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

Optimal. Leaf size=161 \[ \frac{1}{2} b \sin (2 a) \text{CosIntegral}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{1}{2} b \cos (2 a) \text{Si}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac{\cos \left (2 \left (a+b x^2\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2} \]

[Out]

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

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Rubi [A]  time = 0.214949, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {6720, 3403, 3380, 3297, 3303, 3299, 3302} \[ \frac{1}{2} b \sin (2 a) \text{CosIntegral}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{1}{2} b \cos (2 a) \text{Si}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac{\cos \left (2 \left (a+b x^2\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^2]^3)^(2/3)/x^3,x]

[Out]

-(Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/(4*x^2) + (Cos[2*(a + b*x^2)]*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2
]^3)^(2/3))/(4*x^2) + (b*CosIntegral[2*b*x^2]*Csc[a + b*x^2]^2*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3))/2 + (b*Cos
[2*a]*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3)*SinIntegral[2*b*x^2])/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 3403

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rule 3380

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

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{\left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{x^3} \, dx &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \frac{\sin ^2\left (a+b x^2\right )}{x^3} \, dx\\ &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \left (\frac{1}{2 x^3}-\frac{\cos \left (2 a+2 b x^2\right )}{2 x^3}\right ) \, dx\\ &=-\frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}-\frac{1}{2} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \frac{\cos \left (2 a+2 b x^2\right )}{x^3} \, dx\\ &=-\frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}-\frac{1}{4} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\cos (2 a+2 b x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac{\cos \left (2 \left (a+b x^2\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac{1}{2} \left (b \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\sin (2 a+2 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac{\cos \left (2 \left (a+b x^2\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac{1}{2} \left (b \cos (2 a) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\sin (2 b x)}{x} \, dx,x,x^2\right )+\frac{1}{2} \left (b \csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\cos (2 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac{\cos \left (2 \left (a+b x^2\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac{1}{2} b \text{Ci}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{1}{2} b \cos (2 a) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \text{Si}\left (2 b x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.145008, size = 79, normalized size = 0.49 \[ \frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (2 b x^2 \sin (2 a) \text{CosIntegral}\left (2 b x^2\right )+2 b x^2 \cos (2 a) \text{Si}\left (2 b x^2\right )+\cos \left (2 \left (a+b x^2\right )\right )-1\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.095, size = 277, normalized size = 1.7 \begin{align*} -{\frac{1}{8\, \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}{x}^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}}+{\frac{{\frac{i}{4}}{{\rm e}^{2\,ib{x}^{2}}}b{\it Ei} \left ( 1,2\,ib{x}^{2} \right ) }{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}}+{\frac{1}{4\, \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}} \left ( -{\frac{{{\rm e}^{4\,i \left ( b{x}^{2}+a \right ) }}}{2\,{x}^{2}}}-ib{\it Ei} \left ( 1,-2\,ib{x}^{2} \right ){{\rm e}^{2\,i \left ( b{x}^{2}+2\,a \right ) }} \right ) }+{\frac{{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{4\, \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}{x}^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(2/3)/(exp(2*I*(b*x^2+a))-1)^2/x^2+1/4*I*(I*c*(exp(2*I
*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(2/3)/(exp(2*I*(b*x^2+a))-1)^2*exp(2*I*b*x^2)*b*Ei(1,2*I*b*x^2)+1/4*(I*c
*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(2/3)/(exp(2*I*(b*x^2+a))-1)^2*(-1/2/x^2*exp(4*I*(b*x^2+a))-I*b
*Ei(1,-2*I*b*x^2)*exp(2*I*(b*x^2+2*a)))+1/4*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(2/3)/(exp(2*I*
(b*x^2+a))-1)^2/x^2*exp(2*I*(b*x^2+a))

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Maxima [C]  time = 1.69055, size = 86, normalized size = 0.53 \begin{align*} -\frac{{\left ({\left ({\left (i \, \Gamma \left (-1, 2 i \, b x^{2}\right ) - i \, \Gamma \left (-1, -2 i \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) +{\left (\Gamma \left (-1, 2 i \, b x^{2}\right ) + \Gamma \left (-1, -2 i \, b x^{2}\right )\right )} \sin \left (2 \, a\right )\right )} b x^{2} - 1\right )} c^{\frac{2}{3}}}{8 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(((I*gamma(-1, 2*I*b*x^2) - I*gamma(-1, -2*I*b*x^2))*cos(2*a) + (gamma(-1, 2*I*b*x^2) + gamma(-1, -2*I*b*
x^2))*sin(2*a))*b*x^2 - 1)*c^(2/3)/x^2

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Fricas [A]  time = 1.69334, size = 366, normalized size = 2.27 \begin{align*} -\frac{4^{\frac{2}{3}}{\left (2 \cdot 4^{\frac{1}{3}} b x^{2} \cos \left (2 \, a\right ) \operatorname{Si}\left (2 \, b x^{2}\right ) + 2 \cdot 4^{\frac{1}{3}} \cos \left (b x^{2} + a\right )^{2} +{\left (4^{\frac{1}{3}} b x^{2} \operatorname{Ci}\left (2 \, b x^{2}\right ) + 4^{\frac{1}{3}} b x^{2} \operatorname{Ci}\left (-2 \, b x^{2}\right )\right )} \sin \left (2 \, a\right ) - 2 \cdot 4^{\frac{1}{3}}\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac{2}{3}}}{16 \,{\left (x^{2} \cos \left (b x^{2} + a\right )^{2} - x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*4^(2/3)*(2*4^(1/3)*b*x^2*cos(2*a)*sin_integral(2*b*x^2) + 2*4^(1/3)*cos(b*x^2 + a)^2 + (4^(1/3)*b*x^2*co
s_integral(2*b*x^2) + 4^(1/3)*b*x^2*cos_integral(-2*b*x^2))*sin(2*a) - 2*4^(1/3))*(-(c*cos(b*x^2 + a)^2 - c)*s
in(b*x^2 + a))^(2/3)/(x^2*cos(b*x^2 + 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^{2} \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**2+a)**3)**(2/3)/x**3,x)

[Out]

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

[Out]

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