\(\int \frac {\sin (a+b (c+d x)^{2/3})}{(c e+d e x)^{5/3}} \, dx\) [241]

Optimal result

Integrand size = 27, antiderivative size = 126 \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx=\frac {3 b (c+d x)^{2/3} \cos (a) \operatorname {CosIntegral}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac {3 b (c+d x)^{2/3} \sin (a) \text {Si}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69 \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx=-\frac {3 \left (-b (c+d x)^{2/3} \cos (a) \operatorname {CosIntegral}\left (b (c+d x)^{2/3}\right )+\sin \left (a+b (c+d x)^{2/3}\right )+b (c+d x)^{2/3} \sin (a) \text {Si}\left (b (c+d x)^{2/3}\right )\right )}{2 d e (e (c+d x))^{2/3}} \] Input:

Integrate[Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(5/3),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3916, 3862, 3860, 3042, 3778, 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.

\(\displaystyle \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx\)

\(\Big \downarrow \) 3916

\(\displaystyle \frac {\int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(e (c+d x))^{5/3}}d(c+d x)}{d}\)

\(\Big \downarrow \) 3862

\(\displaystyle \frac {(c+d x)^{2/3} \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c+d x)^{5/3}}d(c+d x)}{d e (e (c+d x))^{2/3}}\)

\(\Big \downarrow \) 3860

\(\displaystyle \frac {3 (c+d x)^{2/3} \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c+d x)^{4/3}}d(c+d x)^{2/3}}{2 d e (e (c+d x))^{2/3}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 (c+d x)^{2/3} \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c+d x)^{4/3}}d(c+d x)^{2/3}}{2 d e (e (c+d x))^{2/3}}\)

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

Input:

Int[Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(5/3),x]
 

Output:

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

Defintions of rubi rules used

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 3860

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ 
(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ 
(m + 1)/n], 0]))
 

rule 3862

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, n, p}, x] && Int 
egerQ[Simplify[(m + 1)/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]

\[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{\left (d e x +c e \right )^{\frac {5}{3}}}d x\]

Input:

int(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(5/3),x)
 

Output:

int(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(5/3),x)
 

Fricas [F]

\[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx=\int { \frac {\sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac {5}{3}}} \,d x } \] Input:

integrate(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(5/3),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx=\int \frac {\sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{3}}}\, dx \] Input:

integrate(sin(a+b*(d*x+c)**(2/3))/(d*e*x+c*e)**(5/3),x)
 

Output:

Integral(sin(a + b*(c + d*x)**(2/3))/(e*(c + d*x))**(5/3), x)
 

Maxima [C] (verification not implemented)

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

Time = 0.23 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00 \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx=\frac {3 \, {\left ({\left (\Gamma \left (-1, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-1, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-1, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right ) + \Gamma \left (-1, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (a\right ) + {\left (-i \, \Gamma \left (-1, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + i \, \Gamma \left (-1, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - i \, \Gamma \left (-1, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right ) + i \, \Gamma \left (-1, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (a\right )\right )} b}{8 \, d e^{\frac {5}{3}}} \] Input:

integrate(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(5/3),x, algorithm="maxima")
 

Output:

3/8*((gamma(-1, I*b*conjugate((d*x + c)^(2/3))) + gamma(-1, -I*b*conjugate 
((d*x + c)^(2/3))) + gamma(-1, I*(d*x + c)^(2/3)*b) + gamma(-1, -I*(d*x + 
c)^(2/3)*b))*cos(a) + (-I*gamma(-1, I*b*conjugate((d*x + c)^(2/3))) + I*ga 
mma(-1, -I*b*conjugate((d*x + c)^(2/3))) - I*gamma(-1, I*(d*x + c)^(2/3)*b 
) + I*gamma(-1, -I*(d*x + c)^(2/3)*b))*sin(a))*b/(d*e^(5/3))
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(5/3),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. \[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx=\int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{5/3}} \,d x \] Input:

int(sin(a + b*(c + d*x)^(2/3))/(c*e + d*e*x)^(5/3),x)
 

Output:

int(sin(a + b*(c + d*x)^(2/3))/(c*e + d*e*x)^(5/3), x)
 

Reduce [F]

\[ \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{5/3}} \, dx=\frac {\int \frac {\sin \left (\left (d x +c \right )^{\frac {2}{3}} b +a \right )}{\left (d x +c \right )^{\frac {2}{3}} c +\left (d x +c \right )^{\frac {2}{3}} d x}d x}{e^{\frac {5}{3}}} \] Input:

int(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(5/3),x)
 

Output:

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