\(\int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^3} \, dx\) [317]

Optimal result

Integrand size = 18, antiderivative size = 116 \[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^3} \, dx=-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{2 x^2}-\frac {b \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 x}-\frac {1}{2} b^2 \operatorname {CosIntegral}(b x) \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}-\frac {1}{2} b^2 \cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \text {Si}(b x) \] Output:

-1/2*(c*sin(b*x+a)^3)^(1/3)/x^2-1/2*b*cot(b*x+a)*(c*sin(b*x+a)^3)^(1/3)/x- 
1/2*b^2*Ci(b*x)*csc(b*x+a)*sin(a)*(c*sin(b*x+a)^3)^(1/3)-1/2*b^2*cos(a)*cs 
c(b*x+a)*(c*sin(b*x+a)^3)^(1/3)*Si(b*x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^3} \, dx=-\frac {\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \left (b x \cos (a+b x)+b^2 x^2 \operatorname {CosIntegral}(b x) \sin (a)+\sin (a+b x)+b^2 x^2 \cos (a) \text {Si}(b x)\right )}{2 x^2} \] Input:

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

Output:

-1/2*(Csc[a + b*x]*(c*Sin[a + b*x]^3)^(1/3)*(b*x*Cos[a + b*x] + b^2*x^2*Co 
sIntegral[b*x]*Sin[a] + Sin[a + b*x] + b^2*x^2*Cos[a]*SinIntegral[b*x]))/x 
^2
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.60, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {7271, 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.

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

\(\Big \downarrow \) 7271

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 3783

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

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 = 1.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06

method	result	size
risch	\(-\frac {\left (i c \,{\mathrm e}^{-3 i \left (b x +a \right )} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3}\right )^{\frac {1}{3}} \left ({\mathrm e}^{i b x} \operatorname {expIntegral}_{1}\left (i b x \right ) x^{2} b^{2}-\operatorname {expIntegral}_{1}\left (-i b x \right ) {\mathrm e}^{i \left (b x +2 a \right )} x^{2} b^{2}+i {\mathrm e}^{2 i \left (b x +a \right )} x b +i b x +{\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) x^{2}}\)	\(123\)

Input:

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

Output:

-1/4*(I*c*exp(-3*I*(b*x+a))*(exp(2*I*(b*x+a))-1)^3)^(1/3)*(exp(I*b*x)*Ei(1 
,I*b*x)*x^2*b^2-Ei(1,-I*b*x)*exp(I*(b*x+2*a))*x^2*b^2+I*exp(2*I*(b*x+a))*x 
*b+I*b*x+exp(2*I*(b*x+a))-1)/(exp(2*I*(b*x+a))-1)/x^2
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^3} \, dx=\frac {{\left (i \, b^{2} x^{2} {\rm Ei}\left (i \, b x\right ) e^{\left (i \, a\right )} - i \, b^{2} x^{2} {\rm Ei}\left (-i \, b x\right ) e^{\left (-i \, a\right )} - 2 \, b x \cos \left (b x + a\right ) - 2 \, \sin \left (b x + a\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {1}{3}}}{4 \, x^{2} \sin \left (b x + a\right )} \] Input:

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

Output:

1/4*(I*b^2*x^2*Ei(I*b*x)*e^(I*a) - I*b^2*x^2*Ei(-I*b*x)*e^(-I*a) - 2*b*x*c 
os(b*x + a) - 2*sin(b*x + a))*(-(c*cos(b*x + a)^2 - c)*sin(b*x + a))^(1/3) 
/(x^2*sin(b*x + a))
 

Sympy [F]

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

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

Output:

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^3} \, dx=-\frac {{\left ({\left ({\left (\sqrt {3} - i\right )} E_{3}\left (i \, b x\right ) + {\left (\sqrt {3} + i\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right )^{3} + {\left ({\left (\sqrt {3} - i\right )} E_{3}\left (i \, b x\right ) + {\left (\sqrt {3} + i\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right ) \sin \left (a\right )^{2} + {\left ({\left (-i \, \sqrt {3} - 1\right )} E_{3}\left (i \, b x\right ) + {\left (i \, \sqrt {3} - 1\right )} E_{3}\left (-i \, b x\right )\right )} \sin \left (a\right )^{3} - {\left ({\left (\sqrt {3} + i\right )} E_{3}\left (i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right ) + {\left ({\left ({\left (-i \, \sqrt {3} - 1\right )} E_{3}\left (i \, b x\right ) + {\left (i \, \sqrt {3} - 1\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right )^{2} + {\left (i \, \sqrt {3} - 1\right )} E_{3}\left (i \, b x\right ) + {\left (-i \, \sqrt {3} - 1\right )} E_{3}\left (-i \, b x\right )\right )} \sin \left (a\right )\right )} b^{2} c^{\frac {1}{3}}}{8 \, {\left (a^{2} \cos \left (a\right )^{2} + a^{2} \sin \left (a\right )^{2} + {\left (b x + a\right )}^{2} {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} - 2 \, {\left (a \cos \left (a\right )^{2} + a \sin \left (a\right )^{2}\right )} {\left (b x + a\right )}\right )}} \] Input:

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

Output:

-1/8*(((sqrt(3) - I)*exp_integral_e(3, I*b*x) + (sqrt(3) + I)*exp_integral 
_e(3, -I*b*x))*cos(a)^3 + ((sqrt(3) - I)*exp_integral_e(3, I*b*x) + (sqrt( 
3) + I)*exp_integral_e(3, -I*b*x))*cos(a)*sin(a)^2 + ((-I*sqrt(3) - 1)*exp 
_integral_e(3, I*b*x) + (I*sqrt(3) - 1)*exp_integral_e(3, -I*b*x))*sin(a)^ 
3 - ((sqrt(3) + I)*exp_integral_e(3, I*b*x) + (sqrt(3) - I)*exp_integral_e 
(3, -I*b*x))*cos(a) + (((-I*sqrt(3) - 1)*exp_integral_e(3, I*b*x) + (I*sqr 
t(3) - 1)*exp_integral_e(3, -I*b*x))*cos(a)^2 + (I*sqrt(3) - 1)*exp_integr 
al_e(3, I*b*x) + (-I*sqrt(3) - 1)*exp_integral_e(3, -I*b*x))*sin(a))*b^2*c 
^(1/3)/(a^2*cos(a)^2 + a^2*sin(a)^2 + (b*x + a)^2*(cos(a)^2 + sin(a)^2) - 
2*(a*cos(a)^2 + a*sin(a)^2)*(b*x + a))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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