Optimal. Leaf size=361 \[ -\frac {5 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {3 \left (1-a^2 x^2\right )^{7/2}}{2 a^8 x^7 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (1-a x)^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {11 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (a x+1)^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{24 a^8 x^7 (a x+1)^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {19 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{32 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {51 \left (1-a^2 x^2\right )^{7/2} \log (a x+1)}{32 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]
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Rubi [A] time = 0.24, antiderivative size = 361, 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 = {6160, 6150, 88} \[ -\frac {5 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {3 \left (1-a^2 x^2\right )^{7/2}}{2 a^8 x^7 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (1-a x)^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {11 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (a x+1)^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{24 a^8 x^7 (a x+1)^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {19 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{32 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {51 \left (1-a^2 x^2\right )^{7/2} \log (a x+1)}{32 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac {\left (1-a^2 x^2\right )^{7/2} \int \frac {e^{-\tanh ^{-1}(a x)} x^7}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {\left (1-a^2 x^2\right )^{7/2} \int \frac {x^7}{(1-a x)^3 (1+a x)^4} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac {\left (1-a^2 x^2\right )^{7/2} \int \left (-\frac {1}{a^7}-\frac {1}{16 a^7 (-1+a x)^3}-\frac {5}{16 a^7 (-1+a x)^2}-\frac {19}{32 a^7 (-1+a x)}-\frac {1}{8 a^7 (1+a x)^4}+\frac {11}{16 a^7 (1+a x)^3}-\frac {3}{2 a^7 (1+a x)^2}+\frac {51}{32 a^7 (1+a x)}\right ) \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}+\frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^2}-\frac {5 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{7/2}}{24 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)^3}-\frac {11 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)^2}+\frac {3 \left (1-a^2 x^2\right )^{7/2}}{2 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)}-\frac {19 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {51 \left (1-a^2 x^2\right )^{7/2} \log (1+a x)}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 145, normalized size = 0.40 \[ \frac {\sqrt {1-a^2 x^2} \left (96 a^6 x^6+96 a^5 x^5-366 a^4 x^4-222 a^3 x^3+338 a^2 x^2+122 a x+57 (a x-1)^2 (a x+1)^3 \log (1-a x)-153 (a x-1)^2 (a x+1)^3 \log (a x+1)-88\right )}{96 a^2 x (a x-1)^2 \sqrt {c-\frac {c}{a^2 x^2}} (a c x+c)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} a^{8} x^{8} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{9} c^{4} x^{9} + a^{8} c^{4} x^{8} - 4 \, a^{7} c^{4} x^{7} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{5} c^{4} x^{5} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{3} c^{4} x^{3} - 4 \, a^{2} c^{4} x^{2} + a c^{4} x + c^{4}}, x\right ) \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 239, normalized size = 0.66 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \left (a x -1\right ) \left (96 x^{6} a^{6}+57 \ln \left (a x -1\right ) x^{5} a^{5}-153 \ln \left (a x +1\right ) x^{5} a^{5}+96 x^{5} a^{5}+57 \ln \left (a x -1\right ) x^{4} a^{4}-153 \ln \left (a x +1\right ) x^{4} a^{4}-366 x^{4} a^{4}-114 \ln \left (a x -1\right ) x^{3} a^{3}+306 a^{3} x^{3} \ln \left (a x +1\right )-222 x^{3} a^{3}-114 \ln \left (a x -1\right ) x^{2} a^{2}+306 \ln \left (a x +1\right ) x^{2} a^{2}+338 a^{2} x^{2}+57 \ln \left (a x -1\right ) x a -153 a x \ln \left (a x +1\right )+122 a x +57 \ln \left (a x -1\right )-153 \ln \left (a x +1\right )-88\right )}{96 a^{8} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-a^2\,x^2}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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