Optimal. Leaf size=34 \[ -\frac {2}{3} a \left (1+\frac {1}{a^2 x^2}\right )^{3/2}-\frac {2}{3 a^2 x^3}-\frac {1}{x} \]
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Rubi [A]
time = 0.13, antiderivative size = 54, normalized size of antiderivative = 1.59, number of steps
used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6473, 6847,
2142} \begin {gather*} -\frac {1}{6} a \left (\sqrt {\frac {1}{a^2 x^2}+1}+\frac {1}{a x}\right )^3-\frac {1}{2} a \sqrt {\frac {1}{a^2 x^2}+1}-\frac {1}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2142
Rule 6473
Rule 6847
Rubi steps
\begin {align*} \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^2} \, dx &=\int \frac {\left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2}{x^2} \, dx\\ &=-\text {Subst}\left (\int \left (\frac {x}{a}+\sqrt {1+\frac {x^2}{a^2}}\right )^2 \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {1}{2} a \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )\right )\\ &=-\frac {1}{2} a \sqrt {1+\frac {1}{a^2 x^2}}-\frac {1}{6} a \left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^3-\frac {1}{2 x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 46, normalized size = 1.35 \begin {gather*} -\frac {2+3 a^2 x^2+2 a \sqrt {1+\frac {1}{a^2 x^2}} x \left (1+a^2 x^2\right )}{3 a^2 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. \(62\) vs.
\(2(28)=56\).
time = 0.04, size = 63, normalized size = 1.85
method | result | size |
trager | \(\frac {-\frac {3 a^{2} x^{2}+2}{3 x^{3}}-\frac {2 a \left (a^{2} x^{2}+1\right ) \sqrt {-\frac {-a^{2} x^{2}-1}{a^{2} x^{2}}}}{3 x^{2}}}{a^{2}}\) | \(56\) |
default | \(\frac {-\frac {a^{2}}{x}-\frac {1}{3 x^{3}}}{a^{2}}-\frac {2 \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} x^{2}+1\right )}{3 a \,x^{2}}-\frac {1}{3 a^{2} x^{3}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 28, normalized size = 0.82 \begin {gather*} -\frac {2}{3} \, a {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - \frac {1}{x} - \frac {2}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (28) = 56\).
time = 0.37, size = 57, normalized size = 1.68 \begin {gather*} -\frac {2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 2}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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. 75 vs.
\(2 (28) = 56\).
time = 0.43, size = 75, normalized size = 2.21 \begin {gather*} \frac {4 \, {\left (3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{4} a \mathrm {sgn}\left (x\right ) + a \mathrm {sgn}\left (x\right )\right )}}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} - \frac {3 \, a^{2} x^{2} + 2}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.21, size = 51, normalized size = 1.50 \begin {gather*} -\frac {\frac {2}{3\,a^2}+\frac {2\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3\,a}}{x^3}-\frac {\frac {2\,a\,x\,\sqrt {\frac {1}{a^2\,x^2}+1}}{3}+1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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