Optimal. Leaf size=65 \[ -\frac {1}{4 a x^4}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1+\frac {1}{a^2 x^2}}}{8 x}+\frac {1}{8} a^3 \text {csch}^{-1}(a x) \]
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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6471, 30, 342,
285, 327, 221} \begin {gather*} \frac {1}{8} a^3 \text {csch}^{-1}(a x)-\frac {a^2 \sqrt {\frac {1}{a^2 x^2}+1}}{8 x}-\frac {\sqrt {\frac {1}{a^2 x^2}+1}}{4 x^3}-\frac {1}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 221
Rule 285
Rule 327
Rule 342
Rule 6471
Rubi steps
\begin {align*} \int \frac {e^{\text {csch}^{-1}(a x)}}{x^4} \, dx &=\frac {\int \frac {1}{x^5} \, dx}{a}+\int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x^4} \, dx\\ &=-\frac {1}{4 a x^4}-\text {Subst}\left (\int x^2 \sqrt {1+\frac {x^2}{a^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{4 a x^4}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{4 x^3}-\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{4 a x^4}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1+\frac {1}{a^2 x^2}}}{8 x}+\frac {1}{8} a^2 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{4 a x^4}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{4 x^3}-\frac {a^2 \sqrt {1+\frac {1}{a^2 x^2}}}{8 x}+\frac {1}{8} a^3 \text {csch}^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 53, normalized size = 0.82 \begin {gather*} \frac {-2-a \sqrt {1+\frac {1}{a^2 x^2}} x \left (2+a^2 x^2\right )+a^4 x^4 \sinh ^{-1}\left (\frac {1}{a x}\right )}{8 a x^4} \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. \(172\) vs.
\(2(53)=106\).
time = 0.04, size = 173, normalized size = 2.66
method | result | size |
default | \(\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, a^{2} \left (\left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}\, a^{2} x^{2}-\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, \sqrt {\frac {1}{a^{2}}}\, a^{2} x^{4}+\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right ) x^{4}-2 \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}\right )}{8 x^{3} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, \sqrt {\frac {1}{a^{2}}}}-\frac {1}{4 a \,x^{4}}\) | \(173\) |
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. 129 vs.
\(2 (53) = 106\).
time = 0.26, size = 129, normalized size = 1.98 \begin {gather*} \frac {1}{16} \, a^{3} \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) - \frac {1}{16} \, a^{3} \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right ) - \frac {a^{6} x^{3} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + a^{4} x \sqrt {\frac {1}{a^{2} x^{2}} + 1}}{8 \, {\left (a^{4} x^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{2} - 2 \, a^{2} x^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} + 1\right )}} - \frac {1}{4 \, a x^{4}} \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. 113 vs.
\(2 (53) = 106\).
time = 0.35, size = 113, normalized size = 1.74 \begin {gather*} \frac {a^{4} x^{4} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) - {\left (a^{3} x^{3} + 2 \, a x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - 2}{8 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.90, size = 83, normalized size = 1.28 \begin {gather*} \frac {a^{3} \operatorname {asinh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{2}}{8 x \sqrt {1 + \frac {1}{a^{2} x^{2}}}} - \frac {3}{8 x^{3} \sqrt {1 + \frac {1}{a^{2} x^{2}}}} - \frac {1}{4 a x^{4}} - \frac {1}{4 a^{2} x^{5} \sqrt {1 + \frac {1}{a^{2} x^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 103, normalized size = 1.58 \begin {gather*} \frac {a^{6} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) \mathrm {sgn}\left (x\right ) - a^{6} {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) \mathrm {sgn}\left (x\right ) - \frac {2 \, {\left ({\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + \sqrt {a^{2} x^{2} + 1} a^{6} {\left | a \right |} \mathrm {sgn}\left (x\right ) + 2 \, a^{7}\right )}}{a^{4} x^{4}}}{16 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.54, size = 61, normalized size = 0.94 \begin {gather*} \frac {\mathrm {asinh}\left (\frac {\sqrt {\frac {1}{a^2}}}{x}\right )}{8\,{\left (\frac {1}{a^2}\right )}^{3/2}}-\frac {\sqrt {\frac {1}{a^2\,x^2}+1}}{4\,x^3}-\frac {1}{4\,a\,x^4}-\frac {a^2\,\sqrt {\frac {1}{a^2\,x^2}+1}}{8\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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