Optimal. Leaf size=68 \[ -\frac {\sqrt {-\frac {a e^2}{b}-e^2 x^2}}{2 e \sqrt {a+b x^2} \text {ArcTan}\left (\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}}\right )^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 68, 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 = {5265, 5263}
\begin {gather*} -\frac {\sqrt {e^2 \left (-x^2\right )-\frac {a e^2}{b}}}{2 e \sqrt {a+b x^2} \text {ArcTan}\left (\frac {e x}{\sqrt {e^2 \left (-x^2\right )-\frac {a e^2}{b}}}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 5263
Rule 5265
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 )^3} \, 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 )^3} \, dx}{\sqrt {a+b x^2}}\\ &=-\frac {\sqrt {-\frac {a e^2}{b}-e^2 x^2}}{2 e \sqrt {a+b x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {-\frac {a e^2}{b}-e^2 x^2}}\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 62, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {-\frac {e^2 \left (a+b x^2\right )}{b}}}{2 e \sqrt {a+b x^2} \text {ArcTan}\left (\frac {e x}{\sqrt {-\frac {e^2 \left (a+b x^2\right )}{b}}}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\arctan \left (\frac {e x}{\sqrt {-\frac {a \,e^{2}}{b}-e^{2} x^{2}}}\right )^{3} \sqrt {x^{2} b +a}}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.58, size = 62, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {-\frac {{\left (b x^{2} + a\right )} e^{2}}{b}} e^{\left (-1\right )}}{2 \, \sqrt {b x^{2} + a} \arctan \left (\frac {b x \sqrt {-\frac {{\left (b x^{2} + a\right )} e^{2}}{b}} e^{\left (-1\right )}}{b x^{2} + a}\right )^{2}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a + b x^{2}} \operatorname {atan}^{3}{\left (\frac {e x}{\sqrt {- \frac {a e^{2}}{b} - e^{2} x^{2}}} \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\mathrm {atan}\left (\frac {e\,x}{\sqrt {-e^2\,x^2-\frac {a\,e^2}{b}}}\right )}^3\,\sqrt {b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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