Optimal. Leaf size=112 \[ -\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {(1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{2 x}+\frac {3 a^2 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{2 \sqrt {1-a x} \sqrt {a x+1}} \]
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Rubi [A] time = 0.52, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6167, 6159, 6129, 94, 92, 208} \[ -\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {(1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}{2 x}+\frac {3 a^2 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{2 \sqrt {1-a x} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 92
Rule 94
Rule 208
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^2} \, dx\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^3} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{x^3 \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}{2 x}+\frac {\left (3 a \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {\sqrt {1-a x}}{x^2 \sqrt {1+a x}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}{2 x}-\frac {\left (3 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}{2 x}+\frac {\left (3 a^3 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {3}{2} a \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}{2 x}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{2 \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 78, normalized size = 0.70 \[ -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left ((4 a x-1) \sqrt {a^2 x^2-1}+3 a^2 x^2 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )\right )}{2 x \sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 176, normalized size = 1.57 \[ \left [\frac {3 \, a \sqrt {-c} x \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (4 \, a x - 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, x}, -\frac {3 \, a \sqrt {c} x \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (4 \, a x - 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 194, normalized size = 1.73 \[ {\left (3 \, \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x) - \frac {{\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a c \mathrm {sgn}\relax (x) + 4 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} c^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a c^{2} \mathrm {sgn}\relax (x) + 4 \, c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 348, normalized size = 3.11 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (-4 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{3} a^{3} c +4 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x \,a^{3}+4 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{2} a -4 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{2} a +4 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{2} a^{2} c -3 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{2} a^{2} c -a^{2} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}-3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{2} c^{2}\right )}{2 x \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c} \]
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 \[ \int \frac {{\left (a x - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x-1\right )}{x^2\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x - 1\right )}{x^{2} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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