Optimal. Leaf size=234 \[ -\frac {c^4 (1-a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{9 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (1-a x)^8 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}+\frac {32 c^4 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {8 c^4 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 234, 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} \[ -\frac {c^4 (1-a x)^{10} \sqrt {c-a^2 c x^2}}{10 a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{9 a \sqrt {1-a^2 x^2}}-\frac {3 c^4 (1-a x)^8 \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}+\frac {32 c^4 (1-a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}-\frac {8 c^4 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6143
Rubi steps
\begin {align*} \int e^{-\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^{-\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)^5 (1+a x)^4 \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c^4 \sqrt {c-a^2 c x^2}\right ) \int \left (16 (1-a x)^5-32 (1-a x)^6+24 (1-a x)^7-8 (1-a x)^8+(1-a x)^9\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {8 c^4 (1-a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {32 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}}{a \sqrt {1-a^2 x^2}}+\frac {8 c^4 (1-a x)^9 \sqrt {c-a^2 c x^2}}{9 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 = 0.08, size = 76, normalized size = 0.32 \[ -\frac {c^4 (a x-1)^6 \left (63 a^4 x^4+308 a^3 x^3+588 a^2 x^2+528 a x+193\right ) \sqrt {c-a^2 c x^2}}{630 a \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 142, normalized size = 0.61 \[ \frac {{\left (63 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} + 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} - 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} + 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} - 630 \, c^{4} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{630 \, {\left (a^{2} x^{2} - 1\right )}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} \sqrt {-a^{2} x^{2} + 1}}{a x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 113, normalized size = 0.48 \[ \frac {x \left (63 a^{9} x^{9}-70 x^{8} a^{8}-315 a^{7} x^{7}+360 x^{6} a^{6}+630 x^{5} a^{5}-756 x^{4} a^{4}-630 x^{3} a^{3}+840 a^{2} x^{2}+315 a x -630\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}} \sqrt {-a^{2} x^{2}+1}}{630 \left (a x +1\right )^{5} \left (a x -1\right )^{5}} \]
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 \[ \int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} \sqrt {-a^{2} x^{2} + 1}}{a x + 1}\,{d x} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{9/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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