Integrand size = 14, antiderivative size = 85 \[ \int \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {6 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {6 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d} \]
6*cos(a+b*(d*x+c)^(1/3))/b^3/d-3*(d*x+c)^(2/3)*cos(a+b*(d*x+c)^(1/3))/b/d+ 6*(d*x+c)^(1/3)*sin(a+b*(d*x+c)^(1/3))/b^2/d
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {\left (6-3 b^2 (c+d x)^{2/3}\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )+6 b \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d} \]
((6 - 3*b^2*(c + d*x)^(2/3))*Cos[a + b*(c + d*x)^(1/3)] + 6*b*(c + d*x)^(1 /3)*Sin[a + b*(c + d*x)^(1/3)])/(b^3*d)
Time = 0.37 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3842, 3042, 3777, 3042, 3777, 25, 3042, 3118}
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.
\(\displaystyle \int \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx\) |
\(\Big \downarrow \) 3842 |
\(\displaystyle \frac {3 \int (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \int (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{d}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {3 \left (\frac {2 \int \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \left (\frac {2 \int \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}+\frac {\pi }{2}\right )d\sqrt [3]{c+d x}}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d}\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\int -\sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}+\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {\int \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}-\frac {\int \sin \left (a+b \sqrt [3]{c+d x}\right )d\sqrt [3]{c+d x}}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {3 \left (\frac {2 \left (\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{b^2}+\frac {\sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{b}-\frac {(c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b}\right )}{d}\) |
(3*(-(((c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/b) + (2*(Cos[a + b*(c + d*x)^(1/3)]/b^2 + ((c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/b))/b))/d
3.3.9.3.1 Defintions of rubi rules used
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 = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.58
method | result | size |
derivativedivides | \(\frac {-3 a^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 a \left (\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{d \,b^{3}}\) | \(134\) |
default | \(\frac {-3 a^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 a \left (\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{d \,b^{3}}\) | \(134\) |
3/d/b^3*(-a^2*cos(a+b*(d*x+c)^(1/3))-2*a*(sin(a+b*(d*x+c)^(1/3))-(a+b*(d*x +c)^(1/3))*cos(a+b*(d*x+c)^(1/3)))-(a+b*(d*x+c)^(1/3))^2*cos(a+b*(d*x+c)^( 1/3))+2*cos(a+b*(d*x+c)^(1/3))+2*(a+b*(d*x+c)^(1/3))*sin(a+b*(d*x+c)^(1/3) ))
Time = 0.27 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.68 \[ \int \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} b \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \]
3*(2*(d*x + c)^(1/3)*b*sin((d*x + c)^(1/3)*b + a) - ((d*x + c)^(2/3)*b^2 - 2)*cos((d*x + c)^(1/3)*b + a))/(b^3*d)
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.11 \[ \int \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\begin {cases} x \sin {\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \sin {\left (a + b \sqrt [3]{c} \right )} & \text {for}\: d = 0 \\- \frac {3 \left (c + d x\right )^{\frac {2}{3}} \cos {\left (a + b \sqrt [3]{c + d x} \right )}}{b d} + \frac {6 \sqrt [3]{c + d x} \sin {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{2} d} + \frac {6 \cos {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{3} d} & \text {otherwise} \end {cases} \]
Piecewise((x*sin(a), Eq(b, 0) & (Eq(b, 0) | Eq(d, 0))), (x*sin(a + b*c**(1 /3)), Eq(d, 0)), (-3*(c + d*x)**(2/3)*cos(a + b*(c + d*x)**(1/3))/(b*d) + 6*(c + d*x)**(1/3)*sin(a + b*(c + d*x)**(1/3))/(b**2*d) + 6*cos(a + b*(c + d*x)**(1/3))/(b**3*d), True))
Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.41 \[ \int \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=-\frac {3 \, {\left (a^{2} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \]
-3*(a^2*cos((d*x + c)^(1/3)*b + a) - 2*(((d*x + c)^(1/3)*b + a)*cos((d*x + c)^(1/3)*b + a) - sin((d*x + c)^(1/3)*b + a))*a + (((d*x + c)^(1/3)*b + a )^2 - 2)*cos((d*x + c)^(1/3)*b + a) - 2*((d*x + c)^(1/3)*b + a)*sin((d*x + c)^(1/3)*b + a))/(b^3*d)
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96 \[ \int \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3 \, {\left (\frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b} - \frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + a^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{2}}\right )}}{b d} \]
3*(2*(d*x + c)^(1/3)*sin((d*x + c)^(1/3)*b + a)/b - (((d*x + c)^(1/3)*b + a)^2 - 2*((d*x + c)^(1/3)*b + a)*a + a^2 - 2)*cos((d*x + c)^(1/3)*b + a)/b ^2)/(b*d)
Time = 5.80 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx=\frac {3\,\left (2\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )+2\,b\,\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{1/3}-b^2\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{2/3}\right )}{b^3\,d} \]