Optimal. Leaf size=78 \[ -\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {3}{2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A] time = 0.35, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6167, 6152, 1807, 807, 266, 63, 208} \[ -\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {3}{2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1807
Rule 6152
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx\\ &=-\left (c \int \frac {(1-a x)^2}{x^3 \sqrt {c-a^2 c x^2}} \, dx\right )\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}+\frac {1}{2} \int \frac {4 a c-3 a^2 c x}{x^2 \sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {1}{2} \left (3 a^2 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{x}-\frac {1}{4} \left (3 a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{x}+\frac {3}{2} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 76, normalized size = 0.97 \[ \frac {1}{2} \left (\frac {(1-4 a x) \sqrt {c-a^2 c x^2}}{x^2}+3 a^2 \sqrt {c} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )-3 a^2 \sqrt {c} \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 149, normalized size = 1.91 \[ \left [\frac {3 \, a^{2} \sqrt {c} x^{2} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 1\right )}}{4 \, x^{2}}, \frac {3 \, a^{2} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 1\right )}}{2 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 200, normalized size = 2.56 \[ -\frac {3 \, a^{2} c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} + \frac {{\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{2} c + 4 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a \sqrt {-c} c {\left | a \right |} + {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{2} c^{2} - 4 \, a \sqrt {-c} c^{2} {\left | a \right |}}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 231, normalized size = 2.96 \[ \frac {3 \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right ) \sqrt {c}\, a^{2}}{2}-\frac {3 \sqrt {-a^{2} c \,x^{2}+c}\, a^{2}}{2}-\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{3} x \sqrt {-a^{2} c \,x^{2}+c}-\frac {2 a^{3} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}+\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}+2 a^{2} \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}+\frac {2 a^{3} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 1\right )}}{{\left (a x + 1\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-1\right )}{x^3\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x - 1\right )}{x^{3} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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