Optimal. Leaf size=38 \[ -\frac {1}{a^2 x^2}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{a x}-\text {csch}^{-1}(a x)+\log (x) \]
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Rubi [A]
time = 0.14, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6473, 6874,
342, 201, 221} \begin {gather*} -\frac {\sqrt {\frac {1}{a^2 x^2}+1}}{a x}-\frac {1}{a^2 x^2}-\text {csch}^{-1}(a x)+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 342
Rule 6473
Rule 6874
Rubi steps
\begin {align*} \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x} \, dx &=\int \frac {\left (\sqrt {1+\frac {1}{a^2 x^2}}+\frac {1}{a x}\right )^2}{x} \, dx\\ &=\int \left (\frac {2}{a^2 x^3}+\frac {2 \sqrt {1+\frac {1}{a^2 x^2}}}{a x^2}+\frac {1}{x}\right ) \, dx\\ &=-\frac {1}{a^2 x^2}+\log (x)+\frac {2 \int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x^2} \, dx}{a}\\ &=-\frac {1}{a^2 x^2}+\log (x)-\frac {2 \text {Subst}\left (\int \sqrt {1+\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {1}{a^2 x^2}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{a x}+\log (x)-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {1}{a^2 x^2}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{a x}-\text {csch}^{-1}(a x)+\log (x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 39, normalized size = 1.03 \begin {gather*} -\frac {1+a \sqrt {1+\frac {1}{a^2 x^2}} x}{a^2 x^2}-\sinh ^{-1}\left (\frac {1}{a x}\right )+\log (x) \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. \(163\) vs.
\(2(36)=72\).
time = 0.04, size = 164, normalized size = 4.32
method | result | size |
default | \(\frac {-\frac {1}{2 x^{2}}+a^{2} \ln \left (x \right )}{a^{2}}-\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}-\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2} x^{2}+\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right ) x^{2}\right )}{a x \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}}-\frac {1}{2 a^{2} x^{2}}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs.
\(2 (36) = 72\).
time = 0.26, size = 93, normalized size = 2.45 \begin {gather*} -\frac {\frac {2 \, a^{2} x \sqrt {\frac {1}{a^{2} x^{2}} + 1}}{a^{2} x^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} - 1} + a \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) - a \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} - \frac {1}{a^{2} x^{2}} + \log \left (x\right ) \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. 112 vs.
\(2 (36) = 72\).
time = 0.35, size = 112, normalized size = 2.95 \begin {gather*} -\frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) - a^{2} x^{2} \log \left (x\right ) + a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 1}{a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.44, size = 34, normalized size = 0.89 \begin {gather*} \log {\left (x \right )} - \operatorname {asinh}{\left (\frac {1}{a x} \right )} - \frac {\sqrt {1 + \frac {1}{a^{2} x^{2}}}}{a x} - \frac {1}{a^{2} x^{2}} \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. 118 vs.
\(2 (36) = 72\).
time = 0.42, size = 118, normalized size = 3.11 \begin {gather*} -\frac {{\left (a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) - a^{5}\right )} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) - {\left (a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + a^{5}\right )} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) + \frac {2 \, {\left (\sqrt {a^{2} x^{2} + 1} a^{4} {\left | a \right |} \mathrm {sgn}\left (x\right ) + a^{5}\right )}}{{\left (\sqrt {a^{2} x^{2} + 1} + 1\right )} {\left (\sqrt {a^{2} x^{2} + 1} - 1\right )}}}{2 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.25, size = 44, normalized size = 1.16 \begin {gather*} -\ln \left (\frac {1}{x}\right )-\mathrm {asinh}\left (\frac {1}{a\,x}\right )-\frac {1}{a^2\,x^2}-\frac {\sqrt {\frac {1}{a^2\,x^2}+1}}{a\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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