Optimal. Leaf size=68 \[ \frac {2 c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6128, 865, 875, 208} \[ \frac {2 c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
Antiderivative was successfully verified.
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Rule 208
Rule 865
Rule 875
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {c-a c x}}{x} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}} \, dx\\ &=\frac {2 c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}+\int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {2 c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}+\left (2 a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-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 c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.68 \[ \frac {\sqrt {c-a c x} \left (2 \sqrt {a x+1}-2 \tanh ^{-1}\left (\sqrt {a x+1}\right )\right )}{\sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 183, normalized size = 2.69 \[ \left [\frac {{\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a x - 1}, -\frac {2 \, {\left ({\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}\right )}}{a x - 1}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 83, normalized size = 1.22 \[ \frac {2 \, c^{3} {\left (\frac {\arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right ) + \sqrt {2} \sqrt {-c}}{\sqrt {-c} c^{\frac {3}{2}}} + \frac {\sqrt {a c x + c}}{c^{2}}\right )}}{{\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 71, normalized size = 1.04 \[ \frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (\sqrt {c}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right )-\sqrt {c \left (a x +1\right )}\right )}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}} \]
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 \[ \sqrt {c} \int \frac {1}{\sqrt {a x + 1} x}\,{d x} + \frac {2 \, {\left (a \sqrt {c} x + \sqrt {c}\right )}}{\sqrt {a x + 1}} \]
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 \[ \int \frac {\sqrt {c-a\,c\,x}\,\left (a\,x+1\right )}{x\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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