Optimal. Leaf size=98 \[ \frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 98, 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 = {6142, 653, 192, 191} \[ \frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 653
Rule 6142
Rubi steps
\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{9/2}} \, dx\\ &=-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {5}{7} \int \frac {1}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\\ &=-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{7 c}\\ &=-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{21 c^2}\\ &=-\frac {2 (1-a x)}{7 a \left (c-a^2 c x^2\right )^{7/2}}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 96, normalized size = 0.98 \[ -\frac {\sqrt {1-a^2 x^2} \left (8 a^5 x^5+16 a^4 x^4-4 a^3 x^3-24 a^2 x^2-9 a x+6\right )}{21 a c^3 (1-a x)^{3/2} (a x+1)^{7/2} \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 124, normalized size = 1.27 \[ -\frac {{\left (8 \, a^{5} x^{5} + 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 24 \, a^{2} x^{2} - 9 \, a x + 6\right )} \sqrt {-a^{2} c x^{2} + c}}{21 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 300, normalized size = 3.06 \[ \frac {a^{5} {\left (\frac {14 \, {\left (7 \, c - \frac {15 \, c}{a x + 1}\right )}}{a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )} c^{3} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)} + \frac {3 \, a^{30} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c^{42} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\relax (a)^{6} - 21 \, a^{30} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{43} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\relax (a)^{6} - 210 \, a^{30} c^{45} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\relax (a)^{6} - 70 \, a^{30} c^{44} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\relax (a)^{6}}{a^{35} c^{49} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{7} \mathrm {sgn}\relax (a)^{7}}\right )} - \frac {256 \, \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)}{\sqrt {-c} c^{3}}}{672 \, {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 64, normalized size = 0.65 \[ -\frac {\left (a x -1\right )^{2} \left (8 x^{5} a^{5}+16 x^{4} a^{4}-4 x^{3} a^{3}-24 a^{2} x^{2}-9 a x +6\right )}{21 \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 98, normalized size = 1.00 \[ -\frac {2}{7 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a c\right )}} + \frac {8 \, x}{21 \, \sqrt {-a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{21 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {x}{7 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 133, normalized size = 1.36 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {11\,x}{42\,c^4}-\frac {5}{28\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {\sqrt {c-a^2\,c\,x^2}}{28\,a\,c^4\,{\left (a\,x+1\right )}^4}-\frac {\sqrt {c-a^2\,c\,x^2}}{14\,a\,c^4\,{\left (a\,x+1\right )}^3}-\frac {8\,x\,\sqrt {c-a^2\,c\,x^2}}{21\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \]
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 \[ - \int \frac {a x}{- a^{7} c^{3} x^{7} \sqrt {- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt {- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt {- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt {- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt {- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{3} x \sqrt {- a^{2} c x^{2} + c} + c^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{- a^{7} c^{3} x^{7} \sqrt {- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt {- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt {- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt {- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt {- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{3} x \sqrt {- a^{2} c x^{2} + c} + c^{3} \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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