3.4.30 \(\int \sqrt [3]{c \sin ^3(a+b x^n)} \, dx\) [330]

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

[Out]

1/2*I*exp(I*a)*x*csc(a+b*x^n)*GAMMA(1/n,-I*b*x^n)*(c*sin(a+b*x^n)^3)^(1/3)/n/((-I*b*x^n)^(1/n))-1/2*I*x*csc(a+
b*x^n)*GAMMA(1/n,I*b*x^n)*(c*sin(a+b*x^n)^3)^(1/3)/exp(I*a)/n/((I*b*x^n)^(1/n))

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Rubi [A]
time = 0.03, antiderivative size = 135, normalized size of antiderivative = 1.00, 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 = {6852, 3446, 2239} \begin {gather*} \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} \end {gather*}

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 2239

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]

Rule 3446

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 6852

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])

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*}

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Mathematica [A]
time = 0.09, size = 119, normalized size = 0.88 \begin {gather*} \frac {i x \left (b^2 x^{2 n}\right )^{-1/n} \csc \left (a+b x^n\right ) \left (-\left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},i b x^n\right ) (\cos (a)-i \sin (a))+\left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-i b x^n\right ) (\cos (a)+i \sin (a))\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n} \end {gather*}

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 = 0.15, size = 0, normalized size = 0.00 \[\int \left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {1}{3}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{c \sin ^{3}{\left (a + b x^{n} \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,{\sin \left (a+b\,x^n\right )}^3\right )}^{1/3} \,d x \end {gather*}

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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