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

Optimal. Leaf size=116 \[ -\frac{1}{2} b^2 \sin (a) \text{CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-\frac{1}{2} b^2 \cos (a) \text{Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-\frac{\sqrt [3]{c \sin ^3(a+b x)}}{2 x^2}-\frac{b \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.205576, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6720, 3297, 3303, 3299, 3302} \[ -\frac{1}{2} b^2 \sin (a) \text{CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-\frac{1}{2} b^2 \cos (a) \text{Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-\frac{\sqrt [3]{c \sin ^3(a+b x)}}{2 x^2}-\frac{b \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x]^3)^(1/3)/x^3,x]

[Out]

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

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(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]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 3297

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] - Dist[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 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[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 3299

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

Rule 3302

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

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{c \sin ^3(a+b x)}}{x^3} \, dx &=\left (\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac{\sin (a+b x)}{x^3} \, dx\\ &=-\frac{\sqrt [3]{c \sin ^3(a+b x)}}{2 x^2}+\frac{1}{2} \left (b \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac{\cos (a+b x)}{x^2} \, 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} \left (b^2 \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac{\sin (a+b x)}{x} \, 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} \left (b^2 \cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac{\sin (b x)}{x} \, dx-\frac{1}{2} \left (b^2 \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac{\cos (b x)}{x} \, 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 \text{Ci}(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)\\ \end{align*}

Mathematica [A]  time = 0.145025, size = 69, normalized size = 0.59 \[ -\frac{\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \left (b^2 x^2 \sin (a) \text{CosIntegral}(b x)+b^2 x^2 \cos (a) \text{Si}(b x)+\sin (a+b x)+b x \cos (a+b x)\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.082, size = 183, normalized size = 1.6 \begin{align*} -{\frac{{b}^{2}}{2\,{{\rm e}^{2\,i \left ( bx+a \right ) }}-2}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }}} \left ({\frac{{{\rm e}^{2\,i \left ( bx+a \right ) }}}{2\,{x}^{2}{b}^{2}}}+{\frac{{\frac{i}{2}}{{\rm e}^{2\,i \left ( bx+a \right ) }}}{bx}}-{\frac{{\it Ei} \left ( 1,-ibx \right ){{\rm e}^{i \left ( bx+2\,a \right ) }}}{2}} \right ) }+{\frac{{b}^{2}}{2\,{{\rm e}^{2\,i \left ( bx+a \right ) }}-2}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }}} \left ({\frac{1}{2\,{x}^{2}{b}^{2}}}-{\frac{{\frac{i}{2}}}{bx}}-{\frac{{{\rm e}^{ibx}}{\it Ei} \left ( 1,ibx \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [C]  time = 1.67445, size = 365, normalized size = 3.15 \begin{align*} -\frac{{\left ({\left ({\left (8 \, \sqrt{3} - 8 i\right )} E_{3}\left (i \, b x\right ) +{\left (8 \, \sqrt{3} + 8 i\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right )^{3} +{\left ({\left (8 \, \sqrt{3} - 8 i\right )} E_{3}\left (i \, b x\right ) +{\left (8 \, \sqrt{3} + 8 i\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right ) \sin \left (a\right )^{2} + 8 \,{\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 (8 \, \sqrt{3} + 8 i\right )} E_{3}\left (i \, b x\right ) +{\left (8 \, \sqrt{3} - 8 i\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right ) + 8 \,{\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}}}{64 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(((8*sqrt(3) - 8*I)*exp_integral_e(3, I*b*x) + (8*sqrt(3) + 8*I)*exp_integral_e(3, -I*b*x))*cos(a)^3 + (
(8*sqrt(3) - 8*I)*exp_integral_e(3, I*b*x) + (8*sqrt(3) + 8*I)*exp_integral_e(3, -I*b*x))*cos(a)*sin(a)^2 + 8*
((-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 - ((8*sqrt(3)
+ 8*I)*exp_integral_e(3, I*b*x) + (8*sqrt(3) - 8*I)*exp_integral_e(3, -I*b*x))*cos(a) + 8*(((-I*sqrt(3) - 1)*
exp_integral_e(3, I*b*x) + (I*sqrt(3) - 1)*exp_integral_e(3, -I*b*x))*cos(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))*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))

________________________________________________________________________________________

Fricas [A]  time = 1.69489, size = 401, normalized size = 3.46 \begin{align*} -\frac{4^{\frac{1}{3}}{\left (2 \cdot 4^{\frac{2}{3}} \cos \left (b x + a\right )^{2} -{\left (2 \cdot 4^{\frac{2}{3}} b^{2} x^{2} \cos \left (a\right ) \operatorname{Si}\left (b x\right ) + 2 \cdot 4^{\frac{2}{3}} b x \cos \left (b x + a\right ) +{\left (4^{\frac{2}{3}} b^{2} x^{2} \operatorname{Ci}\left (b x\right ) + 4^{\frac{2}{3}} b^{2} x^{2} \operatorname{Ci}\left (-b x\right )\right )} \sin \left (a\right )\right )} \sin \left (b x + a\right ) - 2 \cdot 4^{\frac{2}{3}}\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac{1}{3}}}{16 \,{\left (x^{2} \cos \left (b x + a\right )^{2} - x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*4^(1/3)*(2*4^(2/3)*cos(b*x + a)^2 - (2*4^(2/3)*b^2*x^2*cos(a)*sin_integral(b*x) + 2*4^(2/3)*b*x*cos(b*x
+ a) + (4^(2/3)*b^2*x^2*cos_integral(b*x) + 4^(2/3)*b^2*x^2*cos_integral(-b*x))*sin(a))*sin(b*x + a) - 2*4^(2/
3))*(-(c*cos(b*x + a)^2 - c)*sin(b*x + a))^(1/3)/(x^2*cos(b*x + a)^2 - x^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{c \sin ^{3}{\left (a + b x \right )}}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \sin \left (b x + a\right )^{3}\right )^{\frac{1}{3}}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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