Optimal. Leaf size=148 \[ -\frac {3 a \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{2 x \sqrt {1-a^2 x^2}}+\frac {4 a^2 x \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {4 a^2 x \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.25, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6160, 6150, 88} \[ -\frac {3 a \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{2 x \sqrt {1-a^2 x^2}}+\frac {4 a^2 x \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {4 a^2 x \sqrt {c-\frac {c}{a^2 x^2}} \log (1-a x)}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int \frac {e^{3 \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^{3 \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2}}{x^3} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1+a x)^2}{x^3 (1-a x)} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \left (\frac {1}{x^3}+\frac {3 a}{x^2}+\frac {4 a^2}{x}-\frac {4 a^3}{-1+a x}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {3 a \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{2 x \sqrt {1-a^2 x^2}}+\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x \log (x)}{\sqrt {1-a^2 x^2}}-\frac {4 a^2 \sqrt {c-\frac {c}{a^2 x^2}} x \log (1-a x)}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 64, normalized size = 0.43 \[ \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (4 a^2 \log (x)-4 a^2 \log (1-a x)-\frac {3 a}{x}-\frac {1}{2 x^2}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 492, normalized size = 3.32 \[ \left [\frac {4 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {-c} \log \left (-\frac {4 \, a^{5} c x^{5} - {\left (2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right )} c x^{6} - {\left (4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x - {\left (4 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - {\left (4 \, a^{4} - 6 \, a^{3} + 4 \, a^{2} - a\right )} x^{5} + 4 \, a^{2} x^{2} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left ({\left (6 \, a + 1\right )} x^{2} - 6 \, a x - 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{2} x^{3} - x\right )}}, \frac {8 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {c} \arctan \left (\frac {{\left (2 \, a^{2} x^{2} - {\left (2 \, a^{3} - 2 \, a^{2} + a\right )} x^{3} - a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} - a^{2}\right )} c x^{4} - {\left (a^{2} - 2 \, a + 1\right )} c x^{2} - 2 \, a c x + c}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left ({\left (6 \, a + 1\right )} x^{2} - 6 \, a x - 1\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, {\left (a^{2} x^{3} - x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 0.53 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {-a^{2} x^{2}+1}\, \left (8 a^{2} \ln \relax (x ) x^{2}-8 \ln \left (a x -1\right ) x^{2} a^{2}-6 a x -1\right )}{2 x \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.45, size = 159, normalized size = 1.07 \[ -\frac {1}{2} \, a^{3} {\left (\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a^{2}} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a^{2}}\right )} - \frac {3}{2} \, a^{2} {\left (-\frac {i \, \sqrt {c} \log \left (a x + 1\right )}{a} - \frac {i \, \sqrt {c} \log \left (a x - 1\right )}{a} + \frac {2 i \, \sqrt {c} \log \relax (x)}{a}\right )} + \frac {1}{2} i \, a \sqrt {c} \log \left (a x + 1\right ) + \frac {1}{2} i \, a \sqrt {c} \log \left (a x - 1\right ) - i \, a \sqrt {c} \log \relax (x) - \frac {3}{2} \, {\left (i \, \sqrt {c} \log \left (a x + 1\right ) - i \, \sqrt {c} \log \left (a x - 1\right ) - \frac {2 i \, \sqrt {c}}{a x}\right )} a + \frac {i \, \sqrt {c}}{2 \, a x^{2}} \]
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 \[ \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (a\,x+1\right )}^3}{x^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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