Optimal. Leaf size=42 \[ \frac {\log \left (\left (a+b x^4\right )^2+1\right )}{8 b}+\frac {\left (a+b x^4\right ) \cot ^{-1}\left (a+b x^4\right )}{4 b} \]
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Rubi [A] time = 0.04, antiderivative size = 42, 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, 5040, 4847, 260} \[ \frac {\log \left (\left (a+b x^4\right )^2+1\right )}{8 b}+\frac {\left (a+b x^4\right ) \cot ^{-1}\left (a+b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4847
Rule 5040
Rule 6715
Rubi steps
\begin {align*} \int x^3 \cot ^{-1}\left (a+b x^4\right ) \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \cot ^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac {\operatorname {Subst}\left (\int \cot ^{-1}(x) \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac {\left (a+b x^4\right ) \cot ^{-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 ) \cot ^{-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 = 37, normalized size = 0.88 \[ \frac {\log \left (\left (a+b x^4\right )^2+1\right )+2 \left (a+b x^4\right ) \cot ^{-1}\left (a+b x^4\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 51, normalized size = 1.21 \[ \frac {2 \, b x^{4} \operatorname {arccot}\left (b x^{4} + a\right ) - 2 \, a \arctan \left (b x^{4} + a\right ) + \log \left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2} + 1\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 127, normalized size = 3.02 \[ -\frac {\arctan \left (\frac {1}{b x^{4} + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x^{4} + a}\right )\right )^{2} + \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x^{4} + a}\right )\right )^{2}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x^{4} + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x^{4} + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x^{4} + a}\right )\right ) - \arctan \left (\frac {1}{b x^{4} + a}\right )}{8 \, b \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x^{4} + a}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 46, normalized size = 1.10 \[ \frac {\mathrm {arccot}\left (b \,x^{4}+a \right ) x^{4}}{4}+\frac {\mathrm {arccot}\left (b \,x^{4}+a \right ) a}{4 b}+\frac {\ln \left (1+\left (b \,x^{4}+a \right )^{2}\right )}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 35, normalized size = 0.83 \[ \frac {2 \, {\left (b x^{4} + a\right )} \operatorname {arccot}\left (b x^{4} + a\right ) + \log \left ({\left (b x^{4} + a\right )}^{2} + 1\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 230, normalized size = 5.48 \[ \frac {\ln \left (a^2+2\,a\,b\,x^4+b^2\,x^8+1\right )}{8\,b}+\frac {x^4\,\mathrm {acot}\left (b\,x^4+a\right )}{4}-\frac {a\,\mathrm {atan}\left (\frac {a}{a^6+3\,a^4+3\,a^2+1}+\frac {3\,a^3}{a^6+3\,a^4+3\,a^2+1}+\frac {3\,a^5}{a^6+3\,a^4+3\,a^2+1}+\frac {a^7}{a^6+3\,a^4+3\,a^2+1}+\frac {b\,x^4}{a^6+3\,a^4+3\,a^2+1}+\frac {3\,a^2\,b\,x^4}{a^6+3\,a^4+3\,a^2+1}+\frac {3\,a^4\,b\,x^4}{a^6+3\,a^4+3\,a^2+1}+\frac {a^6\,b\,x^4}{a^6+3\,a^4+3\,a^2+1}\right )}{4\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.34, size = 60, normalized size = 1.43 \[ \begin {cases} \frac {a \operatorname {acot}{\left (a + b x^{4} \right )}}{4 b} + \frac {x^{4} \operatorname {acot}{\left (a + b x^{4} \right )}}{4} + \frac {\log {\left (a^{2} + 2 a b x^{4} + b^{2} x^{8} + 1 \right )}}{8 b} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {acot}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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