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

Optimal. Leaf size=115 \[ -\frac{e^{i a} x^m (-i b x)^{-m} \csc (a+b x) \text{Gamma}(m+1,-i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \csc (a+b x) \text{Gamma}(m+1,i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.287303, antiderivative size = 115, normalized size of antiderivative = 1., 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, 3308, 2181} \[ -\frac{e^{i a} x^m (-i b x)^{-m} \csc (a+b x) \text{Gamma}(m+1,-i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b}-\frac{e^{-i a} x^m (i b x)^{-m} \csc (a+b x) \text{Gamma}(m+1,i b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(c*Sin[a + b*x]^3)^(1/3),x]

[Out]

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

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 3308

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

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

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

Mathematica [A]  time = 0.125477, size = 94, normalized size = 0.82 \[ -\frac{e^{-i a} x^m \left (b^2 x^2\right )^{-m} \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \left (e^{2 i a} (i b x)^m \text{Gamma}(m+1,-i b x)+(-i b x)^m \text{Gamma}(m+1,i b x)\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.154, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt [3]{c \left ( \sin \left ( bx+a \right ) \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*(c*sin(b*x+a)^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 + a\right )^{3}\right )^{\frac{1}{3}} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.74468, size = 213, normalized size = 1.85 \begin{align*} -\frac{{\left (e^{\left (-m \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m + 1, i \, b x\right ) + e^{\left (-m \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m + 1, -i \, b x\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac{1}{3}}}{2 \, b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(e^(-m*log(I*b) - I*a)*gamma(m + 1, I*b*x) + e^(-m*log(-I*b) + I*a)*gamma(m + 1, -I*b*x))*(-(c*cos(b*x +
a)^2 - c)*sin(b*x + a))^(1/3)/(b*sin(b*x + a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*(c*sin(a + b*x)**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 + a\right )^{3}\right )^{\frac{1}{3}} x^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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