\(\int \frac {\sin (a+\frac {b}{\sqrt [3]{c+d x}})}{\sqrt [3]{c e+d e x}} \, dx\) [243]

Optimal result

Integrand size = 27, antiderivative size = 168 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\frac {3 b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b^2 \sqrt [3]{c+d x} \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}}+\frac {3 b^2 \sqrt [3]{c+d x} \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d \sqrt [3]{e (c+d x)}} \] Output:

3/2*b*(d*x+c)^(2/3)*cos(a+b/(d*x+c)^(1/3))/d/(e*(d*x+c))^(1/3)+3/2*b^2*(d* 
x+c)^(1/3)*Ci(b/(d*x+c)^(1/3))*sin(a)/d/(e*(d*x+c))^(1/3)+3/2*(d*x+c)*sin( 
a+b/(d*x+c)^(1/3))/d/(e*(d*x+c))^(1/3)+3/2*b^2*(d*x+c)^(1/3)*cos(a)*Si(b/( 
d*x+c)^(1/3))/d/(e*(d*x+c))^(1/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.78 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\frac {3 \left (b (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b^2 \sqrt [3]{c+d x} \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)+c \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+d x \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+b^2 \sqrt [3]{c+d x} \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{2 d \sqrt [3]{e (c+d x)}} \] Input:

Integrate[Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(1/3),x]
 

Output:

(3*(b*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)] + b^2*(c + d*x)^(1/3)*Cos 
Integral[b/(c + d*x)^(1/3)]*Sin[a] + c*Sin[a + b/(c + d*x)^(1/3)] + d*x*Si 
n[a + b/(c + d*x)^(1/3)] + b^2*(c + d*x)^(1/3)*Cos[a]*SinIntegral[b/(c + d 
*x)^(1/3)]))/(2*d*(e*(c + d*x))^(1/3))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.70, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3912, 30, 3042, 3778, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783}

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.

\(\displaystyle \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c e+d e x}} \, dx\)

\(\Big \downarrow \) 3912

\(\displaystyle -\frac {3 \int \frac {(c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{e (c+d x)}}d\frac {1}{\sqrt [3]{c+d x}}}{d}\)

\(\Big \downarrow \) 30

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \int (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \int (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {1}{2} b \int (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {1}{2} b \int (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {1}{2} b \left (b \int -\sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {1}{2} b \left (-b \int \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {1}{2} b \left (-b \int \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \sqrt [3]{c+d x} \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}+\cos (a) \int \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}\right )-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt [3]{c+d x}}+\cos (a) \int \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )d\frac {1}{\sqrt [3]{c+d x}}\right )-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 3780

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt [3]{c+d x}}+\cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{e (c+d x)}}\)

\(\Big \downarrow \) 3783

\(\displaystyle -\frac {3 \sqrt [3]{c+d x} \left (\frac {1}{2} b \left (-b \left (\sin (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+\cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )-\frac {1}{2} (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{d \sqrt [3]{e (c+d x)}}\)

Input:

Int[Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(1/3),x]
 

Output:

(-3*(c + d*x)^(1/3)*(-1/2*((c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)]) + ( 
b*(-((c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)]) - b*(CosIntegral[b/(c + d 
*x)^(1/3)]*Sin[a] + Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])))/2))/(d*(e*(c 
+ d*x))^(1/3))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c 
 + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1))   Int[( 
c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 
1]
 

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 3912

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f 
_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/(n*f)   Subst[Int[ExpandIntegra 
nd[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m, x], 
 x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p 
, 0] && IntegerQ[1/n]
 

Maple [F]

\[\int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d e x +c e \right )^{\frac {1}{3}}}d x\]

Input:

int(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(1/3),x)
 

Output:

int(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(1/3),x)
 

Fricas [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(1/3),x, algorithm="fricas")
 

Output:

integral(sin((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x + c))/(d*e*x + c*e)^(1 
/3), x)
 

Sympy [F]

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

integrate(sin(a+b/(d*x+c)**(1/3))/(d*e*x+c*e)**(1/3),x)
 

Output:

Integral(sin(a + b/(c + d*x)**(1/3))/(e*(c + d*x))**(1/3), x)
 

Maxima [C] (verification not implemented)

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

Time = 0.24 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.02 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c e+d e x}} \, dx=-\frac {3 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} {\left ({\left (-i \, \Gamma \left (-1, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + i \, \Gamma \left (-1, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) - i \, \Gamma \left (-1, \frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (-1, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (-1, i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + \Gamma \left (-1, -i \, b \overline {\frac {1}{{\left (d x + c\right )}^{\frac {1}{3}}}}\right ) + \Gamma \left (-1, \frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-1, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \left (a\right )\right )} b^{2} - 4 \, {\left (d x + c\right )} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} d e^{\frac {1}{3}}} \] Input:

integrate(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(1/3),x, algorithm="maxima")
 

Output:

-3/8*((d*x + c)^(1/3)*((-I*gamma(-1, I*b*conjugate((d*x + c)^(-1/3))) + I* 
gamma(-1, -I*b*conjugate((d*x + c)^(-1/3))) - I*gamma(-1, I*b/(d*x + c)^(1 
/3)) + I*gamma(-1, -I*b/(d*x + c)^(1/3)))*cos(a) - (gamma(-1, I*b*conjugat 
e((d*x + c)^(-1/3))) + gamma(-1, -I*b*conjugate((d*x + c)^(-1/3))) + gamma 
(-1, I*b/(d*x + c)^(1/3)) + gamma(-1, -I*b/(d*x + c)^(1/3)))*sin(a))*b^2 - 
 4*(d*x + c)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)))/((d*x + c)^(1/3 
)*d*e^(1/3))
 

Giac [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(1/3),x, algorithm="giac")
 

Output:

integrate(sin(a + b/(d*x + c)^(1/3))/(d*e*x + c*e)^(1/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{1/3}} \,d x \] Input:

int(sin(a + b/(c + d*x)^(1/3))/(c*e + d*e*x)^(1/3),x)
 

Output:

int(sin(a + b/(c + d*x)^(1/3))/(c*e + d*e*x)^(1/3), x)
 

Reduce [F]

\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{\sqrt [3]{c e+d e x}} \, dx=\frac {\int \frac {\sin \left (\frac {\left (d x +c \right )^{\frac {1}{3}} a +b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d x +c \right )^{\frac {1}{3}}}d x}{e^{\frac {1}{3}}} \] Input:

int(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(1/3),x)
 

Output:

int(sin(((c + d*x)**(1/3)*a + b)/(c + d*x)**(1/3))/(c + d*x)**(1/3),x)/e** 
(1/3)