Optimal. Leaf size=37 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {c-d} x}{\sqrt {c+d}}\right )}{\sqrt {c-d} \sqrt {c+d}} \]
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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {214}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x \sqrt {c-d}}{\sqrt {c+d}}\right )}{\sqrt {c-d} \sqrt {c+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rubi steps
\begin {align*} \int \frac {1}{-c-d+(c-d) x^2} \, dx &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-d} x}{\sqrt {c+d}}\right )}{\sqrt {c-d} \sqrt {c+d}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 44, normalized size = 1.19 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {c-d} x}{\sqrt {-c-d}}\right )}{\sqrt {-c-d} \sqrt {c-d}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 33, normalized size = 0.89
method | result | size |
default | \(-\frac {\arctanh \left (\frac {\left (c -d \right ) x}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\) | \(33\) |
risch | \(\frac {\ln \left (\left (-c +d \right ) x +\sqrt {c^{2}-d^{2}}\right )}{2 \sqrt {c^{2}-d^{2}}}-\frac {\ln \left (\left (c -d \right ) x +\sqrt {c^{2}-d^{2}}\right )}{2 \sqrt {c^{2}-d^{2}}}\) | \(68\) |
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: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 102, normalized size = 2.76 \begin {gather*} \left [\frac {\log \left (\frac {{\left (c - d\right )} x^{2} - 2 \, \sqrt {c^{2} - d^{2}} x + c + d}{{\left (c - d\right )} x^{2} - c - d}\right )}{2 \, \sqrt {c^{2} - d^{2}}}, \frac {\sqrt {-c^{2} + d^{2}} \arctan \left (\frac {\sqrt {-c^{2} + d^{2}} x}{c + d}\right )}{c^{2} - d^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (31) = 62\).
time = 0.09, size = 87, normalized size = 2.35 \begin {gather*} \frac {\sqrt {\frac {1}{\left (c - d\right ) \left (c + d\right )}} \log {\left (- c \sqrt {\frac {1}{\left (c - d\right ) \left (c + d\right )}} - d \sqrt {\frac {1}{\left (c - d\right ) \left (c + d\right )}} + x \right )}}{2} - \frac {\sqrt {\frac {1}{\left (c - d\right ) \left (c + d\right )}} \log {\left (c \sqrt {\frac {1}{\left (c - d\right ) \left (c + d\right )}} + d \sqrt {\frac {1}{\left (c - d\right ) \left (c + d\right )}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 33, normalized size = 0.89 \begin {gather*} \frac {\arctan \left (\frac {c x - d x}{\sqrt {-c^{2} + d^{2}}}\right )}{\sqrt {-c^{2} + d^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 29, normalized size = 0.78 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {c-d}}{\sqrt {c+d}}\right )}{\sqrt {c+d}\,\sqrt {c-d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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