Optimal. Leaf size=112 \[ \frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}-\frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6128, 879, 873, 875, 208} \[ \frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}-\frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
Antiderivative was successfully verified.
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Rule 208
Rule 873
Rule 875
Rule 879
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx &=\frac {\int \frac {(c-a c x)^{3/2}}{x^3 \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}-\frac {1}{4} (7 a) \int \frac {\sqrt {c-a c x}}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}+\frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}+\frac {1}{8} \left (7 a^2\right ) \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}+\frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}+\frac {1}{4} \left (7 a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ &=-\frac {c \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a c x}}+\frac {7 a c \sqrt {1-a^2 x^2}}{4 x \sqrt {c-a c x}}-\frac {7}{4} a^2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 64, normalized size = 0.57 \[ -\frac {c \sqrt {1-a x} \left (7 a^2 x^2 \tanh ^{-1}\left (\sqrt {a x+1}\right )+(2-7 a x) \sqrt {a x+1}\right )}{4 x^2 \sqrt {c-a c x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 232, normalized size = 2.07 \[ \left [\frac {7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (7 \, a x - 2\right )}}{8 \, {\left (a x^{3} - x^{2}\right )}}, -\frac {7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (7 \, a x - 2\right )}}{4 \, {\left (a x^{3} - x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 117, normalized size = 1.04 \[ \frac {1}{4} \, a^{2} c {\left (\frac {7 \, \arctan \left (\frac {\sqrt {a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {7 \, {\left (a c x + c\right )}^{\frac {3}{2}} - 9 \, \sqrt {a c x + c} c}{a^{2} c^{3} x^{2}}\right )} {\left | c \right |} - \frac {7 \, a^{2} c {\left | c \right |} \arctan \left (\frac {\sqrt {2} \sqrt {c}}{\sqrt {-c}}\right ) + 5 \, \sqrt {2} a^{2} \sqrt {-c} \sqrt {c} {\left | c \right |}}{4 \, \sqrt {-c} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 101, normalized size = 0.90 \[ \frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (7 c \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) x^{2} a^{2}-7 x a \sqrt {c \left (a x +1\right )}\, \sqrt {c}+2 \sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{4 \sqrt {c}\, \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, x^{2}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{{\left (a x + 1\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{x^3\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{3} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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