Optimal. Leaf size=278 \[ -\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{32 (a x+1) \left (c-a^2 c x^2\right )^{7/2}}-\frac {3 a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{12 (1-a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac {5 a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} \tanh ^{-1}(a x)}{32 \left (c-a^2 c x^2\right )^{7/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6192, 6193, 44, 207} \[ -\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{32 (a x+1) \left (c-a^2 c x^2\right )^{7/2}}-\frac {3 a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{12 (1-a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac {5 a^6 x^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} \tanh ^{-1}(a x)}{32 \left (c-a^2 c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 207
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7} \, dx}{\left (c-a^2 c x^2\right )^{7/2}}\\ &=\frac {\left (a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{(-1+a x)^5 (1+a x)^2} \, dx}{\left (c-a^2 c x^2\right )^{7/2}}\\ &=\frac {\left (a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \left (\frac {1}{4 (-1+a x)^5}-\frac {1}{4 (-1+a x)^4}+\frac {3}{16 (-1+a x)^3}-\frac {1}{8 (-1+a x)^2}-\frac {1}{32 (1+a x)^2}+\frac {5}{32 \left (-1+a^2 x^2\right )}\right ) \, dx}{\left (c-a^2 c x^2\right )^{7/2}}\\ &=-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{12 (1-a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac {3 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1+a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac {\left (5 a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{-1+a^2 x^2} \, dx}{32 \left (c-a^2 c x^2\right )^{7/2}}\\ &=-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^4 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{12 (1-a x)^3 \left (c-a^2 c x^2\right )^{7/2}}-\frac {3 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1-a x)^2 \left (c-a^2 c x^2\right )^{7/2}}-\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{8 (1-a x) \left (c-a^2 c x^2\right )^{7/2}}+\frac {a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7}{32 (1+a x) \left (c-a^2 c x^2\right )^{7/2}}-\frac {5 a^6 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} x^7 \tanh ^{-1}(a x)}{32 \left (c-a^2 c x^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 99, normalized size = 0.36 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (-15 a^4 x^4+45 a^3 x^3-35 a^2 x^2-15 a x+15 (a x-1)^4 (a x+1) \tanh ^{-1}(a x)+32\right )}{96 c^3 (a x-1)^4 (a x+1) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 190, normalized size = 0.68 \[ -\frac {15 \, {\left (a^{6} x^{5} - 3 \, a^{5} x^{4} + 2 \, a^{4} x^{3} + 2 \, a^{3} x^{2} - 3 \, a^{2} x + a\right )} \sqrt {-c} \log \left (\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c} \sqrt {-c} x + c}{a^{2} x^{2} - 1}\right ) - 2 \, {\left (15 \, a^{4} x^{4} - 45 \, a^{3} x^{3} + 35 \, a^{2} x^{2} + 15 \, a x - 32\right )} \sqrt {-a^{2} c}}{192 \, {\left (a^{7} c^{4} x^{5} - 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} + 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} \]
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 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 241, normalized size = 0.87 \[ \frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (15 \ln \left (a x -1\right ) x^{5} a^{5}-15 \ln \left (a x +1\right ) x^{5} a^{5}-45 \ln \left (a x -1\right ) x^{4} a^{4}+45 \ln \left (a x +1\right ) x^{4} a^{4}+30 x^{4} a^{4}+30 \ln \left (a x -1\right ) x^{3} a^{3}-30 a^{3} x^{3} \ln \left (a x +1\right )-90 x^{3} a^{3}+30 \ln \left (a x -1\right ) x^{2} a^{2}-30 \ln \left (a x +1\right ) x^{2} a^{2}+70 a^{2} x^{2}-45 \ln \left (a x -1\right ) x a +45 a x \ln \left (a x +1\right )+30 a x +15 \ln \left (a x -1\right )-15 \ln \left (a x +1\right )-64\right )}{192 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (a^{2} x^{2}-1\right ) c^{4} a} \]
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 \[ \int \frac {1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c-a^2\,c\,x^2\right )}^{7/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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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