Optimal. Leaf size=88 \[ \frac {d \sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 e^{3/2}}+\frac {x \sqrt {d+e x^2}}{4 \sqrt {-e}}+\frac {1}{2} x^2 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5151, 321, 217, 206} \[ \frac {d \sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 e^{3/2}}+\frac {x \sqrt {d+e x^2}}{4 \sqrt {-e}}+\frac {1}{2} x^2 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 5151
Rubi steps
\begin {align*} \int x \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{2} x^2 \tan ^{-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 \tan ^{-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 \tan ^{-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 {1}{2} x^2 \tan ^{-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 \sqrt {-e^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 59, normalized size = 0.67 \[ \frac {\left (d+2 e x^2\right ) \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\sqrt {-e} x \sqrt {d+e x^2}}{4 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 49, normalized size = 0.56 \[ -\frac {\sqrt {e x^{2} + d} \sqrt {-e} x - {\left (2 \, e x^{2} + d\right )} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{4 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 62, normalized size = 0.70 \[ \frac {1}{2} \, x^{2} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right ) - \frac {1}{4} \, d \arcsin \left (\frac {x e}{\sqrt {-d e}}\right ) e^{\left (-1\right )} - \frac {1}{4} \, \sqrt {-x^{2} e^{2} - d e} x e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 116, normalized size = 1.32 \[ \frac {x^{2} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{2}+\frac {\sqrt {-e}\, x^{3} \sqrt {e \,x^{2}+d}}{8 d}-\frac {\sqrt {-e}\, x \sqrt {e \,x^{2}+d}}{8 e}+\frac {\sqrt {-e}\, d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{4 e^{\frac {3}{2}}}-\frac {\sqrt {-e}\, x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{8 d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int x\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 71, normalized size = 0.81 \[ \begin {cases} \frac {i d \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{4 e} + \frac {i x^{2} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{2} - \frac {i x \sqrt {d + e x^{2}}}{4 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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