Optimal. Leaf size=44 \[ \frac{\log \left (1-\left (a+b x^4\right )^2\right )}{8 b}+\frac{\left (a+b x^4\right ) \coth ^{-1}\left (a+b x^4\right )}{4 b} \]
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Rubi [A] time = 0.0523872, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6715, 6104, 5911, 260} \[ \frac{\log \left (1-\left (a+b x^4\right )^2\right )}{8 b}+\frac{\left (a+b x^4\right ) \coth ^{-1}\left (a+b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 6715
Rule 6104
Rule 5911
Rule 260
Rubi steps
\begin{align*} \int x^3 \coth ^{-1}\left (a+b x^4\right ) \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \coth ^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac{\operatorname{Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac{\left (a+b x^4\right ) \coth ^{-1}\left (a+b x^4\right )}{4 b}-\frac{\operatorname{Subst}\left (\int \frac{x}{1-x^2} \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac{\left (a+b x^4\right ) \coth ^{-1}\left (a+b x^4\right )}{4 b}+\frac{\log \left (1-\left (a+b x^4\right )^2\right )}{8 b}\\ \end{align*}
Mathematica [A] time = 0.016213, size = 39, normalized size = 0.89 \[ \frac{\log \left (1-\left (a+b x^4\right )^2\right )+2 \left (a+b x^4\right ) \coth ^{-1}\left (a+b x^4\right )}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 46, normalized size = 1.1 \begin{align*}{\frac{{\rm arccoth} \left (b{x}^{4}+a\right ){x}^{4}}{4}}+{\frac{{\rm arccoth} \left (b{x}^{4}+a\right )a}{4\,b}}+{\frac{\ln \left ( \left ( b{x}^{4}+a \right ) ^{2}-1 \right ) }{8\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04667, size = 50, normalized size = 1.14 \begin{align*} \frac{2 \,{\left (b x^{4} + a\right )} \operatorname{arcoth}\left (b x^{4} + a\right ) + \log \left (-{\left (b x^{4} + a\right )}^{2} + 1\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82153, size = 149, normalized size = 3.39 \begin{align*} \frac{b x^{4} \log \left (\frac{b x^{4} + a + 1}{b x^{4} + a - 1}\right ) +{\left (a + 1\right )} \log \left (b x^{4} + a + 1\right ) -{\left (a - 1\right )} \log \left (b x^{4} + a - 1\right )}{8 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.03658, size = 60, normalized size = 1.36 \begin{align*} \begin{cases} \frac{a \operatorname{acoth}{\left (a + b x^{4} \right )}}{4 b} + \frac{x^{4} \operatorname{acoth}{\left (a + b x^{4} \right )}}{4} + \frac{\log{\left (a + b x^{4} + 1 \right )}}{4 b} - \frac{\operatorname{acoth}{\left (a + b x^{4} \right )}}{4 b} & \text{for}\: b \neq 0 \\\frac{x^{4} \operatorname{acoth}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arcoth}\left (b x^{4} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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