Optimal. Leaf size=220 \[ -\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{x^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3 \sqrt {1-a^2 x^2}}+\frac {4 a^4 x \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {4 a^4 x \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{\sqrt {1-a^2 x^2}}+\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.26, antiderivative size = 220, 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} \[ \frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{x^2 \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3 \sqrt {1-a^2 x^2}}+\frac {4 a^4 x \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {4 a^4 x \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{\sqrt {1-a^2 x^2}} \]
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^4} \, 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^5} \, 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^5 (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^5}-\frac {3 a}{x^4}+\frac {4 a^2}{x^3}-\frac {4 a^3}{x^2}+\frac {4 a^4}{x}-\frac {4 a^5}{1+a x}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{4 x^3 \sqrt {1-a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{x^2 \sqrt {1-a^2 x^2}}-\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-a^2 x^2}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} x \log (x)}{\sqrt {1-a^2 x^2}}-\frac {4 a^4 \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 = 0.05, size = 78, normalized size = 0.35 \[ \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (4 a^4 \log (x)-4 a^4 \log (a x+1)+\frac {4 a^3}{x}-\frac {2 a^2}{x^2}+\frac {a}{x^3}-\frac {1}{4 x^4}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 551, normalized size = 2.50 \[ \left [\frac {8 \, {\left (a^{5} x^{5} - a^{3} x^{3}\right )} \sqrt {-c} \log \left (\frac {4 \, a^{5} c x^{5} + {\left (2 \, a^{6} + 4 \, a^{5} + 6 \, a^{4} + 4 \, a^{3} + a^{2}\right )} c x^{6} + {\left (4 \, a^{4} - 4 \, a^{3} - 6 \, a^{2} - 4 \, a - 1\right )} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, a c x - {\left (4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - {\left (4 \, a^{4} + 6 \, a^{3} + 4 \, a^{2} + a\right )} x^{5} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{4} x^{6} + 2 \, a^{3} x^{5} - 2 \, a x^{3} - x^{2}}\right ) - {\left (16 \, a^{3} x^{3} - {\left (16 \, a^{3} - 8 \, a^{2} + 4 \, a - 1\right )} x^{4} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left (a^{2} x^{5} - x^{3}\right )}}, -\frac {16 \, {\left (a^{5} x^{5} - a^{3} x^{3}\right )} \sqrt {c} \arctan \left (-\frac {{\left (2 \, a^{2} x^{2} + {\left (2 \, a^{3} + 2 \, a^{2} + a\right )} x^{3} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} + a^{2}\right )} c x^{4} + {\left (a^{2} + 2 \, a + 1\right )} c x^{2} - 2 \, a c x - c}\right ) + {\left (16 \, a^{3} x^{3} - {\left (16 \, a^{3} - 8 \, a^{2} + 4 \, a - 1\right )} x^{4} - 8 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left (a^{2} x^{5} - x^{3}\right )}}\right ] \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{3} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 94, normalized size = 0.43 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \sqrt {-a^{2} x^{2}+1}\, \left (16 a^{4} \ln \relax (x ) x^{4}-16 \ln \left (a x +1\right ) x^{4} a^{4}+16 x^{3} a^{3}-8 a^{2} x^{2}+4 a x -1\right )}{4 x^{3} \left (a^{2} x^{2}-1\right )} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{3} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^4\,{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}{x^{4} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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