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

Optimal. Leaf size=143 \[ \frac{i e^{i a} \left (-i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x^2}-\frac{i e^{-i a} \left (i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x^2} \]

[Out]

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

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Rubi [A]  time = 0.21195, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6720, 3423, 2218} \[ \frac{i e^{i a} \left (-i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x^2}-\frac{i e^{-i a} \left (i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n x^2} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^n]^3)^(1/3)/x^3,x]

[Out]

((I/2)*E^(I*a)*((-I)*b*x^n)^(2/n)*Csc[a + b*x^n]*Gamma[-2/n, (-I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(n*x^2) -
((I/2)*(I*b*x^n)^(2/n)*Csc[a + b*x^n]*Gamma[-2/n, I*b*x^n]*(c*Sin[a + b*x^n]^3)^(1/3))/(E^(I*a)*n*x^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 3423

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

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.173913, size = 114, normalized size = 0.8 \[ \frac{i \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left ((\cos (a)+i \sin (a)) \left (-i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},-i b x^n\right )-(\cos (a)-i \sin (a)) \left (i b x^n\right )^{2/n} \text{Gamma}\left (-\frac{2}{n},i b x^n\right )\right )}{2 n x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\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^3,x)

[Out]

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

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Maxima [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^{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^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\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^{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^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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^{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^3,x, algorithm="giac")

[Out]

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