Optimal. Leaf size=55 \[ -\frac {1}{3 a x^3}-\frac {8 a^2 \left (\frac {1-a x}{1+a x}\right )^{3/2}}{3 \left (1-\frac {1-a x}{1+a x}\right )^3} \]
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Rubi [C] Result contains higher order function than in optimal. Order 3 vs. order 2 in
optimal.
time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.53, number of steps
used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6470, 30, 105,
12, 97} \begin {gather*} \frac {\sqrt {1-a x}}{6 a x^3 \sqrt {\frac {1}{a x+1}}}+\frac {1}{6 a x^3}-\frac {e^{\text {sech}^{-1}(a x)}}{2 x^2}+\frac {a \sqrt {1-a x}}{3 x \sqrt {\frac {1}{a x+1}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 12
Rule 30
Rule 97
Rule 105
Rule 6470
Rubi steps
\begin {align*} \int \frac {e^{\text {sech}^{-1}(a x)}}{x^3} \, dx &=-\frac {e^{\text {sech}^{-1}(a x)}}{2 x^2}-\frac {\int \frac {1}{x^4} \, dx}{2 a}-\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 a}\\ &=\frac {1}{6 a x^3}-\frac {e^{\text {sech}^{-1}(a x)}}{2 x^2}+\frac {\sqrt {1-a x}}{6 a x^3 \sqrt {\frac {1}{1+a x}}}+\frac {\left (\sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int -\frac {2 a^2}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{6 a}\\ &=\frac {1}{6 a x^3}-\frac {e^{\text {sech}^{-1}(a x)}}{2 x^2}+\frac {\sqrt {1-a x}}{6 a x^3 \sqrt {\frac {1}{1+a x}}}-\frac {1}{3} \left (a \sqrt {\frac {1}{1+a x}} \sqrt {1+a x}\right ) \int \frac {1}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx\\ &=\frac {1}{6 a x^3}-\frac {e^{\text {sech}^{-1}(a x)}}{2 x^2}+\frac {\sqrt {1-a x}}{6 a x^3 \sqrt {\frac {1}{1+a x}}}+\frac {a \sqrt {1-a x}}{3 x \sqrt {\frac {1}{1+a x}}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 43, normalized size = 0.78 \begin {gather*} \frac {-1+(-1+a x) \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 53, normalized size = 0.96
method | result | size |
default | \(\frac {\sqrt {\frac {a x +1}{a x}}\, \sqrt {-\frac {a x -1}{a x}}\, \left (a^{2} x^{2}-1\right )}{3 x^{2}}-\frac {1}{3 a \,x^{3}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 43, normalized size = 0.78 \begin {gather*} \frac {{\left (a^{2} x^{3} - x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{3 \, a x^{4}} - \frac {1}{3 \, a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 52, normalized size = 0.95 \begin {gather*} \frac {{\left (a^{3} x^{3} - a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1}{3 \, a x^{3}} \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*} \frac {\int \frac {1}{x^{4}}\, dx + \int \frac {a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{3}}\, dx}{a} \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 = 1.47, size = 58, normalized size = 1.05 \begin {gather*} -\frac {1}{3\,a\,x^3}-\frac {\left (\frac {\sqrt {\frac {1}{a\,x}+1}}{3}-\frac {a^2\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{3}\right )\,\sqrt {\frac {1}{a\,x}-1}}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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