Optimal. Leaf size=322 \[ \frac {c^4 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^9 x^8 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{7 a^8 x^7 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {8 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 x^5 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.17, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6197, 6193, 88} \[ \frac {c^4 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {8 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 x^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{7 a^8 x^7 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^9 x^8 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} \, dx &=\frac {\left (c^4 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{9/2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c^4 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \frac {(-1+a x)^3 (1+a x)^6}{x^9} \, dx}{a^9 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c^4 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \left (a^9-\frac {1}{x^9}-\frac {3 a}{x^8}+\frac {8 a^3}{x^6}+\frac {6 a^4}{x^5}-\frac {6 a^5}{x^4}-\frac {8 a^6}{x^3}+\frac {3 a^8}{x}\right ) \, dx}{a^9 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{8 a^9 \sqrt {1-\frac {1}{a^2 x^2}} x^8}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{7 a^8 \sqrt {1-\frac {1}{a^2 x^2}} x^7}-\frac {8 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 \sqrt {1-\frac {1}{a^2 x^2}} x^5}-\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {2 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {4 c^4 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {c^4 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 97, normalized size = 0.30 \[ \frac {\left (c-\frac {c}{a^2 x^2}\right )^{9/2} \left (a^9 x+3 a^8 \log (x)+\frac {4 a^6}{x^2}+\frac {2 a^5}{x^3}-\frac {3 a^4}{2 x^4}-\frac {8 a^3}{5 x^5}+\frac {3 a}{7 x^7}+\frac {1}{8 x^8}\right )}{a^9 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 96, normalized size = 0.30 \[ \frac {{\left (280 \, a^{9} c^{4} x^{9} + 840 \, a^{8} c^{4} x^{8} \log \relax (x) + 1120 \, a^{6} c^{4} x^{6} + 560 \, a^{5} c^{4} x^{5} - 420 \, a^{4} c^{4} x^{4} - 448 \, a^{3} c^{4} x^{3} + 120 \, a c^{4} x + 35 \, c^{4}\right )} \sqrt {a^{2} c}}{280 \, a^{10} x^{8}} \]
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 \[ \int \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 112, normalized size = 0.35 \[ \frac {\left (280 a^{9} x^{9}+840 a^{8} \ln \relax (x ) x^{8}+1120 x^{6} a^{6}+560 x^{5} a^{5}-420 x^{4} a^{4}-448 x^{3} a^{3}+120 a x +35\right ) \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {9}{2}} x}{280 \left (a x +1\right )^{3} \left (a^{2} x^{2}-1\right )^{3} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{9/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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