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 = 0.04, 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 = {6128, 264} \[ -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 264
Rule 6128
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 \[ -\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 29, normalized size = 1.32 \[ \frac {{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 124, normalized size = 5.64 \[ \frac {{\left (a^{4} c - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c}{x^{2}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left | a \right |}} + \frac {\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c}{x} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 1.50 \[ -\frac {\left (a x -1\right )^{2} \left (a x +1\right )^{2} c}{3 x^{3} \sqrt {-a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 40, normalized size = 1.82 \[ \frac {\sqrt {-a^{2} x^{2} + 1} a^{2} c}{3 \, x} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{3 \, x^{3}} \]
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 \[ -\frac {c\,{\left (1-a^2\,x^2\right )}^{3/2}}{3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.23, size = 90, normalized size = 4.09 \[ c \left (\begin {cases} \frac {a^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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