Optimal. Leaf size=44 \[ a x+b x \tanh ^{-1}\left (c x^2\right )+\frac {b \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6091, 298, 203, 206} \[ a x+b x \tanh ^{-1}\left (c x^2\right )+\frac {b \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 6091
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (c x^2\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^2\right )-(2 b c) \int \frac {x^2}{1-c^2 x^4} \, dx\\ &=a x+b x \tanh ^{-1}\left (c x^2\right )-b \int \frac {1}{1-c x^2} \, dx+b \int \frac {1}{1+c x^2} \, dx\\ &=a x+\frac {b \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c}}+b x \tanh ^{-1}\left (c x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 1.30 \[ a x+b x \tanh ^{-1}\left (c x^2\right )+\frac {b \left (\log \left (1-\sqrt {c} x\right )-\log \left (\sqrt {c} x+1\right )+2 \tan ^{-1}\left (\sqrt {c} x\right )\right )}{2 \sqrt {c}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 160, normalized size = 3.64 \[ \left [\frac {b c x \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a c x + 2 \, b \sqrt {c} \arctan \left (\sqrt {c} x\right ) + b \sqrt {c} \log \left (\frac {c x^{2} - 2 \, \sqrt {c} x + 1}{c x^{2} - 1}\right )}{2 \, c}, \frac {b c x \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a c x + 2 \, b \sqrt {-c} \arctan \left (\sqrt {-c} x\right ) - b \sqrt {-c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-c} x - 1}{c x^{2} + 1}\right )}{2 \, c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 83, normalized size = 1.89 \[ \frac {1}{2} \, {\left (c {\left (\frac {2 \, \sqrt {{\left | c \right |}} \arctan \left (x \sqrt {{\left | c \right |}}\right )}{c^{2}} - \frac {\sqrt {{\left | c \right |}} \log \left ({\left | x + \frac {1}{\sqrt {{\left | c \right |}}} \right |}\right )}{c^{2}} + \frac {\sqrt {{\left | c \right |}} \log \left ({\left | x - \frac {1}{\sqrt {{\left | c \right |}}} \right |}\right )}{c^{2}}\right )} + x \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 37, normalized size = 0.84 \[ a x +b x \arctanh \left (c \,x^{2}\right )+\frac {b \arctan \left (x \sqrt {c}\right )}{\sqrt {c}}-\frac {b \arctanh \left (x \sqrt {c}\right )}{\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 55, normalized size = 1.25 \[ \frac {1}{2} \, {\left (c {\left (\frac {2 \, \arctan \left (\sqrt {c} x\right )}{c^{\frac {3}{2}}} + \frac {\log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} + 2 \, x \operatorname {artanh}\left (c x^{2}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 55, normalized size = 1.25 \[ a\,x+\frac {b\,\mathrm {atan}\left (\sqrt {c}\,x\right )}{\sqrt {c}}+\frac {b\,x\,\ln \left (c\,x^2+1\right )}{2}-\frac {b\,x\,\ln \left (1-c\,x^2\right )}{2}+\frac {b\,\mathrm {atan}\left (\sqrt {c}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{\sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.61, size = 178, normalized size = 4.05 \[ a x + b \left (\begin {cases} \frac {c \left (\frac {1}{c}\right )^{\frac {3}{2}} \log {\left (x + i \sqrt {\frac {1}{c}} \right )}}{4} - \frac {i c \left (\frac {1}{c}\right )^{\frac {3}{2}} \log {\left (x + i \sqrt {\frac {1}{c}} \right )}}{4} + x \operatorname {atanh}{\left (c x^{2} \right )} - \frac {\sqrt {\frac {1}{c}} \log {\left (x - i \sqrt {\frac {1}{c}} \right )}}{2} - \frac {i \sqrt {\frac {1}{c}} \log {\left (x - i \sqrt {\frac {1}{c}} \right )}}{2} - \frac {3 \sqrt {\frac {1}{c}} \log {\left (x + i \sqrt {\frac {1}{c}} \right )}}{4} + \frac {3 i \sqrt {\frac {1}{c}} \log {\left (x + i \sqrt {\frac {1}{c}} \right )}}{4} + \sqrt {\frac {1}{c}} \log {\left (x - \sqrt {\frac {1}{c}} \right )} + \sqrt {\frac {1}{c}} \operatorname {atanh}{\left (c x^{2} \right )} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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