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 = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 260
Rule 5911
Rule 6104
Rule 6715
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.61, size = 58, normalized size = 1.32 \[ \frac {b x^{4} \log \left (\frac {b x^{4} + a + 1}{b x^{4} + a - 1}\right ) + {\left (a + 1\right )} \log \left (b x^{4} + a + 1\right ) - {\left (a - 1\right )} \log \left (b x^{4} + a - 1\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {arcoth}\left (b x^{4} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 46, normalized size = 1.05 \[ \frac {\mathrm {arccoth}\left (b \,x^{4}+a \right ) x^{4}}{4}+\frac {\mathrm {arccoth}\left (b \,x^{4}+a \right ) a}{4 b}+\frac {\ln \left (\left (b \,x^{4}+a \right )^{2}-1\right )}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 37, normalized size = 0.84 \[ \frac {2 \, {\left (b x^{4} + a\right )} \operatorname {arcoth}\left (b x^{4} + a\right ) + \log \left (-{\left (b x^{4} + a\right )}^{2} + 1\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.56, size = 107, normalized size = 2.43 \[ \frac {x^4\,\ln \left (\frac {b\,x^4+a+1}{b\,x^4+a}\right )}{8}-\frac {x^4\,\ln \left (\frac {b\,x^4+a-1}{b\,x^4+a}\right )}{8}+\frac {\ln \left (b\,x^4+a-1\right )}{8\,b}+\frac {\ln \left (b\,x^4+a+1\right )}{8\,b}-\frac {a\,\ln \left (b\,x^4+a-1\right )}{8\,b}+\frac {a\,\ln \left (b\,x^4+a+1\right )}{8\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.22, size = 60, normalized size = 1.36 \[ \begin {cases} \frac {a \operatorname {acoth}{\left (a + b x^{4} \right )}}{4 b} + \frac {x^{4} \operatorname {acoth}{\left (a + b x^{4} \right )}}{4} + \frac {\log {\left (a + b x^{4} + 1 \right )}}{4 b} - \frac {\operatorname {acoth}{\left (a + b x^{4} \right )}}{4 b} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {acoth}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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