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

Optimal. Leaf size=135 \[ \frac{i e^{i a} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}-\frac{i e^{-i a} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n} \]

[Out]

((I/2)*E^(I*a)*x*Csc[a + b*x^n]*Gamma[n^(-1), (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*((-I)*b*x^n)^n^(-1))
- ((I/2)*x*Csc[a + b*x^n]*Gamma[n^(-1), I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*(I*b*x^n)^n^(-1))

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Rubi [A]  time = 0.0471616, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {6720, 3365, 2208} \[ \frac{i e^{i a} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}-\frac{i e^{-i a} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^n]^3)^(1/3),x]

[Out]

((I/2)*E^(I*a)*x*Csc[a + b*x^n]*Gamma[n^(-1), (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*((-I)*b*x^n)^n^(-1))
- ((I/2)*x*Csc[a + b*x^n]*Gamma[n^(-1), I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*(I*b*x^n)^n^(-1))

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 3365

Int[Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)], x_Symbol] :> Dist[I/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x],
x] - Dist[I/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f, n}, x]

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin{align*} \int \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \, dx &=\left (\csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int \sin \left (a+b x^n\right ) \, dx\\ &=\frac{1}{2} \left (i \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int e^{-i a-i b x^n} \, dx-\frac{1}{2} \left (i \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}\right ) \int e^{i a+i b x^n} \, dx\\ &=\frac{i e^{i a} x \left (-i b x^n\right )^{-1/n} \csc \left (a+b x^n\right ) \Gamma \left (\frac{1}{n},-i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}-\frac{i e^{-i a} x \left (i b x^n\right )^{-1/n} \csc \left (a+b x^n\right ) \Gamma \left (\frac{1}{n},i b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}\\ \end{align*}

Mathematica [A]  time = 0.137111, size = 119, normalized size = 0.88 \[ \frac{i x \left (b^2 x^{2 n}\right )^{-1/n} \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left ((\cos (a)+i \sin (a)) \left (i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-i b x^n\right )-(\cos (a)-i \sin (a)) \left (-i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},i b x^n\right )\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/2)*x*Csc[a + b*x^n]*(-(((-I)*b*x^n)^n^(-1)*Gamma[n^(-1), I*b*x^n]*(Cos[a] - I*Sin[a])) + (I*b*x^n)^n^(-1)*
Gamma[n^(-1), (-I)*b*x^n]*(Cos[a] + I*Sin[a]))*(c*Sin[a + b*x^n]^3)^(1/3))/(n*(b^2*x^(2*n))^n^(-1))

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Maple [F]  time = 0.128, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{c \left ( \sin \left ( a+b{x}^{n} \right ) \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac{1}{3}}, x\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, algorithm="fricas")

[Out]

integral((-(c*cos(b*x^n + a)^2 - c)*sin(b*x^n + a))^(1/3), x)

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

[Out]

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

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

[Out]

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