Optimal. Leaf size=30 \[ -\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x \]
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Rubi [A] time = 0.13, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6167, 6131, 6129, 73, 270} \[ -\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x \]
Antiderivative was successfully verified.
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Rule 73
Rule 270
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx\\ &=\frac {c^4 \int \frac {e^{4 \tanh ^{-1}(a x)} (1-a x)^4}{x^4} \, dx}{a^4}\\ &=\frac {c^4 \int \frac {(1-a x)^2 (1+a x)^2}{x^4} \, dx}{a^4}\\ &=\frac {c^4 \int \frac {\left (1-a^2 x^2\right )^2}{x^4} \, dx}{a^4}\\ &=\frac {c^4 \int \left (a^4+\frac {1}{x^4}-\frac {2 a^2}{x^2}\right ) \, dx}{a^4}\\ &=-\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x\\ \end {align*}
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Mathematica [A] time = 0.20, size = 30, normalized size = 1.00 \[ -\frac {c^4}{3 a^4 x^3}+\frac {2 c^4}{a^2 x}+c^4 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 36, normalized size = 1.20 \[ \frac {3 \, a^{4} c^{4} x^{4} + 6 \, a^{2} c^{4} x^{2} - c^{4}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 59, normalized size = 1.97 \[ \frac {{\left (a x - 1\right )} c^{4}}{a} - \frac {5 \, c^{4} + \frac {9 \, c^{4}}{a x - 1} + \frac {3 \, c^{4}}{{\left (a x - 1\right )}^{2}}}{3 \, a {\left (\frac {1}{a x - 1} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 27, normalized size = 0.90 \[ \frac {c^{4} \left (x \,a^{4}+\frac {2 a^{2}}{x}-\frac {1}{3 x^{3}}\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 31, normalized size = 1.03 \[ c^{4} x + \frac {6 \, a^{2} c^{4} x^{2} - c^{4}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 27, normalized size = 0.90 \[ \frac {c^4\,\left (a^4\,x^4+2\,a^2\,x^2-\frac {1}{3}\right )}{a^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 31, normalized size = 1.03 \[ \frac {a^{4} c^{4} x + \frac {6 a^{2} c^{4} x^{2} - c^{4}}{3 x^{3}}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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