Integrand size = 8, antiderivative size = 62 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=24 \sqrt [4]{-1+x} \cos \left (\sqrt [4]{-1+x}\right )-4 (-1+x)^{3/4} \cos \left (\sqrt [4]{-1+x}\right )-24 \sin \left (\sqrt [4]{-1+x}\right )+12 \sqrt {-1+x} \sin \left (\sqrt [4]{-1+x}\right ) \] Output:
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)
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=-4 \left (-6+\sqrt {-1+x}\right ) \sqrt [4]{-1+x} \cos \left (\sqrt [4]{-1+x}\right )+12 \left (-2+\sqrt {-1+x}\right ) \sin \left (\sqrt [4]{-1+x}\right ) \] Input:
Integrate[Sin[(-1 + x)^(1/4)],x]
Output:
-4*(-6 + Sqrt[-1 + x])*(-1 + x)^(1/4)*Cos[(-1 + x)^(1/4)] + 12*(-2 + Sqrt[ -1 + x])*Sin[(-1 + x)^(1/4)]
Time = 0.33 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {3842, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin \left (\sqrt [4]{x-1}\right ) \, dx\) |
\(\Big \downarrow \) 3842 |
\(\displaystyle 4 \int (x-1)^{3/4} \sin \left (\sqrt [4]{x-1}\right )d\sqrt [4]{x-1}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 \int (x-1)^{3/4} \sin \left (\sqrt [4]{x-1}\right )d\sqrt [4]{x-1}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 4 \left (3 \int \sqrt {x-1} \cos \left (\sqrt [4]{x-1}\right )d\sqrt [4]{x-1}-(x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 \left (3 \int \sqrt {x-1} \sin \left (\sqrt [4]{x-1}+\frac {\pi }{2}\right )d\sqrt [4]{x-1}-(x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 4 \left (3 \left (2 \int -\sqrt [4]{x-1} \sin \left (\sqrt [4]{x-1}\right )d\sqrt [4]{x-1}+\sqrt {x-1} \sin \left (\sqrt [4]{x-1}\right )\right )-(x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 4 \left (3 \left (\sqrt {x-1} \sin \left (\sqrt [4]{x-1}\right )-2 \int \sqrt [4]{x-1} \sin \left (\sqrt [4]{x-1}\right )d\sqrt [4]{x-1}\right )-(x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 \left (3 \left (\sqrt {x-1} \sin \left (\sqrt [4]{x-1}\right )-2 \int \sqrt [4]{x-1} \sin \left (\sqrt [4]{x-1}\right )d\sqrt [4]{x-1}\right )-(x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle 4 \left (3 \left (\sqrt {x-1} \sin \left (\sqrt [4]{x-1}\right )-2 \left (\int \cos \left (\sqrt [4]{x-1}\right )d\sqrt [4]{x-1}-\sqrt [4]{x-1} \cos \left (\sqrt [4]{x-1}\right )\right )\right )-(x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 \left (3 \left (\sqrt {x-1} \sin \left (\sqrt [4]{x-1}\right )-2 \left (\int \sin \left (\sqrt [4]{x-1}+\frac {\pi }{2}\right )d\sqrt [4]{x-1}-\sqrt [4]{x-1} \cos \left (\sqrt [4]{x-1}\right )\right )\right )-(x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle 4 \left (3 \left (\sqrt {x-1} \sin \left (\sqrt [4]{x-1}\right )-2 \left (\sin \left (\sqrt [4]{x-1}\right )-\sqrt [4]{x-1} \cos \left (\sqrt [4]{x-1}\right )\right )\right )-(x-1)^{3/4} \cos \left (\sqrt [4]{x-1}\right )\right )\) |
Input:
Int[Sin[(-1 + x)^(1/4)],x]
Output:
4*(-((-1 + x)^(3/4)*Cos[(-1 + x)^(1/4)]) + 3*(Sqrt[-1 + x]*Sin[(-1 + x)^(1 /4)] - 2*(-((-1 + x)^(1/4)*Cos[(-1 + x)^(1/4)]) + Sin[(-1 + x)^(1/4)])))
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_S ymbol] :> Simp[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] && Intege rQ[1/n]
