3.53 \(\int \sin \left (\sqrt [4]{-1+x}\right ) \, dx\)

Optimal. Leaf size=62 \[ 12 \sqrt{x-1} \sin \left (\sqrt [4]{x-1}\right )-24 \sin \left (\sqrt [4]{x-1}\right )-4 (x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )+24 \sqrt [4]{x-1} \cos \left (\sqrt [4]{x-1}\right ) \]

[Out]

24*(-1 + x)^(1/4)*Cos[(-1 + x)^(1/4)] - 4*(-1 + x)^(3/4)*Cos[(-1 + x)^(1/4)] - 2
4*Sin[(-1 + x)^(1/4)] + 12*Sqrt[-1 + x]*Sin[(-1 + x)^(1/4)]

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Rubi [A]  time = 0.06974, 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 \[ 12 \sqrt{x-1} \sin \left (\sqrt [4]{x-1}\right )-24 \sin \left (\sqrt [4]{x-1}\right )-4 (x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )+24 \sqrt [4]{x-1} \cos \left (\sqrt [4]{x-1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sin[(-1 + x)^(1/4)],x]

[Out]

24*(-1 + x)^(1/4)*Cos[(-1 + x)^(1/4)] - 4*(-1 + x)^(3/4)*Cos[(-1 + x)^(1/4)] - 2
4*Sin[(-1 + x)^(1/4)] + 12*Sqrt[-1 + x]*Sin[(-1 + x)^(1/4)]

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Rubi in Sympy [A]  time = 2.24162, size = 60, normalized size = 0.97 \[ - 4 \left (x - 1\right )^{\frac{3}{4}} \cos{\left (\sqrt [4]{x - 1} \right )} + 24 \sqrt [4]{x - 1} \cos{\left (\sqrt [4]{x - 1} \right )} + 12 \sqrt{x - 1} \sin{\left (\sqrt [4]{x - 1} \right )} - 24 \sin{\left (\sqrt [4]{x - 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(sin((-1+x)**(1/4)),x)

[Out]

-4*(x - 1)**(3/4)*cos((x - 1)**(1/4)) + 24*(x - 1)**(1/4)*cos((x - 1)**(1/4)) +
12*sqrt(x - 1)*sin((x - 1)**(1/4)) - 24*sin((x - 1)**(1/4))

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Mathematica [A]  time = 0.0282817, 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.

[In]  Integrate[Sin[(-1 + x)^(1/4)],x]

[Out]

-4*(-6 + Sqrt[-1 + x])*(-1 + x)^(1/4)*Cos[(-1 + x)^(1/4)] + 12*(-2 + Sqrt[-1 + x
])*Sin[(-1 + x)^(1/4)]

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Maple [A]  time = 0.007, size = 49, normalized size = 0.8 \[ 24\,\sqrt [4]{-1+x}\cos \left ( \sqrt [4]{-1+x} \right ) -4\, \left ( -1+x \right ) ^{3/4}\cos \left ( \sqrt [4]{-1+x} \right ) -24\,\sin \left ( \sqrt [4]{-1+x} \right ) +12\,\sin \left ( \sqrt [4]{-1+x} \right ) \sqrt{-1+x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(sin((-1+x)^(1/4)),x)

[Out]

24*(-1+x)^(1/4)*cos((-1+x)^(1/4))-4*(-1+x)^(3/4)*cos((-1+x)^(1/4))-24*sin((-1+x)
^(1/4))+12*sin((-1+x)^(1/4))*(-1+x)^(1/2)

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Maxima [A]  time = 1.38576, size = 50, normalized size = 0.81 \[ -4 \,{\left ({\left (x - 1\right )}^{\frac{3}{4}} - 6 \,{\left (x - 1\right )}^{\frac{1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) + 12 \,{\left (\sqrt{x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin((x - 1)^(1/4)),x, algorithm="maxima")

[Out]

-4*((x - 1)^(3/4) - 6*(x - 1)^(1/4))*cos((x - 1)^(1/4)) + 12*(sqrt(x - 1) - 2)*s
in((x - 1)^(1/4))

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Fricas [A]  time = 0.214825, size = 50, normalized size = 0.81 \[ -4 \,{\left ({\left (x - 1\right )}^{\frac{3}{4}} - 6 \,{\left (x - 1\right )}^{\frac{1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) + 12 \,{\left (\sqrt{x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin((x - 1)^(1/4)),x, algorithm="fricas")

[Out]

-4*((x - 1)^(3/4) - 6*(x - 1)^(1/4))*cos((x - 1)^(1/4)) + 12*(sqrt(x - 1) - 2)*s
in((x - 1)^(1/4))

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Sympy [A]  time = 3.38887, size = 60, normalized size = 0.97 \[ - 4 \left (x - 1\right )^{\frac{3}{4}} \cos{\left (\sqrt [4]{x - 1} \right )} + 24 \sqrt [4]{x - 1} \cos{\left (\sqrt [4]{x - 1} \right )} + 12 \sqrt{x - 1} \sin{\left (\sqrt [4]{x - 1} \right )} - 24 \sin{\left (\sqrt [4]{x - 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin((-1+x)**(1/4)),x)

[Out]

-4*(x - 1)**(3/4)*cos((x - 1)**(1/4)) + 24*(x - 1)**(1/4)*cos((x - 1)**(1/4)) +
12*sqrt(x - 1)*sin((x - 1)**(1/4)) - 24*sin((x - 1)**(1/4))

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GIAC/XCAS [A]  time = 0.217693, size = 50, normalized size = 0.81 \[ -4 \,{\left ({\left (x - 1\right )}^{\frac{3}{4}} - 6 \,{\left (x - 1\right )}^{\frac{1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) + 12 \,{\left (\sqrt{x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac{1}{4}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin((x - 1)^(1/4)),x, algorithm="giac")

[Out]

-4*((x - 1)^(3/4) - 6*(x - 1)^(1/4))*cos((x - 1)^(1/4)) + 12*(sqrt(x - 1) - 2)*s
in((x - 1)^(1/4))