Integrand size = 14, antiderivative size = 130 \[ \int \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=-\frac {3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{2 b^{3/2} d}-\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{2 b^{3/2} d} \]
-3/2*(d*x+c)^(1/3)*cos(a+b*(d*x+c)^(2/3))/b/d+3/4*cos(a)*FresnelC((d*x+c)^ (1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)/d-3/4*FresnelS((d *x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)/b^(3/2)/d
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=-\frac {3 \left (2 \sqrt {b} \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )-\sqrt {2 \pi } \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+\sqrt {2 \pi } \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)\right )}{4 b^{3/2} d} \]
(-3*(2*Sqrt[b]*(c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(2/3)] - Sqrt[2*Pi]*Cos [a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)] + Sqrt[2*Pi]*FresnelS[Sqr t[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a]))/(4*b^(3/2)*d)
Time = 0.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3844, 3866, 3835, 3832, 3833}
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.
\(\displaystyle \int \sin \left (a+b (c+d x)^{2/3}\right ) \, dx\) |
\(\Big \downarrow \) 3844 |
\(\displaystyle \frac {3 \int (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{d}\) |
\(\Big \downarrow \) 3866 |
\(\displaystyle \frac {3 \left (\frac {\int \cos \left (a+b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{2 b}-\frac {\sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d}\) |
\(\Big \downarrow \) 3835 |
\(\displaystyle \frac {3 \left (\frac {\cos (a) \int \cos \left (b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}-\sin (a) \int \sin \left (b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{2 b}-\frac {\sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {3 \left (\frac {\cos (a) \int \cos \left (b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}-\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}}{2 b}-\frac {\sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {3 \left (\frac {\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}}{2 b}-\frac {\sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d}\) |
(3*(-1/2*((c + d*x)^(1/3)*Cos[a + b*(c + d*x)^(2/3)])/b + ((Sqrt[Pi/2]*Cos [a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/Sqrt[b] - (Sqrt[Pi/2]*Fr esnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/Sqrt[b])/(2*b)))/d
3.3.14.3.1 Defintions of rubi rules used
Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[Cos[c] In t[Cos[d*(e + f*x)^2], x], x] - Simp[Sin[c] Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_S ymbol] :> Module[{k = Denominator[n]}, Simp[k/f Subst[Int[x^(k - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && FractionQ[n]
Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^ (n - 1))*(e*x)^(m - n + 1)*(Cos[c + d*x^n]/(d*n)), x] + Simp[e^n*((m - n + 1)/(d*n)) Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {-\frac {3 \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d}\) | \(86\) |
default | \(\frac {-\frac {3 \left (d x +c \right )^{\frac {1}{3}} \cos \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{2 b}+\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \operatorname {C}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \operatorname {S}\left (\frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{4 b^{\frac {3}{2}}}}{d}\) | \(86\) |
3/d*(-1/2/b*(d*x+c)^(1/3)*cos(a+b*(d*x+c)^(2/3))+1/4/b^(3/2)*2^(1/2)*Pi^(1 /2)*(cos(a)*FresnelC((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))-sin(a)*Fresne lS((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))))
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75 \[ \int \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\frac {3 \, {\left (\sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) - \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) - 2 \, {\left (d x + c\right )}^{\frac {1}{3}} b \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{4 \, b^{2} d} \]
3/4*(sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*(d*x + c)^(1/3)*sqrt (b/pi)) - sqrt(2)*pi*sqrt(b/pi)*fresnel_sin(sqrt(2)*(d*x + c)^(1/3)*sqrt(b /pi))*sin(a) - 2*(d*x + c)^(1/3)*b*cos((d*x + c)^(2/3)*b + a))/(b^2*d)
\[ \int \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\int \sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.71 \[ \int \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=-\frac {3 \, {\left (\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {i \, b}\right ) + {\left (-\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left ({\left (d x + c\right )}^{\frac {1}{3}} \sqrt {-i \, b}\right )\right )} b^{\frac {3}{2}} + 8 \, {\left (d x + c\right )}^{\frac {1}{3}} b^{2} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{16 \, b^{3} d} \]
-3/16*(sqrt(2)*sqrt(pi)*(((I - 1)*cos(a) + (I + 1)*sin(a))*erf((d*x + c)^( 1/3)*sqrt(I*b)) + (-(I + 1)*cos(a) - (I - 1)*sin(a))*erf((d*x + c)^(1/3)*s qrt(-I*b)))*b^(3/2) + 8*(d*x + c)^(1/3)*b^2*cos((d*x + c)^(2/3)*b + a))/(b ^3*d)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.32 \[ \int \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=-\frac {3 \, {\left (-\frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} i \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{2} i \, \sqrt {2} {\left (d x + c\right )}^{\frac {1}{3}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} e^{\left (i \, {\left (d x + c\right )}^{\frac {2}{3}} b + i \, a\right )}}{b} + \frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} e^{\left (-i \, {\left (d x + c\right )}^{\frac {2}{3}} b - i \, a\right )}}{b}\right )}}{8 \, d} \]
-3/8*(-I*sqrt(2)*sqrt(pi)*erf(-1/2*I*sqrt(2)*(d*x + c)^(1/3)*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/(b*(I*b/abs(b) + 1)*sqrt(abs(b))) + I*sqrt(2)*sq rt(pi)*erf(1/2*I*sqrt(2)*(d*x + c)^(1/3)*(-I*b/abs(b) + 1)*sqrt(abs(b)))*e ^(-I*a)/(b*(-I*b/abs(b) + 1)*sqrt(abs(b))) + 2*(d*x + c)^(1/3)*e^(I*(d*x + c)^(2/3)*b + I*a)/b + 2*(d*x + c)^(1/3)*e^(-I*(d*x + c)^(2/3)*b - I*a)/b) /d
Timed out. \[ \int \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right ) \,d x \]