Optimal. Leaf size=57 \[ -\frac {\sqrt {-e} \sqrt {d+e x^2}}{2 d x}-\frac {\text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{2 x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5259, 270}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{2 x^2}-\frac {\sqrt {-e} \sqrt {d+e x^2}}{2 d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 5259
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{x^3} \, dx &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{2 x^2}+\frac {1}{2} \sqrt {-e} \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx\\ &=-\frac {\sqrt {-e} \sqrt {d+e x^2}}{2 d x}-\frac {\tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 54, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {-e} x \sqrt {d+e x^2}+d \text {ArcTan}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )}{2 d x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(121\) vs.
\(2(45)=90\).
time = 0.01, size = 122, normalized size = 2.14
method | result | size |
default | \(-\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{2 x^{2}}-\frac {\sqrt {-e}\, \sqrt {e}\, \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 d}+\frac {\sqrt {-e}\, \left (-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+\frac {2 e \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{d}\right )}{2 d}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 64, normalized size = 1.12 \begin {gather*} -\frac {\arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right )}{2 \, x^{2}} - \frac {x^{2} \sqrt {-e} e + d \sqrt {-e}}{2 \, \sqrt {x^{2} e + d} d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 6.19, size = 56, normalized size = 0.98 \begin {gather*} \frac {-2 i \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} - i \, d \log \left (\frac {2 \, x^{2} e + 2 \, \sqrt {x^{2} e + d} x e^{\frac {1}{2}} + d}{d}\right )}{4 \, d x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.02, size = 53, normalized size = 0.93 \begin {gather*} - \frac {\operatorname {atan}{\left (\frac {x \sqrt {- e}}{\sqrt {d + e x^{2}}} \right )}}{2 x^{2}} - \frac {\sqrt {e} \sqrt {- e} \sqrt {\frac {d}{e x^{2}} + 1}}{2 d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs.
\(2 (45) = 90\).
time = 0.49, size = 106, normalized size = 1.86 \begin {gather*} -\frac {e^{4} x}{4 \, {\left (\sqrt {-d e} e - \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}\right )} d {\left | e \right |}} - \frac {\arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{2 \, x^{2}} + \frac {\sqrt {-d e} e - \sqrt {-e^{2} x^{2} - d e} {\left | e \right |}}{4 \, d x {\left | e \right |}} \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 = -0.02 \begin {gather*} \int \frac {\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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