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

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

[Out]

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

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Rubi [A]  time = 0.150866, 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} \[ \frac{\sin (a) \text{CosIntegral}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n}+\frac{\cos (a) \text{Si}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^n]^3)^(1/3)/x,x]

[Out]

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

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

Mathematica [A]  time = 0.0744186, size = 47, normalized size = 0.64 \[ \frac{\csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left (\sin (a) \text{CosIntegral}\left (b x^n\right )+\cos (a) \text{Si}\left (b x^n\right )\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.144, size = 280, normalized size = 3.8 \begin{align*} -{\frac{{\it Ei} \left ( 1,-ib{x}^{n} \right ){{\rm e}^{i \left ( b{x}^{n}+2\,a \right ) }}}{ \left ( 2\,{{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-2 \right ) n}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( a+b{x}^{n} \right ) }}}}-{\frac{{\frac{i}{2}}{{\rm e}^{ib{x}^{n}}}\pi \,{\it csgn} \left ( b{x}^{n} \right ) }{ \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) n}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( a+b{x}^{n} \right ) }}}}+{\frac{i{{\rm e}^{ib{x}^{n}}}{\it Si} \left ( b{x}^{n} \right ) }{ \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) n}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( a+b{x}^{n} \right ) }}}}+{\frac{{{\rm e}^{ib{x}^{n}}}{\it Ei} \left ( 1,-ib{x}^{n} \right ) }{ \left ( 2\,{{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-2 \right ) n}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( a+b{x}^{n} \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( a+b{x}^{n} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: IndexError

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Fricas [A]  time = 1.77753, size = 293, normalized size = 4.01 \begin{align*} -\frac{4^{\frac{1}{3}}{\left (4^{\frac{2}{3}} \operatorname{Ci}\left (b x^{n}\right ) \sin \left (a\right ) + 4^{\frac{2}{3}} \operatorname{Ci}\left (-b x^{n}\right ) \sin \left (a\right ) + 2 \cdot 4^{\frac{2}{3}} \cos \left (a\right ) \operatorname{Si}\left (b x^{n}\right )\right )} \left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac{1}{3}} \sin \left (b x^{n} + a\right )}{8 \,{\left (n \cos \left (b x^{n} + a\right )^{2} - n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{n} \right )}}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c \sin \left (b x^{n} + 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(a+b*x^n)^3)^(1/3)/x,x, algorithm="giac")

[Out]

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