Optimal. Leaf size=66 \[ -\frac {\sqrt {e^2 \left (-x^2\right )-\frac {a e^2}{b}}}{e \sqrt {a+b x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {e^2 \left (-x^2\right )-\frac {a e^2}{b}}}\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5157, 5155} \[ -\frac {\sqrt {e^2 \left (-x^2\right )-\frac {a e^2}{b}}}{e \sqrt {a+b x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {e^2 \left (-x^2\right )-\frac {a e^2}{b}}}\right )} \]
Antiderivative was successfully verified.
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Rule 5155
Rule 5157
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}}\right )^2} \, dx &=\frac {\sqrt {-\frac {a e^2}{b}-e^2 x^2} \int \frac {1}{\sqrt {-\frac {a e^2}{b}-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}}\right )^2} \, dx}{\sqrt {a+b x^2}}\\ &=-\frac {\sqrt {-\frac {a e^2}{b}-e^2 x^2}}{e \sqrt {a+b x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}}\right )}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 60, normalized size = 0.91 \[ \frac {e \sqrt {a+b x^2}}{b \sqrt {-\frac {e^2 \left (a+b x^2\right )}{b}} \tan ^{-1}\left (\frac {e x}{\sqrt {-\frac {e^2 \left (a+b x^2\right )}{b}}}\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 82, normalized size = 1.24 \[ \frac {\sqrt {b x^{2} + a} \sqrt {-\frac {b e^{2} x^{2} + a e^{2}}{b}}}{{\left (b e x^{2} + a e\right )} \arctan \left (\frac {b x \sqrt {-\frac {b e^{2} x^{2} + a e^{2}}{b}}}{b e x^{2} + a e}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x^{2} + a} \arctan \left (\frac {e x}{\sqrt {-e^{2} x^{2} - \frac {a e^{2}}{b}}}\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{\arctan \left (\frac {e x}{\sqrt {-\frac {a \,e^{2}}{b}-e^{2} x^{2}}}\right )^{2} \sqrt {b \,x^{2}+a}}\, dx \]
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 \[ \int \frac {1}{{\mathrm {atan}\left (\frac {e\,x}{\sqrt {-e^2\,x^2-\frac {a\,e^2}{b}}}\right )}^2\,\sqrt {b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b x^{2}} \operatorname {atan}^{2}{\left (\frac {e x}{\sqrt {- \frac {a e^{2}}{b} - e^{2} x^{2}}} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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