Optimal. Leaf size=44 \[ \frac {c (1+a x)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6278, 6275, 32}
\begin {gather*} \frac {c (a x+1)^4 \sqrt {c-a^2 c x^2}}{4 a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 6275
Rule 6278
Rubi steps
\begin {align*} \int e^{3 \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^{3 \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)^3 \, dx}{\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 = 66, normalized size = 1.50 \begin {gather*} -\frac {c x \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2} \left (4+6 a x+4 a^2 x^2+a^3 x^3\right )}{-4+4 a^2 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.07, size = 47, normalized size = 1.07
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right )^{3} c}{4 \left (a x -1\right ) a}\) | \(47\) |
gosper | \(\frac {x \left (a^{3} x^{3}+4 a^{2} x^{2}+6 a x +4\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 67, normalized size = 1.52 \begin {gather*} -\frac {{\left (a^{3} c x^{4} + 4 \, a^{2} c x^{3} + 6 \, a c x^{2} + 4 \, c x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{4 \, {\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 {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, 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 [B]
time = 1.05, size = 54, normalized size = 1.23 \begin {gather*} \frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {c\,a^3\,x^4}{4}+c\,a^2\,x^3+\frac {3\,c\,a\,x^2}{2}+c\,x\right )}{\sqrt {1-a^2\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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