Optimal. Leaf size=360 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (1-a x)^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x)^4 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]
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Rubi [A] time = 0.21, antiderivative size = 360, 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} \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 (1-a x)^2 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 (1-a x)^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 (1-a x)^4 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (a x+1)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6193
Rule 6197
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a^2 x^2}\right )^{7/2}} \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^7 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \frac {x^7}{(-1+a x)^5 (1+a x)^2} \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\left (a^7 \sqrt {1-\frac {1}{a^2 x^2}}\right ) \int \left (\frac {1}{a^7}+\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)}+\frac {1}{32 a^7 (1+a x)^2}-\frac {9}{64 a^7 (1+a x)}\right ) \, dx}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^4}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^3}-\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 140, normalized size = 0.39 \[ \frac {a^7 \left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (\frac {75}{16 a^8 (1-a x)}-\frac {1}{32 a^8 (a x+1)}-\frac {59}{32 a^8 (1-a x)^2}+\frac {1}{2 a^8 (1-a x)^3}-\frac {1}{16 a^8 (1-a x)^4}+\frac {201 \log (1-a x)}{64 a^8}-\frac {9 \log (a x+1)}{64 a^8}+\frac {x}{a^7}\right )}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 207, normalized size = 0.58 \[ \frac {{\left (64 \, a^{6} x^{6} - 192 \, a^{5} x^{5} - 174 \, a^{4} x^{4} + 618 \, a^{3} x^{3} - 118 \, a^{2} x^{2} - 414 \, a x - 9 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \log \left (a x + 1\right ) + 201 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \log \left (a x - 1\right ) + 208\right )} \sqrt {a^{2} c}}{64 \, {\left (a^{7} c^{4} x^{5} - 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} + 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} \]
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 \[ \int \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 247, normalized size = 0.69 \[ \frac {\left (a x -1\right ) \left (a x +1\right ) \left (64 x^{6} a^{6}+201 \ln \left (a x -1\right ) x^{5} a^{5}-9 \ln \left (a x +1\right ) x^{5} a^{5}-192 x^{5} a^{5}-603 \ln \left (a x -1\right ) x^{4} a^{4}+27 \ln \left (a x +1\right ) x^{4} a^{4}-174 x^{4} a^{4}+402 \ln \left (a x -1\right ) x^{3} a^{3}-18 a^{3} x^{3} \ln \left (a x +1\right )+618 x^{3} a^{3}+402 \ln \left (a x -1\right ) x^{2} a^{2}-18 \ln \left (a x +1\right ) x^{2} a^{2}-118 a^{2} x^{2}-603 \ln \left (a x -1\right ) x a +27 a x \ln \left (a x +1\right )-414 a x +201 \ln \left (a x -1\right )-9 \ln \left (a x +1\right )+208\right )}{64 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a^{8} x^{7} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {1}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
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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