234, normalized size = 4.11 \begin {gather*} -\frac {\frac {{\left (b + \frac {3 \, b {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}}\right )} \log \left (-\frac {c + x}{c - x}\right )}{\frac {{\left (c + x\right )}^{3} c^{2}}{{\left (c - x\right )}^{3}} - \frac {3 \, {\left (c + x\right )}^{2} c^{2}}{{\left (c - x\right )}^{2}} + \frac {3 \, {\left (c + x\right )} c^{2}}{c - x} - c^{2}} + \frac {2 \, {\left (a + \frac {3 \, a {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {b {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} - \frac {b {\left (c + x\right )}}{c - x}\right )}}{\frac {{\left (c + x\right )}^{3} c^{2}}{{\left (c - x\right )}^{3}} - \frac {3 \, {\left (c + x\right )}^{2} c^{2}}{{\left (c - x\right )}^{2}} + \frac {3 \, {\left (c + x\right )} c^{2}}{c - x} - c^{2}} - \frac {b \log \left (-\frac {c + x}{c - x} + 1\right )}{c^{2}} + \frac {b \log \left (-\frac {c + x}{c - x}\right )}{c^{2}}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.75, size = 59, normalized size = 1.04 \begin {gather*} -\frac {\frac {a}{3}+\frac {b\,\mathrm {atanh}\left (\frac {c}{x}\right )}{3}}{x^3}-\frac {\frac {b\,x^3\,\ln \left (x^2-c^2\right )}{6}-\frac {b\,x^3\,\ln \left (x\right )}{3}+\frac {b\,c^2\,x}{6}}{c^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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