3.345 \(\int x (c \sin ^3(a+b x^2))^{2/3} \, dx\)

Optimal. Leaf size=65 \[ \frac{1}{4} x^2 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b} \]

[Out]

-(Cot[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(2/3))/(4*b) + (x^2*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/4

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Rubi [A]  time = 0.0997761, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6715, 3207, 2635, 8} \[ \frac{1}{4} x^2 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(c*Sin[a + b*x^2]^3)^(2/3),x]

[Out]

-(Cot[a + b*x^2]*(c*Sin[a + b*x^2]^3)^(2/3))/(4*b) + (x^2*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/4

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int x \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \left (c \sin ^3(a+b x)\right )^{2/3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname{Subst}\left (\int \sin ^2(a+b x) \, dx,x,x^2\right )\\ &=-\frac{\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b}+\frac{1}{4} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,x^2\right )\\ &=-\frac{\cot \left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 b}+\frac{1}{4} x^2 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\\ \end{align*}

Mathematica [A]  time = 0.118257, size = 55, normalized size = 0.85 \[ \frac{\left (2 \left (a+b x^2\right )-\sin \left (2 \left (a+b x^2\right )\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3)*(2*(a + b*x^2) - Sin[2*(a + b*x^2)]))/(8*b)

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Maple [C]  time = 0.075, size = 182, normalized size = 2.8 \begin{align*} -{\frac{{x}^{2}{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{4\, \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}}-{\frac{{\frac{i}{16}}{{\rm e}^{4\,i \left ( b{x}^{2}+a \right ) }}}{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}b} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}}+{\frac{{\frac{i}{16}}}{ \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{2}b} \left ( ic \left ({{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}-1 \right ) ^{3}{{\rm e}^{-3\,i \left ( b{x}^{2}+a \right ) }} \right ) ^{{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(2/3)/(exp(2*I*(b*x^2+a))-1)^2*x^2*exp(2*I*(b*x^2+a))-
1/16*I*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(2/3)/(exp(2*I*(b*x^2+a))-1)^2/b*exp(4*I*(b*x^2+a))+
1/16*I*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(2/3)/(exp(2*I*(b*x^2+a))-1)^2/b

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Maxima [A]  time = 1.54614, size = 38, normalized size = 0.58 \begin{align*} -\frac{{\left (2 \, b x^{2} - \sin \left (2 \, b x^{2} + 2 \, a\right )\right )} c^{\frac{2}{3}}}{16 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*b*x^2 - sin(2*b*x^2 + 2*a))*c^(2/3)/b

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Fricas [A]  time = 1.58751, size = 162, normalized size = 2.49 \begin{align*} -\frac{{\left (b x^{2} - \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac{2}{3}}}{4 \,{\left (b \cos \left (b x^{2} + a\right )^{2} - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (c \sin ^{3}{\left (a + b x^{2} \right )}\right )^{\frac{2}{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac{2}{3}} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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