Optimal. Leaf size=142 \[ -\frac {c^3 (1-a x)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {4 c^3 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {2 c^3 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 142, 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} \[ -\frac {c^3 (1-a x)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}+\frac {4 c^3 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {2 c^3 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}} \]
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 )^{7/2} \, dx &=\frac {\left (c^3 \sqrt {c-a^2 c x^2}\right ) \int e^{-3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{7/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^3 \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^5 (1+a x)^2 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^3 \sqrt {c-a^2 c x^2}\right ) \int \left (4 (1-a x)^5-4 (1-a x)^6+(1-a x)^7\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {2 c^3 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {4 c^3 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {c^3 (1-a x)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 0.42 \[ -\frac {c^3 (a x-1)^6 \left (21 a^2 x^2+54 a x+37\right ) \sqrt {c-a^2 c x^2}}{168 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 120, normalized size = 0.85 \[ \frac {{\left (21 \, a^{7} c^{3} x^{8} - 72 \, a^{6} c^{3} x^{7} + 28 \, a^{5} c^{3} x^{6} + 168 \, a^{4} c^{3} x^{5} - 210 \, a^{3} c^{3} x^{4} - 56 \, a^{2} c^{3} x^{3} + 252 \, a c^{3} x^{2} - 168 \, c^{3} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{168 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 97, normalized size = 0.68 \[ \frac {x \left (21 a^{7} x^{7}-72 x^{6} a^{6}+28 x^{5} a^{5}+168 x^{4} a^{4}-210 x^{3} a^{3}-56 a^{2} x^{2}+252 a x -168\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{168 \left (a x +1\right )^{5} \left (a x -1\right )^{5}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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