3.4.10 \(\int x^m \sqrt [3]{c \sin ^3(a+b x)} \, dx\) [310]

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

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 115, 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 = {6852, 3389, 2212} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(c*Sin[a + b*x]^3)^(1/3),x]

[Out]

-1/2*(E^(I*a)*x^m*Csc[a + b*x]*Gamma[1 + m, (-I)*b*x]*(c*Sin[a + b*x]^3)^(1/3))/(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 2212

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

Rule 3389

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 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 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*}

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Mathematica [A]
time = 0.08, size = 94, normalized size = 0.82 \begin {gather*} -\frac {e^{-i a} x^m \left (b^2 x^2\right )^{-m} \csc (a+b x) \left (e^{2 i a} (i b x)^m \Gamma (1+m,-i b x)+(-i b x)^m \Gamma (1+m,i b x)\right ) \sqrt [3]{c \sin ^3(a+b x)}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(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*Si
n[a + b*x]^3)^(1/3))/(b*E^(I*a)*(b^2*x^2)^m)

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int x^{m} \left (c \left (\sin ^{3}\left (b x +a \right )\right )\right )^{\frac {1}{3}}\, dx\]

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 = 0.00, 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(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 = 0.10, size = 80, normalized size = 0.70 \begin {gather*} -\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 {gather*}

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

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 = 0.00, 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(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c*sin(a + b*x)^3)^(1/3),x)

[Out]

int(x^m*(c*sin(a + b*x)^3)^(1/3), x)

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