3.341 \(\int \frac{(c \sin ^3(a+b x))^{2/3}}{x^3} \, dx\)

Optimal. Leaf size=119 \[ b^2 \cos (2 a) \text{CosIntegral}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-b^2 \sin (2 a) \text{Si}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac{\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac{b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x} \]

[Out]

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

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Rubi [A]  time = 0.230095, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {6720, 3314, 29, 3312, 3303, 3299, 3302} \[ b^2 \cos (2 a) \text{CosIntegral}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-b^2 \sin (2 a) \text{Si}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac{\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac{b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x]^3)^(2/3)/x^3,x]

[Out]

-(c*Sin[a + b*x]^3)^(2/3)/(2*x^2) - (b*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(2/3))/x + b^2*Cos[2*a]*CosIntegral[2*b
*x]*Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3) - b^2*Csc[a + b*x]^2*Sin[2*a]*(c*Sin[a + b*x]^3)^(2/3)*SinIntegral
[2*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 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && 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{\left (c \sin ^3(a+b x)\right )^{2/3}}{x^3} \, dx &=\left (\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac{\sin ^2(a+b x)}{x^3} \, dx\\ &=-\frac{\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac{b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+\left (b^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac{1}{x} \, dx-\left (2 b^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac{\sin ^2(a+b x)}{x} \, dx\\ &=-\frac{\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac{b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+b^2 \csc ^2(a+b x) \log (x) \left (c \sin ^3(a+b x)\right )^{2/3}-\left (2 b^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \left (\frac{1}{2 x}-\frac{\cos (2 a+2 b x)}{2 x}\right ) \, dx\\ &=-\frac{\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac{b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+\left (b^2 \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac{\cos (2 a+2 b x)}{x} \, dx\\ &=-\frac{\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac{b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+\left (b^2 \cos (2 a) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac{\cos (2 b x)}{x} \, dx-\left (b^2 \csc ^2(a+b x) \sin (2 a) \left (c \sin ^3(a+b x)\right )^{2/3}\right ) \int \frac{\sin (2 b x)}{x} \, dx\\ &=-\frac{\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac{b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+b^2 \cos (2 a) \text{Ci}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-b^2 \csc ^2(a+b x) \sin (2 a) \left (c \sin ^3(a+b x)\right )^{2/3} \text{Si}(2 b x)\\ \end{align*}

Mathematica [A]  time = 0.203367, size = 85, normalized size = 0.71 \[ \frac{\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (4 b^2 x^2 \cos (2 a) \text{CosIntegral}(2 b x)-4 b^2 x^2 \sin (2 a) \text{Si}(2 b x)-2 b x \sin (2 (a+b x))+\cos (2 (a+b x))-1\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3)*(-1 + Cos[2*(a + b*x)] + 4*b^2*x^2*Cos[2*a]*CosIntegral[2*b*x] - 2*b*
x*Sin[2*(a + b*x)] - 4*b^2*x^2*Sin[2*a]*SinIntegral[2*b*x]))/(4*x^2)

