3.312 \(\int x^2 \sqrt [3]{c \sin ^3(a+b x)} \, dx\)

Optimal. Leaf size=74 \[ \frac{2 x \sqrt [3]{c \sin ^3(a+b x)}}{b^2}+\frac{2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b^3}-\frac{x^2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]

[Out]

(2*x*(c*Sin[a + b*x]^3)^(1/3))/b^2 + (2*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/b^3 - (x^2*Cot[a + b*x]*(c*Sin[
a + b*x]^3)^(1/3))/b

________________________________________________________________________________________

Rubi [A]  time = 0.182098, antiderivative size = 74, 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, 3296, 2638} \[ \frac{2 x \sqrt [3]{c \sin ^3(a+b x)}}{b^2}+\frac{2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b^3}-\frac{x^2 \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(c*Sin[a + b*x]^3)^(1/3),x]

[Out]

(2*x*(c*Sin[a + b*x]^3)^(1/3))/b^2 + (2*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/b^3 - (x^2*Cot[a + b*x]*(c*Sin[
a + b*x]^3)^(1/3))/b

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.223712, size = 40, normalized size = 0.54 \[ \frac{\left (\left (2-b^2 x^2\right ) \cot (a+b x)+2 b x\right ) \sqrt [3]{c \sin ^3(a+b x)}}{b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(c*Sin[a + b*x]^3)^(1/3),x]

[Out]

((2*b*x + (2 - b^2*x^2)*Cot[a + b*x])*(c*Sin[a + b*x]^3)^(1/3))/b^3

________________________________________________________________________________________

Maple [C]  time = 0.075, size = 133, normalized size = 1.8 \begin{align*}{\frac{-{\frac{i}{2}} \left ({x}^{2}{b}^{2}+2\,ibx-2 \right ){{\rm e}^{2\,i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ){b}^{3}}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }}}}-{\frac{{\frac{i}{2}} \left ({x}^{2}{b}^{2}-2\,ibx-2 \right ) }{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ){b}^{3}}\sqrt [3]{ic \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(c*sin(b*x+a)^3)^(1/3),x)

[Out]

-1/2*I*(I*c*(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(1/3)/(exp(2*I*(b*x+a))-1)*(x^2*b^2+2*I*b*x-2)/b^3*exp(2
*I*(b*x+a))-1/2*I*(I*c*(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(1/3)/(exp(2*I*(b*x+a))-1)*(x^2*b^2-2*I*b*x-2
)/b^3

________________________________________________________________________________________

Maxima [A]  time = 1.54939, size = 134, normalized size = 1.81 \begin{align*} -\frac{2 \,{\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} a c^{\frac{1}{3}} -{\left ({\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 2 \,{\left (b x + a\right )} \sin \left (b x + a\right )\right )} c^{\frac{1}{3}} + \frac{4 \, a^{2} c^{\frac{1}{3}}}{\frac{\sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1}}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*sin(b*x+a)^3)^(1/3),x, algorithm="maxima")

[Out]

-1/2*(2*((b*x + a)*cos(b*x + a) - sin(b*x + a))*a*c^(1/3) - (((b*x + a)^2 - 2)*cos(b*x + a) - 2*(b*x + a)*sin(
b*x + a))*c^(1/3) + 4*a^2*c^(1/3)/(sin(b*x + a)^2/(cos(b*x + a) + 1)^2 + 1))/b^3

________________________________________________________________________________________

Fricas [A]  time = 1.75473, size = 155, normalized size = 2.09 \begin{align*} \frac{{\left (2 \, b x \sin \left (b x + a\right ) -{\left (b^{2} x^{2} - 2\right )} \cos \left (b x + a\right )\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac{1}{3}}}{b^{3} \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*sin(b*x+a)^3)^(1/3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 11.8304, size = 117, normalized size = 1.58 \begin{align*} \begin{cases} \frac{x^{3} \sqrt [3]{c \sin ^{3}{\left (a \right )}}}{3} & \text{for}\: b = 0 \\0 & \text{for}\: a = - b x \vee a = - b x + \pi \\- \frac{\sqrt [3]{c} x^{2} \sqrt [3]{\sin ^{3}{\left (a + b x \right )}} \cos{\left (a + b x \right )}}{b \sin{\left (a + b x \right )}} + \frac{2 \sqrt [3]{c} x \sqrt [3]{\sin ^{3}{\left (a + b x \right )}}}{b^{2}} + \frac{2 \sqrt [3]{c} \sqrt [3]{\sin ^{3}{\left (a + b x \right )}} \cos{\left (a + b x \right )}}{b^{3} \sin{\left (a + b x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(c*sin(b*x+a)**3)**(1/3),x)

[Out]

Piecewise((x**3*(c*sin(a)**3)**(1/3)/3, Eq(b, 0)), (0, Eq(a, -b*x) | Eq(a, -b*x + pi)), (-c**(1/3)*x**2*(sin(a
+ b*x)**3)**(1/3)*cos(a + b*x)/(b*sin(a + b*x)) + 2*c**(1/3)*x*(sin(a + b*x)**3)**(1/3)/b**2 + 2*c**(1/3)*(si
n(a + b*x)**3)**(1/3)*cos(a + b*x)/(b**3*sin(a + b*x)), True))

________________________________________________________________________________________

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^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(c*sin(b*x+a)^3)^(1/3),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a)^3)^(1/3)*x^2, x)