Optimal. Leaf size=122 \[ \frac {23 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{4 a^2}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^2}+\frac {1}{2} x^2 \sqrt {c-\frac {c}{a x}}-\frac {9 x \sqrt {c-\frac {c}{a x}}}{4 a} \]
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Rubi [A] time = 0.32, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6167, 6133, 25, 514, 446, 98, 151, 156, 63, 208} \[ \frac {23 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{4 a^2}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^2}+\frac {1}{2} x^2 \sqrt {c-\frac {c}{a x}}-\frac {9 x \sqrt {c-\frac {c}{a x}}}{4 a} \]
Antiderivative was successfully verified.
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Rule 25
Rule 63
Rule 98
Rule 151
Rule 156
Rule 208
Rule 446
Rule 514
Rule 6133
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx\\ &=-\int \frac {\sqrt {c-\frac {c}{a x}} x (1-a x)}{1+a x} \, dx\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^2}{1+a x} \, dx}{c}\\ &=\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x}{a+\frac {1}{x}} \, dx}{c}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x^3 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {1}{2} \sqrt {c-\frac {c}{a x}} x^2+\frac {\operatorname {Subst}\left (\int \frac {\frac {9 c^2}{2}-\frac {7 c^2 x}{2 a}}{x^2 (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {9 \sqrt {c-\frac {c}{a x}} x}{4 a}+\frac {1}{2} \sqrt {c-\frac {c}{a x}} x^2-\frac {\operatorname {Subst}\left (\int \frac {\frac {23 c^3}{4}-\frac {9 c^3 x}{4 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^2}\\ &=-\frac {9 \sqrt {c-\frac {c}{a x}} x}{4 a}+\frac {1}{2} \sqrt {c-\frac {c}{a x}} x^2-\frac {(23 c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}+\frac {(4 c) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^2}\\ &=-\frac {9 \sqrt {c-\frac {c}{a x}} x}{4 a}+\frac {1}{2} \sqrt {c-\frac {c}{a x}} x^2+\frac {23 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{4 a}-\frac {8 \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{a}\\ &=-\frac {9 \sqrt {c-\frac {c}{a x}} x}{4 a}+\frac {1}{2} \sqrt {c-\frac {c}{a x}} x^2+\frac {23 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{4 a^2}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 100, normalized size = 0.82 \[ \frac {a x (2 a x-9) \sqrt {c-\frac {c}{a x}}+23 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-16 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{4 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 239, normalized size = 1.96 \[ \left [\frac {16 \, \sqrt {2} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 2 \, {\left (2 \, a^{2} x^{2} - 9 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} + 23 \, \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{8 \, a^{2}}, \frac {16 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + {\left (2 \, a^{2} x^{2} - 9 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 23 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{4 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 216, normalized size = 1.77 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (4 \sqrt {\frac {1}{a}}\, \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} x -2 \sqrt {\frac {1}{a}}\, \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}}-16 \sqrt {\frac {1}{a}}\, a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}+16 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right ) a^{\frac {3}{2}}+24 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2}-\ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2}\right )}{8 \sqrt {\left (a x -1\right ) x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}} \]
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 \[ \int \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a x}} x}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x-1\right )}{a\,x+1} \,d x \]
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 \[ \int \frac {x \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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