Optimal. Leaf size=125 \[ \frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right ) \]
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Rubi [A] time = 0.24, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6178, 665, 661, 208} \[ \frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 665
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^2} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}-\left (2 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{\left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-(4 c) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {c x}{a}} \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}+\frac {\left (8 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {2 c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a}\\ &=\frac {2 a c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {4 a c \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}-4 \sqrt {2} a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {2} \sqrt {c-\frac {c}{a x}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.25, size = 155, normalized size = 1.24 \[ \frac {2 a \left (\sqrt {1-\frac {1}{a^2 x^2}} (7 a x+1) \sqrt {c-\frac {c}{a x}}-3 \sqrt {2} \sqrt {c} (a x-1) \log \left (2 \sqrt {2} a^2 \sqrt {c} x^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )+3 \sqrt {2} \sqrt {c} (a x-1) \log \left ((a x-1)^2\right )\right )}{3 a x-3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 353, normalized size = 2.82 \[ \left [\frac {3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \, {\left (7 \, a^{2} x^{2} + 8 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}}, \frac {2 \, {\left (3 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) + {\left (7 \, a^{2} x^{2} + 8 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}\right )}}{3 \, {\left (a x^{2} - x\right )}}\right ] \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 141, normalized size = 1.13 \[ -\frac {2 \left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (3 a \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) x^{2}-7 a \sqrt {\frac {1}{a}}\, x \sqrt {\left (a x +1\right ) x}-\sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\right )}{3 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a x}}}{x^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c-\frac {c}{a\,x}}}{x^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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