Optimal. Leaf size=83 \[ \frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}}-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}} \]
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Rubi [A] time = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6127, 665, 661, 208} \[ \frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}}-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 665
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+2 \int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}-(4 a c) \operatorname {Subst}\left (\int \frac {1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.75 \[ -\frac {2 \sqrt {c-a c x} \left (\sqrt {a x+1}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )\right )}{a c \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.50, size = 215, normalized size = 2.59 \[ \left [\frac {\frac {\sqrt {2} {\left (a c x - c\right )} \log \left (-\frac {a^{2} x^{2} + 2 \, a x - \frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{\sqrt {c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{\sqrt {c}} + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a^{2} c x - a c}, \frac {2 \, {\left (\sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-\frac {1}{c}}}{a^{2} x^{2} - 1}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}\right )}}{a^{2} c x - a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 88, normalized size = 1.06 \[ -\frac {2 \, c {\left (\frac {\frac {\sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + \sqrt {a c x + c}}{c} - \frac {\sqrt {2} {\left (c \arctan \left (\frac {\sqrt {c}}{\sqrt {-c}}\right ) + \sqrt {-c} \sqrt {c}\right )}}{\sqrt {-c} c}\right )}}{a {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 84, normalized size = 1.01 \[ -\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {c}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-\sqrt {c \left (a x +1\right )}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}\,{d x} \]
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 \[ \int \frac {a\,x+1}{\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}} \,d x \]
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 \[ \int \frac {a x + 1}{\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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