Optimal. Leaf size=62 \[ 24 \sqrt [4]{-1+x} \cos \left (\sqrt [4]{-1+x}\right )-4 (-1+x)^{3/4} \cos \left (\sqrt [4]{-1+x}\right )-24 \sin \left (\sqrt [4]{-1+x}\right )+12 \sqrt {-1+x} \sin \left (\sqrt [4]{-1+x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 = {3442, 3377,
2717} \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3377
Rule 3442
Rubi steps
\begin {align*} \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx &=4 \text {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 \text {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 \text {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 \text {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*}
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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.74 \begin {gather*} -4 \left (-6+\sqrt {-1+x}\right ) \sqrt [4]{-1+x} \cos \left (\sqrt [4]{-1+x}\right )+12 \left (-2+\sqrt {-1+x}\right ) \sin \left (\sqrt [4]{-1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 49, normalized size = 0.79
method | result | size |
derivativedivides | \(24 \left (-1+x \right )^{\frac {1}{4}} \cos \left (\left (-1+x \right )^{\frac {1}{4}}\right )-4 \left (-1+x \right )^{\frac {3}{4}} \cos \left (\left (-1+x \right )^{\frac {1}{4}}\right )-24 \sin \left (\left (-1+x \right )^{\frac {1}{4}}\right )+12 \sin \left (\left (-1+x \right )^{\frac {1}{4}}\right ) \sqrt {-1+x}\) | \(49\) |
default | \(24 \left (-1+x \right )^{\frac {1}{4}} \cos \left (\left (-1+x \right )^{\frac {1}{4}}\right )-4 \left (-1+x \right )^{\frac {3}{4}} \cos \left (\left (-1+x \right )^{\frac {1}{4}}\right )-24 \sin \left (\left (-1+x \right )^{\frac {1}{4}}\right )+12 \sin \left (\left (-1+x \right )^{\frac {1}{4}}\right ) \sqrt {-1+x}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.89, size = 37, normalized size = 0.60 \begin {gather*} -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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.16, size = 37, normalized size = 0.60 \begin {gather*} -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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 60, normalized size = 0.97 \begin {gather*} - 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 37, normalized size = 0.60 \begin {gather*} -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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 41, normalized size = 0.66 \begin {gather*} 4\,\cos \left ({\left (x-1\right )}^{1/4}\right )\,\left (6\,{\left (x-1\right )}^{1/4}-{\left (x-1\right )}^{3/4}\right )+4\,\sin \left ({\left (x-1\right )}^{1/4}\right )\,\left (3\,\sqrt {x-1}-6\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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