Optimal. Leaf size=121 \[ -\frac {16 x}{45 c^4 \sqrt {c-a^2 c x^2}}-\frac {8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}+\frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6167, 6142, 653, 192, 191} \[ -\frac {16 x}{45 c^4 \sqrt {c-a^2 c x^2}}-\frac {8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}+\frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 653
Rule 6142
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\\ &=-\left (c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{11/2}} \, dx\right )\\ &=\frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {7}{9} \int \frac {1}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\\ &=\frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx}{3 c}\\ &=\frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{15 c^2}\\ &=\frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {16 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{45 c^3}\\ &=\frac {2 (1-a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {16 x}{45 c^4 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 112, normalized size = 0.93 \[ \frac {\sqrt {1-a^2 x^2} \left (-16 a^7 x^7-32 a^6 x^6+24 a^5 x^5+80 a^4 x^4+10 a^3 x^3-60 a^2 x^2-25 a x+10\right )}{45 a c^4 (1-a x)^{5/2} (a x+1)^{9/2} \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.32, size = 152, normalized size = 1.26 \[ \frac {{\left (16 \, a^{7} x^{7} + 32 \, a^{6} x^{6} - 24 \, a^{5} x^{5} - 80 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 60 \, a^{2} x^{2} + 25 \, a x - 10\right )} \sqrt {-a^{2} c x^{2} + c}}{45 \, {\left (a^{9} c^{5} x^{8} + 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} - 2 \, a^{2} c^{5} x - a c^{5}\right )}} \]
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 \[ \int \frac {a x - 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 80, normalized size = 0.66 \[ -\frac {\left (a x -1\right )^{2} \left (16 a^{7} x^{7}+32 x^{6} a^{6}-24 x^{5} a^{5}-80 x^{4} a^{4}-10 x^{3} a^{3}+60 a^{2} x^{2}+25 a x -10\right )}{45 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 117, normalized size = 0.97 \[ \frac {2}{9 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} a^{2} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} a c\right )}} - \frac {16 \, x}{45 \, \sqrt {-a^{2} c x^{2} + c} c^{4}} - \frac {8 \, x}{45 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{3}} - \frac {2 \, x}{15 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c^{2}} - \frac {x}{9 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 177, normalized size = 1.46 \[ \frac {5\,\sqrt {c-a^2\,c\,x^2}}{144\,a\,c^5\,{\left (a\,x+1\right )}^4}+\frac {\sqrt {c-a^2\,c\,x^2}}{72\,a\,c^5\,{\left (a\,x+1\right )}^5}+\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {31\,x}{120\,c^5}-\frac {5}{24\,a\,c^5}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {8\,x}{45\,c^5}+\frac {5}{144\,a\,c^5}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}+\frac {16\,x\,\sqrt {c-a^2\,c\,x^2}}{45\,c^5\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \]
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 \[ \int \frac {a x - 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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