Optimal. Leaf size=59 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x}-\frac {\sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d}} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5151, 266, 63, 208} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x}-\frac {\sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 5151
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^2} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x}+\sqrt {-e} \int \frac {1}{x \sqrt {d+e x^2}} \, dx\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x}+\frac {1}{2} \sqrt {-e} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{\sqrt {-e}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x}-\frac {\sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{\sqrt {d}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 86, normalized size = 1.46 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x}+\frac {i \sqrt {e} \log \left (-\frac {2 \sqrt {-e} \sqrt {d+e x^2}}{e x}+\frac {2 i \sqrt {d}}{\sqrt {e} x}\right )}{\sqrt {d}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 148, normalized size = 2.51 \[ \left [\frac {x \sqrt {-\frac {e}{d}} \log \left (-\frac {e^{2} x^{2} + 2 \, \sqrt {e x^{2} + d} d \sqrt {-e} \sqrt {-\frac {e}{d}} + 2 \, d e}{x^{2}}\right ) - 2 \, \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{2 \, x}, -\frac {x \sqrt {\frac {e}{d}} \arctan \left (\frac {\sqrt {e x^{2} + d} d \sqrt {-e} \sqrt {\frac {e}{d}}}{e^{2} x^{2} + d e}\right ) + \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 54, normalized size = 0.92 \[ -\frac {\arctan \left (\frac {\sqrt {-x^{2} e^{2} - d e} e^{\left (-\frac {1}{2}\right )}}{\sqrt {d}}\right ) e^{\frac {1}{2}}}{\sqrt {d}} - \frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 57, normalized size = 0.97 \[ -\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{x}-\frac {\sqrt {-e}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{\sqrt {d}} \]
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 = -0.02 \[ \int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.54, size = 60, normalized size = 1.02 \[ - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{x} + \frac {\sqrt {- e} \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{d}} \sqrt {d + e x^{2}}} \right )}}{d \sqrt {- \frac {1}{d}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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