Time = 0.03 (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 | \(24 \left (-1+x \right )^{\frac {1}{4}} \cos \left (\left (-1+x \right )^{\frac {1}{4}}\right )-4 \left (-1+x \right )^{\frac {3}{4}} \cos \left (\left (-1+x \right )^{\frac {1}{4}}\right )-24 \sin \left (\left (-1+x \right )^{\frac {1}{4}}\right )+12 \sin \left (\left (-1+x \right )^{\frac {1}{4}}\right ) \sqrt {-1+x}\) | \(49\) |
Input:
int(sin((-1+x)^(1/4)),x,method=_RETURNVERBOSE)
Output:
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)
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=-4 \, {\left ({\left (x - 1\right )}^{\frac {3}{4}} - 6 \, {\left (x - 1\right )}^{\frac {1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) + 12 \, {\left (\sqrt {x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) \] Input:
integrate(sin((-1+x)^(1/4)),x, algorithm="fricas")
Output:
-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))
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.97 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=- 4 \left (x - 1\right )^{\frac {3}{4}} \cos {\left (\sqrt [4]{x - 1} \right )} + 24 \sqrt [4]{x - 1} \cos {\left (\sqrt [4]{x - 1} \right )} + 12 \sqrt {x - 1} \sin {\left (\sqrt [4]{x - 1} \right )} - 24 \sin {\left (\sqrt [4]{x - 1} \right )} \] Input:
integrate(sin((-1+x)**(1/4)),x)
Output:
-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))
Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=-4 \, {\left ({\left (x - 1\right )}^{\frac {3}{4}} - 6 \, {\left (x - 1\right )}^{\frac {1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) + 12 \, {\left (\sqrt {x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) \] Input:
integrate(sin((-1+x)^(1/4)),x, algorithm="maxima")
Output:
-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))
Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.60 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=-4 \, {\left ({\left (x - 1\right )}^{\frac {3}{4}} - 6 \, {\left (x - 1\right )}^{\frac {1}{4}}\right )} \cos \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) + 12 \, {\left (\sqrt {x - 1} - 2\right )} \sin \left ({\left (x - 1\right )}^{\frac {1}{4}}\right ) \] Input:
integrate(sin((-1+x)^(1/4)),x, algorithm="giac")
Output:
-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))
Time = 0.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=4\,\cos \left ({\left (x-1\right )}^{1/4}\right )\,\left (6\,{\left (x-1\right )}^{1/4}-{\left (x-1\right )}^{3/4}\right )+4\,\sin \left ({\left (x-1\right )}^{1/4}\right )\,\left (3\,\sqrt {x-1}-6\right ) \] Input:
int(sin((x - 1)^(1/4)),x)
Output:
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)
Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76 \[ \int \sin \left (\sqrt [4]{-1+x}\right ) \, dx=-4 \left (x -1\right )^{\frac {3}{4}} \cos \left (\left (x -1\right )^{\frac {1}{4}}\right )+24 \left (x -1\right )^{\frac {1}{4}} \cos \left (\left (x -1\right )^{\frac {1}{4}}\right )+12 \sqrt {x -1}\, \sin \left (\left (x -1\right )^{\frac {1}{4}}\right )-24 \sin \left (\left (x -1\right )^{\frac {1}{4}}\right ) \] Input:
int(sin((-1+x)^(1/4)),x)
Output:
4*( - (x - 1)**(3/4)*cos((x - 1)**(1/4)) + 6*(x - 1)**(1/4)*cos((x - 1)**( 1/4)) + 3*sqrt(x - 1)*sin((x - 1)**(1/4)) - 6*sin((x - 1)**(1/4)))