Optimal. Leaf size=93 \[ -\frac {2 (1+a x)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}+\frac {(1+a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6327, 6328, 45}
\begin {gather*} \frac {(a x+1)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {2 (a x+1)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6327
Rule 6328
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=\frac {\left (c-a^2 c x^2\right )^{3/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3 \, dx}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (c-a^2 c x^2\right )^{3/2} \int (-1+a x) (1+a x)^2 \, dx}{a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}\\ &=\frac {\left (c-a^2 c x^2\right )^{3/2} \int \left (-2 (1+a x)^2+(1+a x)^3\right ) \, dx}{a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {2 (1+a x)^3 \left (c-a^2 c x^2\right )^{3/2}}{3 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}+\frac {(1+a x)^4 \left (c-a^2 c x^2\right )^{3/2}}{4 a^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 53, normalized size = 0.57 \begin {gather*} -\frac {c (1+a x)^3 (-5+3 a x) \sqrt {c-a^2 c x^2}}{12 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 63, normalized size = 0.68
method | result | size |
default | \(-\frac {\left (3 a^{3} x^{3}+4 a^{2} x^{2}-6 a x -12\right ) x c \sqrt {-c \left (a^{2} x^{2}-1\right )}}{12 \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(63\) |
gosper | \(\frac {x \left (3 a^{3} x^{3}+4 a^{2} x^{2}-6 a x -12\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{12 \left (a x -1\right ) \left (a x +1\right )^{2} \sqrt {\frac {a x -1}{a x +1}}}\) | \(68\) |
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 = 43, normalized size = 0.46 \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}}{12 \, a} \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}}}{\sqrt {\frac {a x - 1}{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 {\frac {a\,x-1}{a\,x+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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