Optimal. Leaf size=189 \[ \frac {c^4 (1-a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}-\frac {2 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {3 c^4 (1-a x)^8 \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-a^2 x^2}}-\frac {8 c^4 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 189, 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 = {6143, 6140, 43} \[ \frac {c^4 (1-a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}-\frac {2 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {3 c^4 (1-a x)^8 \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-a^2 x^2}}-\frac {8 c^4 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{9/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^6 (1+a x)^3 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int \left (8 (1-a x)^6-12 (1-a x)^7+6 (1-a x)^8-(1-a x)^9\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {8 c^4 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {3 c^4 (1-a x)^8 \sqrt {c-a^2 c x^2}}{2 a \sqrt {1-a^2 x^2}}-\frac {2 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {c^4 (1-a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 68, normalized size = 0.36 \[ \frac {c^4 (a x-1)^7 \left (21 a^3 x^3+77 a^2 x^2+98 a x+44\right ) \sqrt {c-a^2 c x^2}}{210 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 120, normalized size = 0.63 \[ -\frac {{\left (21 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} + 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} - 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} + 210 \, c^{4} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{210 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 97, normalized size = 0.51 \[ \frac {x \left (21 a^{9} x^{9}-70 x^{8} a^{8}+240 x^{6} a^{6}-210 x^{5} a^{5}-252 x^{4} a^{4}+420 x^{3} a^{3}-315 a x +210\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{210 \left (a x +1\right )^{6} \left (a x -1\right )^{6}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{9/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
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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