\(\int \sin (\sqrt [4]{-1+x}) \, dx\) [53]

Optimal result

Integrand size = 8, antiderivative size = 62 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=24 \sqrt [4]{-1+x} \cos \left (\sqrt [4]{-1+x}\right )-4 (-1+x)^{3/4} \cos \left (\sqrt [4]{-1+x}\right )-24 \sin \left (\sqrt [4]{-1+x}\right )+12 \sqrt {-1+x} \sin \left (\sqrt [4]{-1+x}\right ) \]

[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)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3442, 3377, 2717} \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=12 \sqrt {x-1} \sin \left (\sqrt [4]{x-1}\right )-24 \sin \left (\sqrt [4]{x-1}\right )-4 (x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )+24 \sqrt [4]{x-1} \cos \left (\sqrt [4]{x-1}\right ) \]

[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 2717

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

Rule 3377

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 3442

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]

Rubi steps \begin{align*} \text {integral}& = 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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=-4 \left (-6+\sqrt {-1+x}\right ) \sqrt [4]{-1+x} \cos \left (\sqrt [4]{-1+x}\right )+12 \left (-2+\sqrt {-1+x}\right ) \sin \left (\sqrt [4]{-1+x}\right ) \]

[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] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 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	\(24 \left (-1+x \right )^{\frac {1}{4}} \cos \left (\left (-1+x \right )^{\frac {1}{4}}\right )-4 \left (-1+x \right )^{\frac {3}{4}} \cos \left (\left (-1+x \right )^{\frac {1}{4}}\right )-24 \sin \left (\left (-1+x \right )^{\frac {1}{4}}\right )+12 \sin \left (\left (-1+x \right )^{\frac {1}{4}}\right ) \sqrt {-1+x}\)	\(49\)

[In]

int(sin((-1+x)^(1/4)),x,method=_RETURNVERBOSE)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=-4 \, {\left ({\left (x - 1\right )}^{\frac {3}{4}} - 6 \, {\left (x - 1\right )}^{\frac {1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) + 12 \, {\left (\sqrt {x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) \]

[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] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.97 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=- 4 \left (x - 1\right )^{\frac {3}{4}} \cos {\left (\sqrt [4]{x - 1} \right )} + 24 \sqrt [4]{x - 1} \cos {\left (\sqrt [4]{x - 1} \right )} + 12 \sqrt {x - 1} \sin {\left (\sqrt [4]{x - 1} \right )} - 24 \sin {\left (\sqrt [4]{x - 1} \right )} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=-4 \, {\left ({\left (x - 1\right )}^{\frac {3}{4}} - 6 \, {\left (x - 1\right )}^{\frac {1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) + 12 \, {\left (\sqrt {x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=-4 \, {\left ({\left (x - 1\right )}^{\frac {3}{4}} - 6 \, {\left (x - 1\right )}^{\frac {1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) + 12 \, {\left (\sqrt {x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=4\,\cos \left ({\left (x-1\right )}^{1/4}\right )\,\left (6\,{\left (x-1\right )}^{1/4}-{\left (x-1\right )}^{3/4}\right )+4\,\sin \left ({\left (x-1\right )}^{1/4}\right )\,\left (3\,\sqrt {x-1}-6\right ) \]

[In]

int(sin((x - 1)^(1/4)),x)

[Out]

4*cos((x - 1)^(1/4))*(6*(x - 1)^(1/4) - (x - 1)^(3/4)) + 4*sin((x - 1)^(1/4))*(3*(x - 1)^(1/2) - 6)