Optimal. Leaf size=45 \[ \frac {\left (a+b x^4\right ) \sinh ^{-1}\left (a+b x^4\right )}{4 b}-\frac {\sqrt {\left (a+b x^4\right )^2+1}}{4 b} \]
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Rubi [A] time = 0.05, antiderivative size = 45, 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, 5863, 5653, 261} \[ \frac {\left (a+b x^4\right ) \sinh ^{-1}\left (a+b x^4\right )}{4 b}-\frac {\sqrt {\left (a+b x^4\right )^2+1}}{4 b} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5653
Rule 5863
Rule 6715
Rubi steps
\begin {align*} \int x^3 \sinh ^{-1}\left (a+b x^4\right ) \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \sinh ^{-1}(a+b x) \, dx,x,x^4\right )\\ &=\frac {\operatorname {Subst}\left (\int \sinh ^{-1}(x) \, dx,x,a+b x^4\right )}{4 b}\\ &=\frac {\left (a+b x^4\right ) \sinh ^{-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 ) \sinh ^{-1}\left (a+b x^4\right )}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.91 \[ \frac {\left (a+b x^4\right ) \sinh ^{-1}\left (a+b x^4\right )-\sqrt {\left (a+b x^4\right )^2+1}}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 66, normalized size = 1.47 \[ \frac {{\left (b x^{4} + a\right )} \log \left (b x^{4} + a + \sqrt {b^{2} x^{8} + 2 \, a b x^{4} + a^{2} + 1}\right ) - \sqrt {b^{2} x^{8} + 2 \, a b x^{4} + a^{2} + 1}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 105, normalized size = 2.33 \[ \frac {1}{4} \, x^{4} \log \left (b x^{4} + a + \sqrt {{\left (b x^{4} + a\right )}^{2} + 1}\right ) - \frac {1}{4} \, b {\left (\frac {a \log \left (-a b - {\left (x^{4} {\left | b \right |} - \sqrt {b^{2} x^{8} + 2 \, a b x^{4} + a^{2} + 1}\right )} {\left | b \right |}\right )}{b {\left | b \right |}} + \frac {\sqrt {b^{2} x^{8} + 2 \, a b x^{4} + a^{2} + 1}}{b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.84 \[ \frac {\left (b \,x^{4}+a \right ) \arcsinh \left (b \,x^{4}+a \right )-\sqrt {1+\left (b \,x^{4}+a \right )^{2}}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 37, normalized size = 0.82 \[ \frac {{\left (b x^{4} + a\right )} \operatorname {arsinh}\left (b x^{4} + a\right ) - \sqrt {{\left (b x^{4} + a\right )}^{2} + 1}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 88, normalized size = 1.96 \[ \frac {x^4\,\mathrm {asinh}\left (b\,x^4+a\right )}{4}-\frac {\sqrt {a^2+2\,a\,b\,x^4+b^2\,x^8+1}}{4\,b}+\frac {a\,\ln \left (\sqrt {a^2+2\,a\,b\,x^4+b^2\,x^8+1}+\frac {b^2\,x^4+a\,b}{\sqrt {b^2}}\right )}{4\,\sqrt {b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 61, normalized size = 1.36 \[ \begin {cases} \frac {a \operatorname {asinh}{\left (a + b x^{4} \right )}}{4 b} + \frac {x^{4} \operatorname {asinh}{\left (a + b x^{4} \right )}}{4} - \frac {\sqrt {a^{2} + 2 a b x^{4} + b^{2} x^{8} + 1}}{4 b} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {asinh}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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