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 = 0.0485548, antiderivative size = 45, normalized size of antiderivative = 1., 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 6715
Rule 5863
Rule 5653
Rule 261
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*}
Mathematica [A] time = 0.0245283, 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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Maple [A] time = 0.003, size = 38, normalized size = 0.8 \begin{align*}{\frac{1}{4\,b} \left ( \left ( b{x}^{4}+a \right ){\it Arcsinh} \left ( b{x}^{4}+a \right ) -\sqrt{1+ \left ( b{x}^{4}+a \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11223, size = 50, normalized size = 1.11 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42898, size = 151, normalized size = 3.36 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.12979, size = 61, normalized size = 1.36 \begin{align*} \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}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39041, size = 142, normalized size = 3.16 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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