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

Optimal. Leaf size=25 \[ -\frac{\cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]

[Out]

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

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Rubi [A]  time = 0.0175031, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3207, 2638} \[ -\frac{\cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x]^3)^(1/3),x]

[Out]

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

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

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

Mathematica [A]  time = 0.0627671, size = 25, normalized size = 1. \[ -\frac{\cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x]^3)^(1/3),x]

[Out]

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

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Maple [C]  time = 0.111, size = 105, normalized size = 4.2 \begin{align*}{\frac{-{\frac{i}{2}}{{\rm e}^{2\,i \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) b}\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 ({{\rm e}^{2\,i \left ( bx+a \right ) }}-1 \right ) b}\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((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)/b*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)/b

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Maxima [A]  time = 1.45972, size = 42, normalized size = 1.68 \begin{align*} -\frac{2 \, c^{\frac{1}{3}}}{b{\left (\frac{\sin \left (b x + a\right )^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.69497, size = 104, normalized size = 4.16 \begin{align*} -\frac{\left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac{1}{3}} \cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.94314, size = 53, normalized size = 2.12 \begin{align*} \begin{cases} x \sqrt [3]{c \sin ^{3}{\left (a \right )}} & \text{for}\: b = 0 \\0 & \text{for}\: a = - b x \vee a = - b x + \pi \\- \frac{\sqrt [3]{c} \sqrt [3]{\sin ^{3}{\left (a + b x \right )}} \cos{\left (a + b x \right )}}{b \sin{\left (a + b x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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