Optimal. Leaf size=321 \[ \frac {c^3 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^7 x^6 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 x^5 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.15, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6197, 6193, 88} \[ \frac {c^3 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 x^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^7 x^6 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx &=\frac {\left (c^3 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{7/2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c^3 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \frac {(-1+a x)^3 (1+a x)^4}{x^7} \, dx}{a^7 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c^3 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \left (a^7-\frac {1}{x^7}-\frac {a}{x^6}+\frac {3 a^2}{x^5}+\frac {3 a^3}{x^4}-\frac {3 a^4}{x^3}-\frac {3 a^5}{x^2}+\frac {a^6}{x}\right ) \, dx}{a^7 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{6 a^7 \sqrt {1-\frac {1}{a^2 x^2}} x^6}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{5 a^6 \sqrt {1-\frac {1}{a^2 x^2}} x^5}-\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {3 c^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^3 \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 = 0.08, size = 96, normalized size = 0.30 \[ \frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} \left (a^7 x+a^6 \log (x)+\frac {3 a^5}{x}+\frac {3 a^4}{2 x^2}-\frac {a^3}{x^3}-\frac {3 a^2}{4 x^4}+\frac {a}{5 x^5}+\frac {1}{6 x^6}\right )}{a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 96, normalized size = 0.30 \[ \frac {{\left (60 \, a^{7} c^{3} x^{7} + 60 \, a^{6} c^{3} x^{6} \log \relax (x) + 180 \, a^{5} c^{3} x^{5} + 90 \, a^{4} c^{3} x^{4} - 60 \, a^{3} c^{3} x^{3} - 45 \, a^{2} c^{3} x^{2} + 12 \, a c^{3} x + 10 \, c^{3}\right )} \sqrt {a^{2} c}}{60 \, a^{8} x^{6}} \]
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 \[ \int \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 112, normalized size = 0.35 \[ \frac {\left (60 a^{7} x^{7}+60 a^{6} \ln \relax (x ) x^{6}+180 x^{5} a^{5}+90 x^{4} a^{4}-60 x^{3} a^{3}-45 a^{2} x^{2}+12 a x +10\right ) \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} x}{60 \left (a x +1\right ) \left (a^{2} x^{2}-1\right )^{3} \sqrt {\frac {a x -1}{a x +1}}} \]
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 \[ \int \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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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