Optimal. Leaf size=91 \[ -\frac {2 c (1-a x)^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {c (1-a x)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 91, 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 = {6278, 6275, 45}
\begin {gather*} \frac {c (1-a x)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}-\frac {2 c (1-a x)^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{3/2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int (1-a x)^2 (1+a x) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int \left (2 (1-a x)^2-(1-a x)^3\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {2 c (1-a x)^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}+\frac {c (1-a x)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.63 \begin {gather*} \frac {c x \sqrt {c-a^2 c x^2} \left (12-6 a x-4 a^2 x^2+3 a^3 x^3\right )}{12 \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.06, size = 64, normalized size = 0.70
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, c x \left (3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +12\right )}{12 \left (a^{2} x^{2}-1\right )}\) | \(64\) |
gosper | \(\frac {x \left (3 a^{3} x^{3}-4 a^{2} x^{2}-6 a x +12\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{12 \left (a x -1\right )^{2} \left (a x +1\right )^{2}}\) | \(65\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 68, normalized size = 0.75 \begin {gather*} -\frac {{\left (3 \, a^{3} c x^{4} - 4 \, a^{2} c x^{3} - 6 \, a c x^{2} + 12 \, c x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{12 \, {\left (a^{2} x^{2} - 1\right )}} \end {gather*}
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 \begin {gather*} \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 {3}{2}}}{a x + 1}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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