Optimal. Leaf size=28 \[ -\frac {1}{4 \sqrt {1+x^8}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1600, 1607,
457, 79, 65, 213} \begin {gather*} -\frac {1}{4 \sqrt {x^8+1}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {x^8+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 79
Rule 213
Rule 457
Rule 1600
Rule 1607
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^8} \left (1+2 x^8\right )}{x+2 x^9+x^{17}} \, dx &=\int \frac {1+2 x^8}{\sqrt {1+x^8} \left (x+x^9\right )} \, dx\\ &=\int \frac {1+2 x^8}{x \left (1+x^8\right )^{3/2}} \, dx\\ &=\frac {1}{8} \text {Subst}\left (\int \frac {1+2 x}{x (1+x)^{3/2}} \, dx,x,x^8\right )\\ &=-\frac {1}{4 \sqrt {1+x^8}}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^8\right )\\ &=-\frac {1}{4 \sqrt {1+x^8}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^8}\right )\\ &=-\frac {1}{4 \sqrt {1+x^8}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 28, normalized size = 1.00 \begin {gather*} -\frac {1}{4 \sqrt {1+x^8}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 27, normalized size = 0.96
method | result | size |
trager | \(-\frac {1}{4 \sqrt {x^{8}+1}}-\frac {\ln \left (\frac {\sqrt {x^{8}+1}+1}{x^{4}}\right )}{4}\) | \(27\) |
risch | \(-\frac {1}{4 \sqrt {x^{8}+1}}+\frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{\sqrt {x^{8}}}\right )}{4}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 52 vs.
\(2 (20) = 40\).
time = 1.04, size = 52, normalized size = 1.86 \begin {gather*} -\frac {{\left (x^{8} + 1\right )} \log \left (\sqrt {x^{8} + 1} + 1\right ) - {\left (x^{8} + 1\right )} \log \left (\sqrt {x^{8} + 1} - 1\right ) + 2 \, \sqrt {x^{8} + 1}}{8 \, {\left (x^{8} + 1\right )}} \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*} \int \frac {2 x^{8} + 1}{x \left (x^{8} + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 34, normalized size = 1.21 \begin {gather*} -\frac {1}{4 \, \sqrt {x^{8} + 1}} - \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {x^8+1}\,\left (2\,x^8+1\right )}{x^{17}+2\,x^9+x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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