Optimal. Leaf size=47 \[ \frac {\sqrt {1-\left (a+b x^4\right )^2}}{4 b}+\frac {\left (a+b x^4\right ) \sin ^{-1}\left (a+b x^4\right )}{4 b} \]
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Rubi [A] time = 0.06, 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, 4803, 4619, 261} \[ \frac {\sqrt {1-\left (a+b x^4\right )^2}}{4 b}+\frac {\left (a+b x^4\right ) \sin ^{-1}\left (a+b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4619
Rule 4803
Rule 6715
Rubi steps
\begin {align*} \int x^3 \sin ^{-1}\left (a+b x^4\right ) \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \sin ^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac {\operatorname {Subst}\left (\int \sin ^{-1}(x) \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac {\left (a+b x^4\right ) \sin ^{-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 ) \sin ^{-1}\left (a+b x^4\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.87 \[ \frac {\sqrt {1-\left (a+b x^4\right )^2}+\left (a+b x^4\right ) \sin ^{-1}\left (a+b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 46, normalized size = 0.98 \[ \frac {{\left (b x^{4} + a\right )} \arcsin \left (b x^{4} + a\right ) + \sqrt {-b^{2} x^{8} - 2 \, a b x^{4} - a^{2} + 1}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.46, size = 37, normalized size = 0.79 \[ \frac {{\left (b x^{4} + a\right )} \arcsin \left (b x^{4} + a\right ) + \sqrt {-{\left (b x^{4} + a\right )}^{2} + 1}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.81 \[ \frac {\left (b \,x^{4}+a \right ) \arcsin \left (b \,x^{4}+a \right )+\sqrt {1-\left (b \,x^{4}+a \right )^{2}}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 37, normalized size = 0.79 \[ \frac {{\left (b x^{4} + a\right )} \arcsin \left (b x^{4} + a\right ) + \sqrt {-{\left (b x^{4} + a\right )}^{2} + 1}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 99, normalized size = 2.11 \[ \frac {x^4\,\mathrm {asin}\left (b\,x^4+a\right )}{4}+\frac {\sqrt {-a^2-2\,a\,b\,x^4-b^2\,x^8+1}}{4\,b}+\frac {a\,\ln \left (\sqrt {-a^2-2\,a\,b\,x^4-b^2\,x^8+1}-\frac {b^2\,x^4+a\,b}{\sqrt {-b^2}}\right )}{4\,\sqrt {-b^2}} \]
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 \[ \begin {cases} \frac {a \operatorname {asin}{\left (a + b x^{4} \right )}}{4 b} + \frac {x^{4} \operatorname {asin}{\left (a + b x^{4} \right )}}{4} + \frac {\sqrt {- a^{2} - 2 a b x^{4} - b^{2} x^{8} + 1}}{4 b} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {asin}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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