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

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

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6720, 3423, 2218} \[ \frac {i e^{i a} x^2 \left (-i b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n}-\frac {i e^{-i a} x^2 \left (i b x^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},i b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[x*(c*Sin[a + b*x^n]^3)^(1/3),x]

[Out]

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

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]

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 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])

Rubi steps

\begin {align*} \int x \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 x \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} x \, 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} x \, dx\\ &=\frac {i e^{i a} x^2 \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}-\frac {i e^{-i a} x^2 \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}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 129, normalized size = 0.90 \[ \frac {i x^2 \left (b^2 x^{2 n}\right )^{-2/n} \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left ((\cos (a)+i \sin (a)) \left (i b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},-i b x^n\right )-(\cos (a)-i \sin (a)) \left (-i b x^n\right )^{2/n} \Gamma \left (\frac {2}{n},i b x^n\right )\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {1}{3}} x, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int x \left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {1}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x*(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 \[ \int \left (c \sin \left (b x^{n} + a\right )^{3}\right )^{\frac {1}{3}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(c*sin(a + b*x^n)^3)^(1/3),x)

[Out]

int(x*(c*sin(a + b*x^n)^3)^(1/3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt [3]{c \sin ^{3}{\left (a + b x^{n} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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