Optimal. Leaf size=62 \[ 12 \sqrt{x-1} \sin \left (\sqrt [4]{x-1}\right )-24 \sin \left (\sqrt [4]{x-1}\right )-4 (x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )+24 \sqrt [4]{x-1} \cos \left (\sqrt [4]{x-1}\right ) \]
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Rubi [A] time = 0.0426793, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {3361, 3296, 2637} \[ 12 \sqrt{x-1} \sin \left (\sqrt [4]{x-1}\right )-24 \sin \left (\sqrt [4]{x-1}\right )-4 (x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )+24 \sqrt [4]{x-1} \cos \left (\sqrt [4]{x-1}\right ) \]
Antiderivative was successfully verified.
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Rule 3361
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx &=4 \operatorname{Subst}\left (\int x^3 \sin (x) \, dx,x,\sqrt [4]{-1+x}\right )\\ &=-4 (-1+x)^{3/4} \cos \left (\sqrt [4]{-1+x}\right )+12 \operatorname{Subst}\left (\int x^2 \cos (x) \, dx,x,\sqrt [4]{-1+x}\right )\\ &=-4 (-1+x)^{3/4} \cos \left (\sqrt [4]{-1+x}\right )+12 \sqrt{-1+x} \sin \left (\sqrt [4]{-1+x}\right )-24 \operatorname{Subst}\left (\int x \sin (x) \, dx,x,\sqrt [4]{-1+x}\right )\\ &=24 \sqrt [4]{-1+x} \cos \left (\sqrt [4]{-1+x}\right )-4 (-1+x)^{3/4} \cos \left (\sqrt [4]{-1+x}\right )+12 \sqrt{-1+x} \sin \left (\sqrt [4]{-1+x}\right )-24 \operatorname{Subst}\left (\int \cos (x) \, dx,x,\sqrt [4]{-1+x}\right )\\ &=24 \sqrt [4]{-1+x} \cos \left (\sqrt [4]{-1+x}\right )-4 (-1+x)^{3/4} \cos \left (\sqrt [4]{-1+x}\right )-24 \sin \left (\sqrt [4]{-1+x}\right )+12 \sqrt{-1+x} \sin \left (\sqrt [4]{-1+x}\right )\\ \end{align*}
Mathematica [A] time = 0.0286381, size = 46, normalized size = 0.74 \[ 12 \left (\sqrt{x-1}-2\right ) \sin \left (\sqrt [4]{x-1}\right )-4 \left (\sqrt{x-1}-6\right ) \sqrt [4]{x-1} \cos \left (\sqrt [4]{x-1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 49, normalized size = 0.8 \begin{align*} 24\,\sqrt [4]{-1+x}\cos \left ( \sqrt [4]{-1+x} \right ) -4\, \left ( -1+x \right ) ^{3/4}\cos \left ( \sqrt [4]{-1+x} \right ) -24\,\sin \left ( \sqrt [4]{-1+x} \right ) +12\,\sin \left ( \sqrt [4]{-1+x} \right ) \sqrt{-1+x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968972, size = 50, normalized size = 0.81 \begin{align*} -4 \,{\left ({\left (x - 1\right )}^{\frac{3}{4}} - 6 \,{\left (x - 1\right )}^{\frac{1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) + 12 \,{\left (\sqrt{x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.512283, size = 134, normalized size = 2.16 \begin{align*} -4 \,{\left ({\left (x - 1\right )}^{\frac{3}{4}} - 6 \,{\left (x - 1\right )}^{\frac{1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) + 12 \,{\left (\sqrt{x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.36792, size = 60, normalized size = 0.97 \begin{align*} - 4 \left (x - 1\right )^{\frac{3}{4}} \cos{\left (\sqrt [4]{x - 1} \right )} + 24 \sqrt [4]{x - 1} \cos{\left (\sqrt [4]{x - 1} \right )} + 12 \sqrt{x - 1} \sin{\left (\sqrt [4]{x - 1} \right )} - 24 \sin{\left (\sqrt [4]{x - 1} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10905, size = 50, normalized size = 0.81 \begin{align*} -4 \,{\left ({\left (x - 1\right )}^{\frac{3}{4}} - 6 \,{\left (x - 1\right )}^{\frac{1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) + 12 \,{\left (\sqrt{x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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