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

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)] - 24*Sin[(-1 + x)^(1/4)] + 12*Sqr
t[-1 + x]*Sin[(-1 + x)^(1/4)]

________________________________________________________________________________________

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.

[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*Sqr
t[-1 + x]*Sin[(-1 + x)^(1/4)]

Rule 3361

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Dist[1/(n*f), Subst[Int[x
^(1/n - 1)*(a + b*Sin[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && In
tegerQ[1/n]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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.

[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)]

________________________________________________________________________________________

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.

[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)

________________________________________________________________________________________

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.

[In]

integrate(sin((-1+x)^(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)*sin((x - 1)^(1/4))

________________________________________________________________________________________

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.

[In]

integrate(sin((-1+x)^(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)*sin((x - 1)^(1/4))

________________________________________________________________________________________

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.

[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))

________________________________________________________________________________________

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.

[In]

integrate(sin((-1+x)^(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)*sin((x - 1)^(1/4))