Optimal. Leaf size=44 \[ a x+b x \tanh ^{-1}\left (\frac {c}{x^2}\right )+b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6091, 263, 298, 203, 206} \[ a x+b x \tanh ^{-1}\left (\frac {c}{x^2}\right )+b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 263
Rule 298
Rule 6091
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (\frac {c}{x^2}\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (\frac {c}{x^2}\right )+(2 b c) \int \frac {1}{\left (1-\frac {c^2}{x^4}\right ) x^2} \, dx\\ &=a x+b x \tanh ^{-1}\left (\frac {c}{x^2}\right )+(2 b c) \int \frac {x^2}{-c^2+x^4} \, dx\\ &=a x+b x \tanh ^{-1}\left (\frac {c}{x^2}\right )-(b c) \int \frac {1}{c-x^2} \, dx+(b c) \int \frac {1}{c+x^2} \, dx\\ &=a x+b \sqrt {c} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+b x \tanh ^{-1}\left (\frac {c}{x^2}\right )-b \sqrt {c} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 1.23 \[ a x+b x \tanh ^{-1}\left (\frac {c}{x^2}\right )+\frac {1}{2} b \sqrt {c} \left (\log \left (\sqrt {c}-x\right )-\log \left (\sqrt {c}+x\right )+2 \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 138, normalized size = 3.14 \[ \left [\frac {1}{2} \, b x \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + b \sqrt {c} \arctan \left (\frac {x}{\sqrt {c}}\right ) + \frac {1}{2} \, b \sqrt {c} \log \left (\frac {x^{2} - 2 \, \sqrt {c} x + c}{x^{2} - c}\right ) + a x, \frac {1}{2} \, b x \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + b \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{c}\right ) + \frac {1}{2} \, b \sqrt {-c} \log \left (\frac {x^{2} + 2 \, \sqrt {-c} x - c}{x^{2} + c}\right ) + a x\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 57, normalized size = 1.30 \[ \frac {1}{2} \, {\left (2 \, c {\left (\frac {\arctan \left (\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}\right )} + x \log \left (-\frac {\frac {c}{x^{2}} + 1}{\frac {c}{x^{2}} - 1}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 39, normalized size = 0.89 \[ a x +b x \arctanh \left (\frac {c}{x^{2}}\right )-\arctanh \left (\frac {\sqrt {c}}{x}\right ) \sqrt {c}\, b +b \arctan \left (\frac {x}{\sqrt {c}}\right ) \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 51, normalized size = 1.16 \[ \frac {1}{2} \, {\left (c {\left (\frac {2 \, \arctan \left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}} + \frac {\log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right )}{\sqrt {c}}\right )} + 2 \, x \operatorname {artanh}\left (\frac {c}{x^{2}}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 52, normalized size = 1.18 \[ a\,x+\frac {b\,x\,\ln \left (x^2+c\right )}{2}+b\,\sqrt {c}\,\mathrm {atan}\left (\frac {x}{\sqrt {c}}\right )-\frac {b\,x\,\ln \left (x^2-c\right )}{2}+b\,\sqrt {c}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.14, size = 520, normalized size = 11.82 \[ a x + b \left (\begin {cases} 0 & \text {for}\: c = 0 \\- \infty x & \text {for}\: c = - x^{2} \\\infty x & \text {for}\: c = x^{2} \\- \frac {2 i c^{\frac {5}{2}} x \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} + \frac {2 i \sqrt {c} x^{5} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} - \frac {2 i c^{3} \log {\left (- \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} - \frac {c^{3} \log {\left (- i \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} + \frac {i c^{3} \log {\left (- i \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} + \frac {c^{3} \log {\left (i \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} + \frac {i c^{3} \log {\left (i \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} - \frac {2 i c^{3} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} + \frac {2 i c x^{4} \log {\left (- \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} + \frac {c x^{4} \log {\left (- i \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} - \frac {i c x^{4} \log {\left (- i \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} - \frac {c x^{4} \log {\left (i \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} - \frac {i c x^{4} \log {\left (i \sqrt {c} + x \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} + \frac {2 i c x^{4} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 i c^{\frac {5}{2}} + 2 i \sqrt {c} x^{4}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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