Optimal. Leaf size=65 \[ -\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {2 b}{3 c x} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6097, 263, 325, 298, 203, 206} \[ -\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {2 b}{3 c x} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 263
Rule 298
Rule 325
Rule 6097
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x^4} \, dx &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {1}{3} (2 b c) \int \frac {1}{\left (1-\frac {c^2}{x^4}\right ) x^6} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {1}{3} (2 b c) \int \frac {1}{x^2 \left (-c^2+x^4\right )} \, dx\\ &=-\frac {2 b}{3 c x}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {(2 b) \int \frac {x^2}{-c^2+x^4} \, dx}{3 c}\\ &=-\frac {2 b}{3 c x}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}+\frac {b \int \frac {1}{c-x^2} \, dx}{3 c}-\frac {b \int \frac {1}{c+x^2} \, dx}{3 c}\\ &=-\frac {2 b}{3 c x}-\frac {b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 90, normalized size = 1.38 \[ -\frac {a}{3 x^3}-\frac {b \log \left (\sqrt {c}-x\right )}{6 c^{3/2}}+\frac {b \log \left (\sqrt {c}+x\right )}{6 c^{3/2}}-\frac {b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {2 b}{3 c x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 189, normalized size = 2.91 \[ \left [-\frac {2 \, b \sqrt {c} x^{3} \arctan \left (\frac {x}{\sqrt {c}}\right ) - b \sqrt {c} x^{3} \log \left (\frac {x^{2} + 2 \, \sqrt {c} x + c}{x^{2} - c}\right ) + 4 \, b c x^{2} + b c^{2} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}}, -\frac {2 \, b \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-c} x}{c}\right ) + b \sqrt {-c} x^{3} \log \left (\frac {x^{2} + 2 \, \sqrt {-c} x - c}{x^{2} + c}\right ) + 4 \, b c x^{2} + b c^{2} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 72, normalized size = 1.11 \[ -\frac {b \arctan \left (\frac {x}{\sqrt {-c}}\right )}{3 \, \sqrt {-c} c} - \frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{3 \, c^{\frac {3}{2}}} - \frac {b \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{6 \, x^{3}} - \frac {2 \, b x^{2} + a c}{3 \, c x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 55, normalized size = 0.85 \[ -\frac {a}{3 x^{3}}-\frac {b \arctanh \left (\frac {c}{x^{2}}\right )}{3 x^{3}}-\frac {2 b}{3 c x}-\frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{3 c^{\frac {3}{2}}}+\frac {b \arctanh \left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 64, normalized size = 0.98 \[ -\frac {1}{6} \, {\left (c {\left (\frac {2 \, \arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {\log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {4}{c^{2} x}\right )} + \frac {2 \, \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 69, normalized size = 1.06 \[ \frac {b\,\ln \left (x^2-c\right )}{6\,x^3}-\frac {2\,b}{3\,c\,x}-\frac {b\,\mathrm {atan}\left (\frac {x}{\sqrt {c}}\right )}{3\,c^{3/2}}-\frac {b\,\ln \left (x^2+c\right )}{6\,x^3}-\frac {a}{3\,x^3}-\frac {b\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{3\,c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.69, size = 733, normalized size = 11.28 \[ \begin {cases} - \frac {a}{3 x^{3}} & \text {for}\: c = 0 \\- \frac {a - \infty b}{3 x^{3}} & \text {for}\: c = - x^{2} \\- \frac {a + \infty b}{3 x^{3}} & \text {for}\: c = x^{2} \\\frac {2 i a c^{9}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} - \frac {2 i a c^{7} x^{4}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} + \frac {2 i b c^{\frac {15}{2}} x^{3} \log {\left (- \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} + \frac {b c^{\frac {15}{2}} x^{3} \log {\left (- i \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} - \frac {i b c^{\frac {15}{2}} x^{3} \log {\left (- i \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} - \frac {b c^{\frac {15}{2}} x^{3} \log {\left (i \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} - \frac {i b c^{\frac {15}{2}} x^{3} \log {\left (i \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} + \frac {2 i b c^{\frac {15}{2}} x^{3} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} - \frac {2 i b c^{\frac {11}{2}} x^{7} \log {\left (- \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} - \frac {b c^{\frac {11}{2}} x^{7} \log {\left (- i \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} + \frac {i b c^{\frac {11}{2}} x^{7} \log {\left (- i \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} + \frac {b c^{\frac {11}{2}} x^{7} \log {\left (i \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} + \frac {i b c^{\frac {11}{2}} x^{7} \log {\left (i \sqrt {c} + x \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} - \frac {2 i b c^{\frac {11}{2}} x^{7} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} + \frac {2 i b c^{9} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} + \frac {4 i b c^{8} x^{2}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} - \frac {2 i b c^{7} x^{4} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} - \frac {4 i b c^{6} x^{6}}{- 6 i c^{9} x^{3} + 6 i c^{7} x^{7}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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