Optimal. Leaf size=359 \[ \frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (a x+1)^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {\left (1-a^2 x^2\right )^{7/2}}{2 a^8 x^7 (a x+1)^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (a x+1)^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {9 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {201 \left (1-a^2 x^2\right )^{7/2} \log (a x+1)}{64 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]
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Rubi [A] time = 0.25, antiderivative size = 359, 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} \[ \frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (1-a x) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 x^7 (a x+1)^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {\left (1-a^2 x^2\right )^{7/2}}{2 a^8 x^7 (a x+1)^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{16 a^8 x^7 (a x+1)^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}+\frac {9 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}-\frac {201 \left (1-a^2 x^2\right )^{7/2} \log (a x+1)}{64 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int \frac {e^{-3 \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^{-3 \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)^2 (1+a x)^5} \, 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}{32 a^7 (-1+a x)^2}+\frac {9}{64 a^7 (-1+a x)}-\frac {1}{4 a^7 (1+a x)^5}+\frac {3}{2 a^7 (1+a x)^4}-\frac {59}{16 a^7 (1+a x)^3}+\frac {75}{16 a^7 (1+a x)^2}-\frac {201}{64 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)}+\frac {\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)^4}-\frac {\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)^3}+\frac {59 \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 {75 \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 {9 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}-\frac {201 \left (1-a^2 x^2\right )^{7/2} \log (1+a x)}{64 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 = 146, normalized size = 0.41 \[ \frac {\sqrt {1-a^2 x^2} \left (-2 \left (32 a^6 x^6+96 a^5 x^5-87 a^4 x^4-309 a^3 x^3-59 a^2 x^2+207 a x+104\right )-9 (a x-1) (a x+1)^4 \log (1-a x)+201 (a x-1) (a x+1)^4 \log (a x+1)\right )}{64 a^2 c^3 x (a x-1) (a x+1)^4 \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} a^{8} x^{8} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{9} c^{4} x^{9} + 3 \, a^{8} c^{4} x^{8} - 8 \, a^{6} c^{4} x^{6} - 6 \, a^{5} c^{4} x^{5} + 6 \, a^{4} c^{4} x^{4} + 8 \, a^{3} c^{4} x^{3} - 3 \, a c^{4} x - c^{4}}, x\right ) \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 248, normalized size = 0.69 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a x -1\right )^{2} \left (64 x^{6} a^{6}+9 \ln \left (a x -1\right ) x^{5} a^{5}-201 \ln \left (a x +1\right ) x^{5} a^{5}+192 x^{5} a^{5}+27 \ln \left (a x -1\right ) x^{4} a^{4}-603 \ln \left (a x +1\right ) x^{4} a^{4}-174 x^{4} a^{4}+18 \ln \left (a x -1\right ) x^{3} a^{3}-402 a^{3} x^{3} \ln \left (a x +1\right )-618 x^{3} a^{3}-18 \ln \left (a x -1\right ) x^{2} a^{2}+402 \ln \left (a x +1\right ) x^{2} a^{2}-118 a^{2} x^{2}-27 \ln \left (a x -1\right ) x a +603 a x \ln \left (a x +1\right )+414 a x -9 \ln \left (a x -1\right )+201 \ln \left (a x +1\right )+208\right )}{64 \left (a x +1\right ) a^{8} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (a\,x+1\right )}^3} \,d x \]
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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