Integrand size = 27, antiderivative size = 133 \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b} d (e (c+d x))^{2/3}}+\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{\sqrt {b} d (e (c+d x))^{2/3}} \]
3/2*(d*x+c)^(2/3)*cos(a)*FresnelS((d*x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))* 2^(1/2)*Pi^(1/2)/d/(e*(d*x+c))^(2/3)/b^(1/2)+3/2*(d*x+c)^(2/3)*FresnelC((d *x+c)^(1/3)*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)/d/(e*(d*x+c) )^(2/3)/b^(1/2)
Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.72 \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \left (\cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+\operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)\right )}{\sqrt {b} d (e (c+d x))^{2/3}} \]
(3*Sqrt[Pi/2]*(c + d*x)^(2/3)*(Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x )^(1/3)] + FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a]))/(Sqrt[b]* d*(e*(c + d*x))^(2/3))
Time = 0.40 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3916, 3898, 3864, 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.
\(\displaystyle \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx\) |
\(\Big \downarrow \) 3916 |
\(\displaystyle \frac {\int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(e (c+d x))^{2/3}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 3898 |
\(\displaystyle \frac {(c+d x)^{2/3} \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c+d x)^{2/3}}d(c+d x)}{d (e (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3864 |
\(\displaystyle \frac {3 (c+d x)^{2/3} \int \sin \left (a+b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}}{d (e (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3834 |
\(\displaystyle \frac {3 (c+d x)^{2/3} \left (\sin (a) \int \cos \left (b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}+\cos (a) \int \sin \left (b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}\right )}{d (e (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3832 |
\(\displaystyle \frac {3 (c+d x)^{2/3} \left (\sin (a) \int \cos \left (b (c+d x)^{2/3}\right )d\sqrt [3]{c+d x}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}\right )}{d (e (c+d x))^{2/3}}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {3 (c+d x)^{2/3} \left (\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}\right )}{d (e (c+d x))^{2/3}}\) |
(3*(c + d*x)^(2/3)*((Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d* x)^(1/3)])/Sqrt[b] + (Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/ 3)]*Sin[a])/Sqrt[b]))/(d*(e*(c + d*x))^(2/3))
3.3.39.3.1 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[(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_)], x_Symbol] :> Simp[2/n Subst[ Int[Sin[a + b*x^2], x], x, x^(n/2)], x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]
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]
\[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{\left (d e x +c e \right )^{\frac {2}{3}}}d x\]
\[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\int { \frac {\sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\int \frac {\sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {2}{3}}}\, dx \]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.48 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.66 \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\frac {3 \, {\left ({\left ({\left (-i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \left (a\right ) + {\left ({\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \left (a\right )\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} b d e^{\frac {2}{3}}} \]
3/8*(((-I*sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x + c)^(2/3)))) - 1) + I*sq rt(pi)*(erf(sqrt(I*(d*x + c)^(2/3)*b)) - 1))*cos(1/4*pi + 1/3*arctan2(0, d *x + c)) + (I*sqrt(pi)*(erf(sqrt(I*b*conjugate((d*x + c)^(2/3)))) - 1) - I *sqrt(pi)*(erf(sqrt(-I*(d*x + c)^(2/3)*b)) - 1))*cos(-1/4*pi + 1/3*arctan2 (0, d*x + c)) + (sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x + c)^(2/3)))) - 1) + sqrt(pi)*(erf(sqrt(I*(d*x + c)^(2/3)*b)) - 1))*sin(1/4*pi + 1/3*arctan2 (0, d*x + c)) - (sqrt(pi)*(erf(sqrt(I*b*conjugate((d*x + c)^(2/3)))) - 1) + sqrt(pi)*(erf(sqrt(-I*(d*x + c)^(2/3)*b)) - 1))*sin(-1/4*pi + 1/3*arctan 2(0, d*x + c)))*cos(a) + ((sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x + c)^(2/ 3)))) - 1) + sqrt(pi)*(erf(sqrt(I*(d*x + c)^(2/3)*b)) - 1))*cos(1/4*pi + 1 /3*arctan2(0, d*x + c)) + (sqrt(pi)*(erf(sqrt(I*b*conjugate((d*x + c)^(2/3 )))) - 1) + sqrt(pi)*(erf(sqrt(-I*(d*x + c)^(2/3)*b)) - 1))*cos(-1/4*pi + 1/3*arctan2(0, d*x + c)) + (I*sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x + c)^ (2/3)))) - 1) - I*sqrt(pi)*(erf(sqrt(I*(d*x + c)^(2/3)*b)) - 1))*sin(1/4*p i + 1/3*arctan2(0, d*x + c)) + (I*sqrt(pi)*(erf(sqrt(I*b*conjugate((d*x + c)^(2/3)))) - 1) - I*sqrt(pi)*(erf(sqrt(-I*(d*x + c)^(2/3)*b)) - 1))*sin(- 1/4*pi + 1/3*arctan2(0, d*x + c)))*sin(a))*sqrt((d*x + c)^(2/3)*b)/((d*x + c)^(1/3)*b*d*e^(2/3))
Exception generated. \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{2/3}} \,d x \]