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Maple [C]  time = 0.088, size = 238, normalized size = 2. \begin{align*} -{\frac{{b}^{2}}{4\, \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }} \right ) ^{{\frac{2}{3}}} \left ({\frac{1}{2\,{x}^{2}{b}^{2}}}-{\frac{i}{bx}}-2\,{{\rm e}^{2\,ibx}}{\it Ei} \left ( 1,2\,ibx \right ) \right ) }-{\frac{{b}^{2}}{4\, \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }} \right ) ^{{\frac{2}{3}}} \left ({\frac{{{\rm e}^{4\,i \left ( bx+a \right ) }}}{2\,{x}^{2}{b}^{2}}}+{\frac{i{{\rm e}^{4\,i \left ( bx+a \right ) }}}{bx}}-2\,{\it Ei} \left ( 1,-2\,ibx \right ){{\rm e}^{2\,i \left ( bx+2\,a \right ) }} \right ) }+{\frac{{{\rm e}^{2\,i \left ( bx+a \right ) }}}{4\, \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{2}{x}^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }} \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.70297, size = 420, normalized size = 3.53 \begin{align*} -\frac{{\left (128 \,{\left ({\left (-i \, \sqrt{3} + 1\right )} E_{3}\left (2 i \, b x\right ) +{\left (i \, \sqrt{3} + 1\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{3} -{\left ({\left (128 \, \sqrt{3} + 128 i\right )} E_{3}\left (2 i \, b x\right ) +{\left (128 \, \sqrt{3} - 128 i\right )} E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + 128 \,{\left ({\left ({\left (-i \, \sqrt{3} + 1\right )} E_{3}\left (2 i \, b x\right ) +{\left (i \, \sqrt{3} + 1\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 2\right )} \sin \left (2 \, a\right )^{2} + 128 \,{\left ({\left (i \, \sqrt{3} + 1\right )} E_{3}\left (2 i \, b x\right ) +{\left (-i \, \sqrt{3} + 1\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 256 \, \cos \left (2 \, a\right )^{2} -{\left ({\left ({\left (128 \, \sqrt{3} + 128 i\right )} E_{3}\left (2 i \, b x\right ) +{\left (128 \, \sqrt{3} - 128 i\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} -{\left (128 \, \sqrt{3} - 128 i\right )} E_{3}\left (2 i \, b x\right ) -{\left (128 \, \sqrt{3} + 128 i\right )} E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} b^{2} c^{\frac{2}{3}}}{2048 \,{\left (a^{2} \cos \left (2 \, a\right )^{2} + a^{2} \sin \left (2 \, a\right )^{2} +{\left (b x + a\right )}^{2}{\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} - 2 \,{\left (a \cos \left (2 \, a\right )^{2} + a \sin \left (2 \, 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)^(2/3)/x^3,x, algorithm="maxima")

[Out]

-1/2048*(128*((-I*sqrt(3) + 1)*exp_integral_e(3, 2*I*b*x) + (I*sqrt(3) + 1)*exp_integral_e(3, -2*I*b*x))*cos(2
*a)^3 - ((128*sqrt(3) + 128*I)*exp_integral_e(3, 2*I*b*x) + (128*sqrt(3) - 128*I)*exp_integral_e(3, -2*I*b*x))
*sin(2*a)^3 + 128*(((-I*sqrt(3) + 1)*exp_integral_e(3, 2*I*b*x) + (I*sqrt(3) + 1)*exp_integral_e(3, -2*I*b*x))
*cos(2*a) - 2)*sin(2*a)^2 + 128*((I*sqrt(3) + 1)*exp_integral_e(3, 2*I*b*x) + (-I*sqrt(3) + 1)*exp_integral_e(
3, -2*I*b*x))*cos(2*a) - 256*cos(2*a)^2 - (((128*sqrt(3) + 128*I)*exp_integral_e(3, 2*I*b*x) + (128*sqrt(3) -
128*I)*exp_integral_e(3, -2*I*b*x))*cos(2*a)^2 - (128*sqrt(3) - 128*I)*exp_integral_e(3, 2*I*b*x) - (128*sqrt(
3) + 128*I)*exp_integral_e(3, -2*I*b*x))*sin(2*a))*b^2*c^(2/3)/(a^2*cos(2*a)^2 + a^2*sin(2*a)^2 + (b*x + a)^2*
(cos(2*a)^2 + sin(2*a)^2) - 2*(a*cos(2*a)^2 + a*sin(2*a)^2)*(b*x + a))

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Fricas [A]  time = 1.84185, size = 404, normalized size = 3.39 \begin{align*} \frac{4^{\frac{2}{3}}{\left (2 \cdot 4^{\frac{1}{3}} b^{2} x^{2} \sin \left (2 \, a\right ) \operatorname{Si}\left (2 \, b x\right ) + 2 \cdot 4^{\frac{1}{3}} b x \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 4^{\frac{1}{3}} \cos \left (b x + a\right )^{2} -{\left (4^{\frac{1}{3}} b^{2} x^{2} \operatorname{Ci}\left (2 \, b x\right ) + 4^{\frac{1}{3}} b^{2} x^{2} \operatorname{Ci}\left (-2 \, b x\right )\right )} \cos \left (2 \, a\right ) + 4^{\frac{1}{3}}\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac{2}{3}}}{8 \,{\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)^(2/3)/x^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*sin(a + b*x)**3)**(2/3)/x**3, 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{2}{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)^(2/3)/x^3,x, algorithm="giac")

[Out]

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