Optimal. Leaf size=183 \[ -\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {a^3}{4 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {3 a^3}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {7 a^3}{3 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}+\frac {2 a^3}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}-\frac {4 a^3}{5 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5} \]
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Rubi [A] time = 0.50, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6337, 1612} \[ -\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )}-\frac {a^3}{4 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )}+\frac {3 a^3}{2 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^2}-\frac {7 a^3}{3 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^3}+\frac {2 a^3}{\left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4}-\frac {4 a^3}{5 \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^5} \]
Antiderivative was successfully verified.
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Rule 1612
Rule 6337
Rubi steps
\begin {align*} \int \frac {e^{2 \text {sech}^{-1}(a x)}}{x^4} \, dx &=\int \frac {\left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )^2}{x^4} \, dx\\ &=-\left ((4 a) \operatorname {Subst}\left (\int \frac {x \left (a+a x^2\right )^2}{(-1+x)^6 (1+x)^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\left ((4 a) \operatorname {Subst}\left (\int \left (\frac {a^2}{(-1+x)^6}+\frac {2 a^2}{(-1+x)^5}+\frac {7 a^2}{4 (-1+x)^4}+\frac {3 a^2}{4 (-1+x)^3}+\frac {a^2}{16 (-1+x)^2}-\frac {a^2}{16 (1+x)^2}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right )\\ &=-\frac {4 a^3}{5 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^5}+\frac {2 a^3}{\left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^4}-\frac {7 a^3}{3 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^3}+\frac {3 a^3}{2 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )^2}-\frac {a^3}{4 \left (1-\sqrt {\frac {1-a x}{1+a x}}\right )}-\frac {a^3}{4 \left (1+\sqrt {\frac {1-a x}{1+a x}}\right )}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 69, normalized size = 0.38 \[ \frac {5 a^2 x^2+2 \sqrt {\frac {1-a x}{a x+1}} (a x+1)^2 \left (2 a^3 x^3-2 a^2 x^2+3 a x-3\right )-6}{15 a^2 x^5} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.74, size = 69, normalized size = 0.38 \[ \frac {5 \, a^{2} x^{2} + 2 \, {\left (2 \, a^{5} x^{5} + a^{3} x^{3} - 3 \, a x\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 6}{15 \, a^{2} x^{5}} \]
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 \[ \int \frac {{\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 84, normalized size = 0.46 \[ \frac {\frac {a^{2}}{3 x^{3}}-\frac {1}{5 x^{5}}}{a^{2}}+\frac {2 \sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (a^{2} x^{2}-1\right ) \left (2 a^{2} x^{2}+3\right )}{15 a \,x^{4}}-\frac {1}{5 a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 56, normalized size = 0.31 \[ \frac {1}{3 \, x^{3}} + \frac {2 \, {\left (2 \, a^{4} x^{5} + a^{2} x^{3} - 3 \, x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{15 \, a^{2} x^{6}} - \frac {2}{5 \, a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.94, size = 86, normalized size = 0.47 \[ \frac {\sqrt {\frac {1}{a\,x}-1}\,\left (\frac {2\,a\,x^2\,\sqrt {\frac {1}{a\,x}+1}}{15}-\frac {2\,\sqrt {\frac {1}{a\,x}+1}}{5\,a}+\frac {4\,a^3\,x^4\,\sqrt {\frac {1}{a\,x}+1}}{15}\right )}{x^4}+\frac {\frac {a^2\,x^2}{3}-\frac {2}{5}}{a^2\,x^5} \]
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 \[ \frac {\int \frac {2}{x^{6}}\, dx + \int \left (- \frac {a^{2}}{x^{4}}\right )\, dx + \int \frac {2 a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}}{x^{5}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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