Optimal. Leaf size=22 \[ -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6263, 270}
\begin {gather*} -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} (c-a c x)}{x^4} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x^4} \, dx\\ &=-\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs.
\(2(18)=36\).
time = 0.47, size = 42, normalized size = 1.91
method | result | size |
trager | \(\frac {c \left (a^{2} x^{2}-1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}\) | \(28\) |
gosper | \(-\frac {c \left (a x +1\right )^{2} \left (a x -1\right )^{2}}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}\) | \(33\) |
risch | \(-\frac {c \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right )}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}\) | \(36\) |
default | \(-c \left (\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}-\frac {a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}\right )\) | \(42\) |
meijerg | \(\frac {a^{2} c \sqrt {-a^{2} x^{2}+1}}{x}-\frac {c \left (2 a^{2} x^{2}+1\right ) \sqrt {-a^{2} x^{2}+1}}{3 x^{3}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs.
\(2 (18) = 36\).
time = 0.47, size = 40, normalized size = 1.82 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} c}{3 \, x} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 29, normalized size = 1.32 \begin {gather*} \frac {{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.85, size = 133, normalized size = 6.05 \begin {gather*} - a^{2} c \left (\begin {cases} - \frac {i \sqrt {a^{2} x^{2} - 1}}{x} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {\sqrt {- a^{2} x^{2} + 1}}{x} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {2 i a^{2} \sqrt {a^{2} x^{2} - 1}}{3 x} - \frac {i \sqrt {a^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {2 a^{2} \sqrt {- a^{2} x^{2} + 1}}{3 x} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 124 vs.
\(2 (18) = 36\).
time = 0.42, size = 124, normalized size = 5.64 \begin {gather*} \frac {{\left (a^{4} c - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c}{x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} + \frac {\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 18, normalized size = 0.82 \begin {gather*} -\frac {c\,{\left (1-a^2\,x^2\right )}^{3/2}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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