Integrand size = 27, antiderivative size = 227 \[ \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=-\frac {3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac {9 \sqrt {\pi } (e (c+d x))^{2/3} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}-\frac {9 \sqrt {\pi } (e (c+d x))^{2/3} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{4 \sqrt {2} b^{5/2} d (c+d x)^{2/3}}+\frac {9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}} \]
-3/2*(d*x+c)^(1/3)*(e*(d*x+c))^(2/3)*cos(a+b*(d*x+c)^(2/3))/b/d+9/4*(e*(d* x+c))^(2/3)*sin(a+b*(d*x+c)^(2/3))/b^2/d/(d*x+c)^(1/3)-9/8*(e*(d*x+c))^(2/ 3)*cos(a)*FresnelS((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(5/2 )/d/(d*x+c)^(2/3)*2^(1/2)-9/8*(e*(d*x+c))^(2/3)*FresnelC((d*x+c)^(1/3)*b^( 1/2)*2^(1/2)/Pi^(1/2))*sin(a)*Pi^(1/2)/b^(5/2)/d/(d*x+c)^(2/3)*2^(1/2)
Time = 0.38 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.70 \[ \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=-\frac {3 (e (c+d x))^{2/3} \left (3 \sqrt {2 \pi } \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+3 \sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)+2 \sqrt {b} \left (2 b (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )-3 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )\right )\right )}{8 b^{5/2} d (c+d x)^{2/3}} \]
(-3*(e*(c + d*x))^(2/3)*(3*Sqrt[2*Pi]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*( c + d*x)^(1/3)] + 3*Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3) ]*Sin[a] + 2*Sqrt[b]*(2*b*(c + d*x)*Cos[a + b*(c + d*x)^(2/3)] - 3*(c + d* x)^(1/3)*Sin[a + b*(c + d*x)^(2/3)])))/(8*b^(5/2)*d*(c + d*x)^(2/3))
Time = 0.56 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.80, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3916, 3898, 3896, 3866, 3867, 3834, 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.
\(\displaystyle \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx\) |
\(\Big \downarrow \) 3916 |
\(\displaystyle \frac {\int (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )d(c+d x)}{d}\) |
\(\Big \downarrow \) 3898 |
\(\displaystyle \frac {(e (c+d x))^{2/3} \int (c+d x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )d(c+d x)}{d (c+d x)^{2/3}}\) |
\(\Big \downarrow \) 3896 |
\(\displaystyle \frac {3 (e (c+d x))^{2/3} \int (c+d x)^{4/3} \sin \left (a+b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{d (c+d x)^{2/3}}\) |
\(\Big \downarrow \) 3866 |
\(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {3 \int (c+d x)^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{2 b}-\frac {(c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d (c+d x)^{2/3}}\) |
\(\Big \downarrow \) 3867 |
\(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {3 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b}-\frac {\int \sin \left (a+b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{2 b}\right )}{2 b}-\frac {(c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d (c+d x)^{2/3}}\) |
\(\Big \downarrow \) 3834 |
\(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {3 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b}-\frac {\sin (a) \int \cos \left (b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}+\cos (a) \int \sin \left (b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{2 b}\right )}{2 b}-\frac {(c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d (c+d x)^{2/3}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {3 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b}-\frac {\sin (a) \int \cos \left (b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}}{2 b}\right )}{2 b}-\frac {(c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d (c+d x)^{2/3}}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {3 (e (c+d x))^{2/3} \left (\frac {3 \left (\frac {\sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )}{2 b}-\frac {\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}}{2 b}\right )}{2 b}-\frac {(c+d x) \cos \left (a+b (c+d x)^{2/3}\right )}{2 b}\right )}{d (c+d x)^{2/3}}\) |
(3*(e*(c + d*x))^(2/3)*(-1/2*((c + d*x)*Cos[a + b*(c + d*x)^(2/3)])/b + (3 *(-1/2*((Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/S qrt[b] + (Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/ Sqrt[b])/b + ((c + d*x)^(1/3)*Sin[a + b*(c + d*x)^(2/3)])/(2*b)))/(2*b)))/ (d*(c + d*x)^(2/3))
