Optimal. Leaf size=75 \[ -\frac {x \sqrt {d+e x^2}}{4 \sqrt {e}}+\frac {1}{2} x^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 e} \]
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Rubi [A] time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6221, 321, 217, 206} \[ -\frac {x \sqrt {d+e x^2}}{4 \sqrt {e}}+\frac {1}{2} x^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 6221
Rubi steps
\begin {align*} \int x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{2} x^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{2} \sqrt {e} \int \frac {x^2}{\sqrt {d+e x^2}} \, dx\\ &=-\frac {x \sqrt {d+e x^2}}{4 \sqrt {e}}+\frac {1}{2} x^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {d \int \frac {1}{\sqrt {d+e x^2}} \, dx}{4 \sqrt {e}}\\ &=-\frac {x \sqrt {d+e x^2}}{4 \sqrt {e}}+\frac {1}{2} x^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )+\frac {d \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {e}}\\ &=-\frac {x \sqrt {d+e x^2}}{4 \sqrt {e}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 e}+\frac {1}{2} x^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 76, normalized size = 1.01 \[ -\frac {x \sqrt {d+e x^2}}{4 \sqrt {e}}+\frac {d \log \left (\sqrt {d+e x^2}+\sqrt {e} x\right )}{4 e}+\frac {1}{2} x^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 59, normalized size = 0.79 \[ -\frac {2 \, \sqrt {e x^{2} + d} \sqrt {e} x - {\left (2 \, e x^{2} + d\right )} \log \left (\frac {2 \, e x^{2} + 2 \, \sqrt {e x^{2} + d} \sqrt {e} x + d}{d}\right )}{8 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 97, normalized size = 1.29 \[ \frac {x^{2} \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )}{2}+\frac {\sqrt {e}\, x^{3} \sqrt {e \,x^{2}+d}}{8 d}-\frac {x \sqrt {e \,x^{2}+d}}{8 \sqrt {e}}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{4 e}-\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{8 \sqrt {e}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, x^{2} \log \left (\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - \frac {1}{4} \, x^{2} \log \left (-\sqrt {e} x + \sqrt {e x^{2} + d}\right ) - \frac {1}{2} \, d \sqrt {e} \int -\frac {\sqrt {e x^{2} + d} x^{2}}{e^{2} x^{4} + d e x^{2} - {\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 66, normalized size = 0.88 \[ \begin {cases} \frac {d \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{4 e} + \frac {x^{2} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{2} - \frac {x \sqrt {d + e x^{2}}}{4 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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