\(\int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx\) [316]

Optimal result

Integrand size = 18, antiderivative size = 77 \[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx=-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{x}+b \cos (a) \operatorname {CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-b \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)} \text {Si}(b x) \] Output:

-(c*sin(b*x+a)^3)^(1/3)/x+b*cos(a)*Ci(b*x)*csc(b*x+a)*(c*sin(b*x+a)^3)^(1/ 
3)-b*csc(b*x+a)*sin(a)*(c*sin(b*x+a)^3)^(1/3)*Si(b*x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx=\frac {\sqrt [3]{c \sin ^3(a+b x)} (-1+b x \cos (a) \operatorname {CosIntegral}(b x) \csc (a+b x)-b x \csc (a+b x) \sin (a) \text {Si}(b x))}{x} \] Input:

Integrate[(c*Sin[a + b*x]^3)^(1/3)/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.66, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {7271, 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.

\(\displaystyle \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx\)

\(\Big \downarrow \) 7271

\(\displaystyle \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \int \frac {\sin (a+b x)}{x^2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \int \frac {\sin (a+b x)}{x^2}dx\)

\(\Big \downarrow \) 3778

\(\displaystyle \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \left (b \int \frac {\cos (a+b x)}{x}dx-\frac {\sin (a+b x)}{x}\right )\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

\(\displaystyle \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \left (b \left (\cos (a) \int \frac {\cos (b x)}{x}dx-\sin (a) \int \frac {\sin (b x)}{x}dx\right )-\frac {\sin (a+b x)}{x}\right )\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

Input:

Int[(c*Sin[a + b*x]^3)^(1/3)/x^2,x]
 

Output:

Csc[a + b*x]*(c*Sin[a + b*x]^3)^(1/3)*(-(Sin[a + b*x]/x) + b*(Cos[a]*CosIn 
tegral[b*x] - Sin[a]*SinIntegral[b*x]))
 

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 7271

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ 
FracPart[p]/v^(m*FracPart[p]))   Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, 
x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&  !(Eq 
Q[v, x] && EqQ[m, 1])
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.32

method	result	size
risch	\(\frac {i \left (i c \,{\mathrm e}^{-3 i \left (b x +a \right )} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}\right )^{\frac {1}{3}} \left (-\operatorname {expIntegral}_{1}\left (-i b x \right ) {\mathrm e}^{i \left (b x +2 a \right )} b x -{\mathrm e}^{i b x} \operatorname {expIntegral}_{1}\left (i b x \right ) b x +i {\mathrm e}^{2 i \left (b x +a \right )}-i\right )}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) x}\)	\(102\)

Input:

int((c*sin(b*x+a)^3)^(1/3)/x^2,x,method=_RETURNVERBOSE)
 

Output:

1/2*I*(I*c*exp(-3*I*(b*x+a))*(exp(2*I*(b*x+a))-1)^3)^(1/3)*(-Ei(1,-I*b*x)* 
exp(I*(b*x+2*a))*b*x-exp(I*b*x)*Ei(1,I*b*x)*b*x+I*exp(2*I*(b*x+a))-I)/(exp 
(2*I*(b*x+a))-1)/x
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx=\frac {{\left (b x {\rm Ei}\left (i \, b x\right ) e^{\left (i \, a\right )} + b x {\rm Ei}\left (-i \, b x\right ) e^{\left (-i \, a\right )} - 2 \, \sin \left (b x + a\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {1}{3}}}{2 \, x \sin \left (b x + a\right )} \] Input:

integrate((c*sin(b*x+a)^3)^(1/3)/x^2,x, algorithm="fricas")
 

Output:

1/2*(b*x*Ei(I*b*x)*e^(I*a) + b*x*Ei(-I*b*x)*e^(-I*a) - 2*sin(b*x + a))*(-( 
c*cos(b*x + a)^2 - c)*sin(b*x + a))^(1/3)/(x*sin(b*x + a))
 

Sympy [F]

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

integrate((c*sin(b*x+a)**3)**(1/3)/x**2,x)
 

Output:

Integral((c*sin(a + b*x)**3)**(1/3)/x**2, x)
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.97 \[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx=\frac {{\left ({\left ({\left (\sqrt {3} - i\right )} E_{2}\left (i \, b x\right ) + {\left (\sqrt {3} + i\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right )^{3} + {\left ({\left (\sqrt {3} - i\right )} E_{2}\left (i \, b x\right ) + {\left (\sqrt {3} + i\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right ) \sin \left (a\right )^{2} + {\left ({\left (-i \, \sqrt {3} - 1\right )} E_{2}\left (i \, b x\right ) + {\left (i \, \sqrt {3} - 1\right )} E_{2}\left (-i \, b x\right )\right )} \sin \left (a\right )^{3} - {\left ({\left (\sqrt {3} + i\right )} E_{2}\left (i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right ) + {\left ({\left ({\left (-i \, \sqrt {3} - 1\right )} E_{2}\left (i \, b x\right ) + {\left (i \, \sqrt {3} - 1\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right )^{2} + {\left (i \, \sqrt {3} - 1\right )} E_{2}\left (i \, b x\right ) + {\left (-i \, \sqrt {3} - 1\right )} E_{2}\left (-i \, b x\right )\right )} \sin \left (a\right )\right )} b c^{\frac {1}{3}}}{8 \, {\left (a \cos \left (a\right )^{2} + a \sin \left (a\right )^{2} - {\left (b x + a\right )} {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )}\right )}} \] Input:

integrate((c*sin(b*x+a)^3)^(1/3)/x^2,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

1/8*(((sqrt(3) - I)*exp_integral_e(2, I*b*x) + (sqrt(3) + I)*exp_integral_ 
e(2, -I*b*x))*cos(a)^3 + ((sqrt(3) - I)*exp_integral_e(2, I*b*x) + (sqrt(3 
) + I)*exp_integral_e(2, -I*b*x))*cos(a)*sin(a)^2 + ((-I*sqrt(3) - 1)*exp_ 
integral_e(2, I*b*x) + (I*sqrt(3) - 1)*exp_integral_e(2, -I*b*x))*sin(a)^3 
 - ((sqrt(3) + I)*exp_integral_e(2, I*b*x) + (sqrt(3) - I)*exp_integral_e( 
2, -I*b*x))*cos(a) + (((-I*sqrt(3) - 1)*exp_integral_e(2, I*b*x) + (I*sqrt 
(3) - 1)*exp_integral_e(2, -I*b*x))*cos(a)^2 + (I*sqrt(3) - 1)*exp_integra 
l_e(2, I*b*x) + (-I*sqrt(3) - 1)*exp_integral_e(2, -I*b*x))*sin(a))*b*c^(1 
/3)/(a*cos(a)^2 + a*sin(a)^2 - (b*x + a)*(cos(a)^2 + sin(a)^2))
 

Giac [F]

\[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )^{3}\right )^{\frac {1}{3}}}{x^{2}} \,d x } \] Input:

integrate((c*sin(b*x+a)^3)^(1/3)/x^2,x, algorithm="giac")
 

Output:

integrate((c*sin(b*x + a)^3)^(1/3)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx=\int \frac {{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{1/3}}{x^2} \,d x \] Input:

int((c*sin(a + b*x)^3)^(1/3)/x^2,x)
 

Output:

int((c*sin(a + b*x)^3)^(1/3)/x^2, x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx=c^{\frac {1}{3}} \left (\int \frac {\sin \left (b x +a \right )}{x^{2}}d x \right ) \] Input:

int((c*sin(b*x+a)^3)^(1/3)/x^2,x)
 

Output:

c**(1/3)*int(sin(a + b*x)/x**2,x)