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

Optimal. Leaf size=73 \[ \frac{1}{2} \sin (a) \text{CosIntegral}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}+\frac{1}{2} \cos (a) \text{Si}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.120985, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 3377, 3376, 3375} \[ \frac{1}{2} \sin (a) \text{CosIntegral}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )}+\frac{1}{2} \cos (a) \text{Si}\left (b x^2\right ) \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^2]^3)^(1/3)/x,x]

[Out]

(CosIntegral[b*x^2]*Csc[a + b*x^2]*Sin[a]*(c*Sin[a + b*x^2]^3)^(1/3))/2 + (Cos[a]*Csc[a + b*x^2]*(c*Sin[a + b*
x^2]^3)^(1/3)*SinIntegral[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 3377

Int[Sin[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Sin[c], Int[Cos[d*x^n]/x, x], x] + Dist[Cos[c], Int[Si
n[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rule 3376

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3375

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.0584489, size = 47, normalized size = 0.64 \[ \frac{1}{2} \csc \left (a+b x^2\right ) \sqrt [3]{c \sin ^3\left (a+b x^2\right )} \left (\sin (a) \text{CosIntegral}\left (b x^2\right )+\cos (a) \text{Si}\left (b x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.083, size = 268, normalized size = 3.7 \begin{align*} -{\frac{{\it Ei} \left ( 1,-ib{x}^{2} \right ){{\rm e}^{i \left ( b{x}^{2}+2\,a \right ) }}}{4\,{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-4}\sqrt [3]{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 ) }}}}-{\frac{{\frac{i}{4}}{{\rm e}^{ib{x}^{2}}}\pi \,{\it csgn} \left ( b{x}^{2} \right ) }{{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1}\sqrt [3]{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 ) }}}}+{\frac{{\frac{i}{2}}{{\rm e}^{ib{x}^{2}}}{\it Si} \left ( b{x}^{2} \right ) }{{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1}\sqrt [3]{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 ) }}}}+{\frac{{{\rm e}^{ib{x}^{2}}}{\it Ei} \left ( 1,-ib{x}^{2} \right ) }{4\,{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-4}\sqrt [3]{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 ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(1/3)/(exp(2*I*(b*x^2+a))-1)*Ei(1,-I*b*x^2)*exp(I*(b*x
^2+2*a))-1/4*I*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(1/3)/(exp(2*I*(b*x^2+a))-1)*exp(I*b*x^2)*Pi
*csgn(b*x^2)+1/2*I*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(1/3)/(exp(2*I*(b*x^2+a))-1)*exp(I*b*x^2
)*Si(b*x^2)+1/4*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(1/3)/(exp(2*I*(b*x^2+a))-1)*exp(I*b*x^2)*E
i(1,-I*b*x^2)

________________________________________________________________________________________

Maxima [C]  time = 1.69687, size = 63, normalized size = 0.86 \begin{align*} \frac{1}{8} \,{\left ({\left (i \,{\rm Ei}\left (i \, b x^{2}\right ) - i \,{\rm Ei}\left (-i \, b x^{2}\right )\right )} \cos \left (a\right ) -{\left ({\rm Ei}\left (i \, b x^{2}\right ) +{\rm Ei}\left (-i \, b x^{2}\right )\right )} \sin \left (a\right )\right )} c^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.74076, size = 285, normalized size = 3.9 \begin{align*} -\frac{4^{\frac{1}{3}}{\left (2 \cdot 4^{\frac{2}{3}} \cos \left (a\right ) \operatorname{Si}\left (b x^{2}\right ) +{\left (4^{\frac{2}{3}} \operatorname{Ci}\left (b x^{2}\right ) + 4^{\frac{2}{3}} \operatorname{Ci}\left (-b x^{2}\right )\right )} \sin \left (a\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac{1}{3}} \sin \left (b x^{2} + a\right )}{16 \,{\left (\cos \left (b x^{2} + a\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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{1}{3}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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