Optimal. Leaf size=60 \[ \frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}}+\frac {\text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}} \]
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Rubi [A]
time = 0.08, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6302, 6277,
667, 223, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}+\frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 667
Rule 6277
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\left (c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\right )\\ &=\frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}}+\int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}}+\text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )\\ &=\frac {2 (1-a x)}{a \sqrt {c-a^2 c x^2}}+\frac {\tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 100, normalized size = 1.67 \begin {gather*} -\frac {2 \sqrt {1-a^2 x^2} \left ((-1+a x) \sqrt {1+a x}+\sqrt {1-a x} (1+a x) \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{a \sqrt {1-a x} (1+a x) \sqrt {c-a^2 c x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 73, normalized size = 1.22
method | result | size |
default | \(\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}+\frac {2 \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+2 \left (x +\frac {1}{a}\right ) a c}}{a^{2} c \left (x +\frac {1}{a}\right )}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 39, normalized size = 0.65 \begin {gather*} \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x + a c} + \frac {\arcsin \left (a x\right )}{a \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 151, normalized size = 2.52 \begin {gather*} \left [-\frac {{\left (a x + 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 4 \, \sqrt {-a^{2} c x^{2} + c}}{2 \, {\left (a^{2} c x + a c\right )}}, -\frac {{\left (a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{2} c x + a c}\right ] \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 {a x - 1}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\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 = -0.02 \begin {gather*} \int \frac {a\,x-1}{\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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