Optimal. Leaf size=51 \[ x \left (-\sqrt {c-\frac {c}{a x}}\right )-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
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Rubi [A] time = 0.10, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6133, 25, 514, 375, 78, 63, 208} \[ x \left (-\sqrt {c-\frac {c}{a x}}\right )-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 25
Rule 63
Rule 78
Rule 208
Rule 375
Rule 514
Rule 6133
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} (1+a x)}{1-a x} \, dx\\ &=-\frac {c \int \frac {1+a x}{\sqrt {c-\frac {c}{a x}} x} \, dx}{a}\\ &=-\frac {c \int \frac {a+\frac {1}{x}}{\sqrt {c-\frac {c}{a x}}} \, dx}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \frac {a+x}{x^2 \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\sqrt {c-\frac {c}{a x}} x+\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\sqrt {c-\frac {c}{a x}} x-3 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )\\ &=-\sqrt {c-\frac {c}{a x}} x-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 1.00 \[ x \left (-\sqrt {c-\frac {c}{a x}}\right )-\frac {3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 125, normalized size = 2.45 \[ \left [-\frac {2 \, a x \sqrt {\frac {a c x - c}{a x}} - 3 \, \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{2 \, a}, -\frac {a x \sqrt {\frac {a c x - c}{a x}} - 3 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.94, size = 97, normalized size = 1.90 \[ -\frac {3 \, \sqrt {c} \log \left ({\left | a \right |} {\left | c \right |}\right ) \mathrm {sgn}\relax (x)}{2 \, a} + \frac {3 \, \sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - a c x}\right )} \sqrt {c} {\left | a \right |} + a c \right |}\right )}{2 \, a \mathrm {sgn}\relax (x)} - \frac {\sqrt {a^{2} c x^{2} - a c x} {\left | a \right |}}{a^{2} \mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 120, normalized size = 2.35 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 \sqrt {a \,x^{2}-x}\, \sqrt {a}-4 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}-\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )-2 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int -\frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
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 \[ - \int \frac {\sqrt {c - \frac {c}{a x}}}{a x - 1}\, dx - \int \frac {a x \sqrt {c - \frac {c}{a x}}}{a x - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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