Optimal. Leaf size=44 \[ \frac {\log \left (1-\left (a+b x^4\right )^2\right )}{8 b}+\frac {\left (a+b x^4\right ) \tanh ^{-1}\left (a+b x^4\right )}{4 b} \]
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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, 6103, 5910, 260} \[ \frac {\log \left (1-\left (a+b x^4\right )^2\right )}{8 b}+\frac {\left (a+b x^4\right ) \tanh ^{-1}\left (a+b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 260
Rule 5910
Rule 6103
Rule 6715
Rubi steps
\begin {align*} \int x^3 \tanh ^{-1}\left (a+b x^4\right ) \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \tanh ^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac {\operatorname {Subst}\left (\int \tanh ^{-1}(x) \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac {\left (a+b x^4\right ) \tanh ^{-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 ) \tanh ^{-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 \[ \frac {\log \left (1-\left (a+b x^4\right )^2\right )+2 \left (a+b x^4\right ) \tanh ^{-1}\left (a+b x^4\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 59, normalized size = 1.34 \[ \frac {b x^{4} \log \left (-\frac {b x^{4} + a + 1}{b x^{4} + a - 1}\right ) + {\left (a + 1\right )} \log \left (b x^{4} + a + 1\right ) - {\left (a - 1\right )} \log \left (b x^{4} + a - 1\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 223, normalized size = 5.07 \[ \frac {1}{8} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {\log \left (\frac {{\left | -b x^{4} - a - 1 \right |}}{{\left | b x^{4} + a - 1 \right |}}\right )}{b^{2}} - \frac {\log \left ({\left | -\frac {b x^{4} + a + 1}{b x^{4} + a - 1} + 1 \right |}\right )}{b^{2}} + \frac {\log \left (-\frac {a - \frac {{\left (\frac {{\left (b x^{4} + a + 1\right )} {\left (a - 1\right )}}{b x^{4} + a - 1} - a - 1\right )} b}{\frac {{\left (b x^{4} + a + 1\right )} b}{b x^{4} + a - 1} - b} + 1}{a - \frac {{\left (\frac {{\left (b x^{4} + a + 1\right )} {\left (a - 1\right )}}{b x^{4} + a - 1} - a - 1\right )} b}{\frac {{\left (b x^{4} + a + 1\right )} b}{b x^{4} + a - 1} - b} - 1}\right )}{b^{2} {\left (\frac {b x^{4} + a + 1}{b x^{4} + a - 1} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 48, normalized size = 1.09 \[ \frac {\arctanh \left (b \,x^{4}+a \right ) x^{4}}{4}+\frac {\arctanh \left (b \,x^{4}+a \right ) a}{4 b}+\frac {\ln \left (1-\left (b \,x^{4}+a \right )^{2}\right )}{8 b} \]
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 \[ \frac {2 \, {\left (b x^{4} + a\right )} \operatorname {artanh}\left (b x^{4} + a\right ) + \log \left (-{\left (b x^{4} + a\right )}^{2} + 1\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 90, normalized size = 2.05 \[ \frac {\ln \left (b\,x^4+a-1\right )}{8\,b}-\frac {x^4\,\ln \left (-b\,x^4-a+1\right )}{8}+\frac {\ln \left (b\,x^4+a+1\right )}{8\,b}+\frac {x^4\,\ln \left (b\,x^4+a+1\right )}{8}-\frac {a\,\ln \left (b\,x^4+a-1\right )}{8\,b}+\frac {a\,\ln \left (b\,x^4+a+1\right )}{8\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.18, size = 60, normalized size = 1.36 \[ \begin {cases} \frac {a \operatorname {atanh}{\left (a + b x^{4} \right )}}{4 b} + \frac {x^{4} \operatorname {atanh}{\left (a + b x^{4} \right )}}{4} + \frac {\log {\left (a + b x^{4} + 1 \right )}}{4 b} - \frac {\operatorname {atanh}{\left (a + b x^{4} \right )}}{4 b} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {atanh}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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