Optimal. Leaf size=47 \[ \frac {\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )}{4 b}-\frac {\sqrt {1-\left (a+b x^4\right )^2}}{4 b} \]
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Rubi [A] time = 0.05, antiderivative size = 47, 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, 4804, 4620, 261} \[ \frac {\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )}{4 b}-\frac {\sqrt {1-\left (a+b x^4\right )^2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4620
Rule 4804
Rule 6715
Rubi steps
\begin {align*} \int x^3 \cos ^{-1}\left (a+b x^4\right ) \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \cos ^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac {\operatorname {Subst}\left (\int \cos ^{-1}(x) \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac {\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )}{4 b}+\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2}} \, dx,x,a+b x^4\right )}{4 b}\\ &=-\frac {\sqrt {1-\left (a+b x^4\right )^2}}{4 b}+\frac {\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 0.91 \[ \frac {\left (a+b x^4\right ) \cos ^{-1}\left (a+b x^4\right )-\sqrt {1-\left (a+b x^4\right )^2}}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 48, normalized size = 1.02 \[ \frac {{\left (b x^{4} + a\right )} \arccos \left (b x^{4} + a\right ) - \sqrt {-b^{2} x^{8} - 2 \, a b x^{4} - a^{2} + 1}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 39, normalized size = 0.83 \[ \frac {{\left (b x^{4} + a\right )} \arccos \left (b x^{4} + a\right ) - \sqrt {-{\left (b x^{4} + a\right )}^{2} + 1}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 40, normalized size = 0.85 \[ \frac {\left (b \,x^{4}+a \right ) \arccos \left (b \,x^{4}+a \right )-\sqrt {1-\left (b \,x^{4}+a \right )^{2}}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 39, normalized size = 0.83 \[ \frac {{\left (b x^{4} + a\right )} \arccos \left (b x^{4} + a\right ) - \sqrt {-{\left (b x^{4} + a\right )}^{2} + 1}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 99, normalized size = 2.11 \[ \frac {x^4\,\mathrm {acos}\left (b\,x^4+a\right )}{4}-\frac {\sqrt {-a^2-2\,a\,b\,x^4-b^2\,x^8+1}}{4\,b}-\frac {a\,\ln \left (\sqrt {-a^2-2\,a\,b\,x^4-b^2\,x^8+1}-\frac {b^2\,x^4+a\,b}{\sqrt {-b^2}}\right )}{4\,\sqrt {-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 61, normalized size = 1.30 \[ \begin {cases} \frac {a \operatorname {acos}{\left (a + b x^{4} \right )}}{4 b} + \frac {x^{4} \operatorname {acos}{\left (a + b x^{4} \right )}}{4} - \frac {\sqrt {- a^{2} - 2 a b x^{4} - b^{2} x^{8} + 1}}{4 b} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {acos}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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