Optimal. Leaf size=58 \[ -\sqrt {c-d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c-d}}\right )+\sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c+d}}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {714, 1144, 212}
\begin {gather*} \sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c+d}}\right )-\sqrt {c-d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c-d}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 714
Rule 1144
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x}}{1-x^2} \, dx &=(2 d) \text {Subst}\left (\int \frac {x^2}{-c^2+d^2+2 c x^2-x^4} \, dx,x,\sqrt {c+d x}\right )\\ &=-\left ((c-d) \text {Subst}\left (\int \frac {1}{c-d-x^2} \, dx,x,\sqrt {c+d x}\right )\right )+(c+d) \text {Subst}\left (\int \frac {1}{c+d-x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\sqrt {c-d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c-d}}\right )+\sqrt {c+d} \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c+d}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 66, normalized size = 1.14 \begin {gather*} \sqrt {-c-d} \tan ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {-c-d}}\right )-\sqrt {-c+d} \tan ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {-c+d}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 57, normalized size = 0.98
method | result | size |
derivativedivides | \(-2 d \left (-\frac {\sqrt {c +d}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c +d}}\right )}{2 d}+\frac {\sqrt {-c +d}\, \arctan \left (\frac {\sqrt {d x +c}}{\sqrt {-c +d}}\right )}{2 d}\right )\) | \(57\) |
default | \(-2 d \left (-\frac {\sqrt {c +d}\, \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c +d}}\right )}{2 d}+\frac {\sqrt {-c +d}\, \arctan \left (\frac {\sqrt {d x +c}}{\sqrt {-c +d}}\right )}{2 d}\right )\) | \(57\) |
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 = 2.30, size = 295, normalized size = 5.09 \begin {gather*} \left [\frac {1}{2} \, \sqrt {c - d} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c - d} + 2 \, c - d}{x + 1}\right ) + \frac {1}{2} \, \sqrt {c + d} \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c + d} + 2 \, c + d}{x - 1}\right ), -\sqrt {-c + d} \arctan \left (-\frac {\sqrt {d x + c} \sqrt {-c + d}}{c - d}\right ) + \frac {1}{2} \, \sqrt {c + d} \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c + d} + 2 \, c + d}{x - 1}\right ), -\sqrt {-c - d} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c - d}}{c + d}\right ) + \frac {1}{2} \, \sqrt {c - d} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c - d} + 2 \, c - d}{x + 1}\right ), -\sqrt {-c + d} \arctan \left (-\frac {\sqrt {d x + c} \sqrt {-c + d}}{c - d}\right ) - \sqrt {-c - d} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c - d}}{c + d}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.14, size = 61, normalized size = 1.05 \begin {gather*} \frac {2 \left (\frac {d \left (c - d\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c + d}} \right )}}{2 \sqrt {- c + d}} - \frac {d \left (c + d\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c - d}} \right )}}{2 \sqrt {- c - d}}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.55, size = 54, normalized size = 0.93 \begin {gather*} -\sqrt {-c + d} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c + d}}\right ) + \sqrt {-c - d} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c - d}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 46, normalized size = 0.79 \begin {gather*} \mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c+d}}\right )\,\sqrt {c+d}-\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c-d}}\right )\,\sqrt {c-d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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