3.3.36.3.1 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[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[Sin[c] In t[Cos[d*(e + f*x)^2], x], x] + Simp[Cos[c] Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]
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]
Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Sin[c + d*x^n]/(d*n)), x] - Simp[e^n*((m - n + 1)/ (d*n)) Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Module[{k = Denominator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p] && FractionQ[n]
Int[((e_)*(x_))^(m_)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_ Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IntegerQ[ p] && FractionQ[n]
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/f Subst[Int[(h*(x/f))^m*(a + b*Sin[c + d*x^n])^p, x], x, e + f*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && EqQ[f*g - e*h, 0]
\[\int \left (d e x +c e \right )^{\frac {2}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )d x\]
\[ \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\int { {\left (d e x + c e\right )}^{\frac {2}{3}} \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right ) \,d x } \]
\[ \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\int \left (e \left (c + d x\right )\right )^{\frac {2}{3}} \sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}\, dx \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.51 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.87 \[ \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=-\frac {3 \, {\left (3 \, {\left (d x + c\right )}^{\frac {2}{3}} {\left ({\left ({\left (\Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {3}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) + {\left (\Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {3}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) + {\left (i \, \Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - i \, \Gamma \left (\frac {3}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) + {\left (i \, \Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - i \, \Gamma \left (\frac {3}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right )\right )} \cos \left (a\right ) + {\left ({\left (i \, \Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - i \, \Gamma \left (\frac {3}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) + {\left (-i \, \Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (\frac {3}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) - {\left (\Gamma \left (\frac {3}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {3}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right ) + {\left (\Gamma \left (\frac {3}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (\frac {3}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \arctan \left (0, d x + c\right )\right )\right )} \sin \left (a\right )\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b} e^{\frac {2}{3}} + 8 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} e^{\frac {2}{3}} \cos \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )\right )}}{16 \, {\left (b^{3} d^{2} x + b^{3} c d\right )}} \]
-3/16*(3*(d*x + c)^(2/3)*(((gamma(3/2, -I*b*conjugate((d*x + c)^(2/3))) + gamma(3/2, I*(d*x + c)^(2/3)*b))*cos(3/4*pi + arctan2(0, d*x + c)) + (gamm a(3/2, I*b*conjugate((d*x + c)^(2/3))) + gamma(3/2, -I*(d*x + c)^(2/3)*b)) *cos(-3/4*pi + arctan2(0, d*x + c)) + (I*gamma(3/2, -I*b*conjugate((d*x + c)^(2/3))) - I*gamma(3/2, I*(d*x + c)^(2/3)*b))*sin(3/4*pi + arctan2(0, d* x + c)) + (I*gamma(3/2, I*b*conjugate((d*x + c)^(2/3))) - I*gamma(3/2, -I* (d*x + c)^(2/3)*b))*sin(-3/4*pi + arctan2(0, d*x + c)))*cos(a) + ((I*gamma (3/2, -I*b*conjugate((d*x + c)^(2/3))) - I*gamma(3/2, I*(d*x + c)^(2/3)*b) )*cos(3/4*pi + arctan2(0, d*x + c)) + (-I*gamma(3/2, I*b*conjugate((d*x + c)^(2/3))) + I*gamma(3/2, -I*(d*x + c)^(2/3)*b))*cos(-3/4*pi + arctan2(0, d*x + c)) - (gamma(3/2, -I*b*conjugate((d*x + c)^(2/3))) + gamma(3/2, I*(d *x + c)^(2/3)*b))*sin(3/4*pi + arctan2(0, d*x + c)) + (gamma(3/2, I*b*conj ugate((d*x + c)^(2/3))) + gamma(3/2, -I*(d*x + c)^(2/3)*b))*sin(-3/4*pi + arctan2(0, d*x + c)))*sin(a))*sqrt((d*x + c)^(2/3)*b)*e^(2/3) + 8*(b^2*d^2 *x^2 + 2*b^2*c*d*x + b^2*c^2)*e^(2/3)*cos((d*x + c)^(2/3)*b + a))/(b^3*d^2 *x + b^3*c*d)
Exception generated. \[ \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{2/3} \,d x \]