\(\int \frac {\sin (a+b (c+d x)^{2/3})}{(c e+d e x)^{2/3}} \, dx\) [239]

Optimal result

Integrand size = 27, antiderivative size = 133 \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b} d (e (c+d x))^{2/3}}+\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{\sqrt {b} d (e (c+d x))^{2/3}} \] Output:

3/2*2^(1/2)*Pi^(1/2)*(d*x+c)^(2/3)*cos(a)*FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2 
)*(d*x+c)^(1/3))/b^(1/2)/d/(e*(d*x+c))^(2/3)+3/2*2^(1/2)*Pi^(1/2)*(d*x+c)^ 
(2/3)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/3))*sin(a)/b^(1/2)/d/(e 
*(d*x+c))^(2/3)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.72 \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \left (\cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+\operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)\right )}{\sqrt {b} d (e (c+d x))^{2/3}} \] Input:

Integrate[Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(2/3),x]
 

Output:

(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))
 

Rubi [A] (verified)

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

\(\displaystyle \frac {3 (c+d x)^{2/3} \left (\frac {\sqrt {\frac {\pi }{2}} \sin (a) \operatorname {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}+\frac {\sqrt {\frac {\pi }{2}} \cos (a) \operatorname {FresnelS}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt {b}}\right )}{d (e (c+d x))^{2/3}}\)

Input:

Int[Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(2/3),x]
 

Output:

(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))
 

Defintions of rubi rules used

rule 3832

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]
 

rule 3833

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]
 

rule 3834

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]
 

rule 3864

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]
 

rule 3898

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]
 

rule 3916

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]
 

Maple [F]

\[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{\left (d e x +c e \right )^{\frac {2}{3}}}d x\]

Input:

int(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(2/3),x)
 

Output:

int(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(2/3),x)
 

Fricas [F]

\[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\int { \frac {\sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac {2}{3}}} \,d x } \] Input:

integrate(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(2/3),x, algorithm="fricas")
 

Output:

integral(sin((d*x + c)^(2/3)*b + a)/(d*e*x + c*e)^(2/3), x)
 

Sympy [F]

\[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\int \frac {\sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {2}{3}}}\, dx \] Input:

integrate(sin(a+b*(d*x+c)**(2/3))/(d*e*x+c*e)**(2/3),x)
 

Output:

Integral(sin(a + b*(c + d*x)**(2/3))/(e*(c + d*x))**(2/3), x)
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.35 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.66 \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx =\text {Too large to display} \] Input:

integrate(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(2/3),x, algorithm="maxima")
 

Output:

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))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(2/3),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{2/3}} \,d x \] Input:

int(sin(a + b*(c + d*x)^(2/3))/(c*e + d*e*x)^(2/3),x)
 

Output:

int(sin(a + b*(c + d*x)^(2/3))/(c*e + d*e*x)^(2/3), x)
 

Reduce [F]

\[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{2/3}} \, dx=\frac {\int \frac {\sin \left (\left (d x +c \right )^{\frac {2}{3}} b +a \right )}{\left (d x +c \right )^{\frac {2}{3}}}d x}{e^{\frac {2}{3}}} \] Input:

int(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(2/3),x)
 

Output:

int(sin((c + d*x)**(2/3)*b + a)/(c + d*x)**(2/3),x)/e**(2/3)