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

Optimal. Leaf size=77 \[ b \cos (a) \text{CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-b \sin (a) \text{Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-\frac{\sqrt [3]{c \sin ^3(a+b x)}}{x} \]

[Out]

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

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Rubi [A]  time = 0.176734, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ b \cos (a) \text{CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-b \sin (a) \text{Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-\frac{\sqrt [3]{c \sin ^3(a+b x)}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x]^3)^(1/3)/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.179418, size = 51, normalized size = 0.66 \[ \frac{\sqrt [3]{c \sin ^3(a+b x)} (b x \cos (a) \text{CosIntegral}(b x) \csc (a+b x)-b x \sin (a) \text{Si}(b x) \csc (a+b x)-1)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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

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Maple [C]  time = 0.083, size = 155, normalized size = 2. \begin{align*}{\frac{{\frac{i}{2}}b}{{{\rm e}^{2\,i \left ( bx+a \right ) }}-1}\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{i{{\rm e}^{2\,i \left ( bx+a \right ) }}}{bx}}-{\it Ei} \left ( 1,-ibx \right ){{\rm e}^{i \left ( bx+2\,a \right ) }} \right ) }-{\frac{{\frac{i}{2}}b}{{{\rm e}^{2\,i \left ( bx+a \right ) }}-1}\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{i}{bx}}+{{\rm e}^{ibx}}{\it Ei} \left ( 1,ibx \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.66097, size = 328, normalized size = 4.26 \begin{align*} \frac{{\left ({\left ({\left (8 \, \sqrt{3} - 8 i\right )} E_{2}\left (i \, b x\right ) +{\left (8 \, \sqrt{3} + 8 i\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right )^{3} +{\left ({\left (8 \, \sqrt{3} - 8 i\right )} E_{2}\left (i \, b x\right ) +{\left (8 \, \sqrt{3} + 8 i\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right ) \sin \left (a\right )^{2} + 8 \,{\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 (8 \, \sqrt{3} + 8 i\right )} E_{2}\left (i \, b x\right ) +{\left (8 \, \sqrt{3} - 8 i\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right ) + 8 \,{\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}}}{64 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.79112, size = 339, normalized size = 4.4 \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 x \sin \left (a\right ) \operatorname{Si}\left (b x\right ) -{\left (4^{\frac{2}{3}} b x \operatorname{Ci}\left (b x\right ) + 4^{\frac{2}{3}} b x \operatorname{Ci}\left (-b x\right )\right )} \cos \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}}}{8 \,{\left (x \cos \left (b x + a\right )^{2} - x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*4^(1/3)*(2*4^(2/3)*cos(b*x + a)^2 - (2*4^(2/3)*b*x*sin(a)*sin_integral(b*x) - (4^(2/3)*b*x*cos_integral(b
*x) + 4^(2/3)*b*x*cos_integral(-b*x))*cos(a))*sin(b*x + a) - 2*4^(2/3))*(-(c*cos(b*x + a)^2 - c)*sin(b*x + a))
^(1/3)/(x*cos(b*x + a)^2 - x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{c \sin ^{3}{\left (a + b x \right )}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{2}}\,{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^2,x, algorithm="giac")

[Out]

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