Optimal. Leaf size=31 \[ \frac{1}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{3 x^3} \]
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Rubi [A] time = 0.0106171, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \[ \frac{1}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{1+x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1+x^3}}{3 x^3}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1+x^3}}{3 x^3}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^3}\right )\\ &=-\frac{\sqrt{1+x^3}}{3 x^3}+\frac{1}{3} \tanh ^{-1}\left (\sqrt{1+x^3}\right )\\ \end{align*}
Mathematica [A] time = 0.0076484, size = 31, normalized size = 1. \[ \frac{1}{3} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 24, normalized size = 0.8 \begin{align*}{\frac{1}{3}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) }-{\frac{1}{3\,{x}^{3}}\sqrt{{x}^{3}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970765, size = 50, normalized size = 1.61 \begin{align*} -\frac{\sqrt{x^{3} + 1}}{3 \, x^{3}} + \frac{1}{6} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) - \frac{1}{6} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50203, size = 115, normalized size = 3.71 \begin{align*} \frac{x^{3} \log \left (\sqrt{x^{3} + 1} + 1\right ) - x^{3} \log \left (\sqrt{x^{3} + 1} - 1\right ) - 2 \, \sqrt{x^{3} + 1}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.40328, size = 26, normalized size = 0.84 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{\sqrt{1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12145, size = 51, normalized size = 1.65 \begin{align*} -\frac{\sqrt{x^{3} + 1}}{3 \, x^{3}} + \frac{1}{6} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) - \frac{1}{6} \, \log \left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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