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

Optimal. Leaf size=195 \[ \frac{\sqrt{\pi } \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}+\frac{\sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}+\frac{1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{x \sin \left (2 a+2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b} \]

[Out]

(x^3*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/6 + (Sqrt[Pi]*Cos[2*a]*Csc[a + b*x^2]^2*FresnelS[(2*Sqrt[b]*
x)/Sqrt[Pi]]*(c*Sin[a + b*x^2]^3)^(2/3))/(16*b^(3/2)) + (Sqrt[Pi]*Csc[a + b*x^2]^2*FresnelC[(2*Sqrt[b]*x)/Sqrt
[Pi]]*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3))/(16*b^(3/2)) - (x*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3)*Sin[2
*a + 2*b*x^2])/(8*b)

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Rubi [A]  time = 0.230888, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6720, 3403, 3386, 3353, 3352, 3351} \[ \frac{\sqrt{\pi } \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}+\frac{\sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}+\frac{1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{x \sin \left (2 a+2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(c*Sin[a + b*x^2]^3)^(2/3),x]

[Out]

(x^3*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/6 + (Sqrt[Pi]*Cos[2*a]*Csc[a + b*x^2]^2*FresnelS[(2*Sqrt[b]*
x)/Sqrt[Pi]]*(c*Sin[a + b*x^2]^3)^(2/3))/(16*b^(3/2)) + (Sqrt[Pi]*Csc[a + b*x^2]^2*FresnelC[(2*Sqrt[b]*x)/Sqrt
[Pi]]*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3))/(16*b^(3/2)) - (x*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3)*Sin[2
*a + 2*b*x^2])/(8*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 3403

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rule 3386

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*
x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x
] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3353

Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[
Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin{align*} \int x^2 \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \, dx &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int x^2 \sin ^2\left (a+b x^2\right ) \, dx\\ &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \left (\frac{x^2}{2}-\frac{1}{2} x^2 \cos \left (2 a+2 b x^2\right )\right ) \, dx\\ &=\frac{1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\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 ) \int x^2 \cos \left (2 a+2 b x^2\right ) \, dx\\ &=\frac{1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \sin \left (2 a+2 b x^2\right )}{8 b}+\frac{\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \sin \left (2 a+2 b x^2\right ) \, dx}{8 b}\\ &=\frac{1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}-\frac{x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \sin \left (2 a+2 b x^2\right )}{8 b}+\frac{\left (\cos (2 a) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \sin \left (2 b x^2\right ) \, dx}{8 b}+\frac{\left (\csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \cos \left (2 b x^2\right ) \, dx}{8 b}\\ &=\frac{1}{6} x^3 \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{\sqrt{\pi } \cos (2 a) \csc ^2\left (a+b x^2\right ) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}+\frac{\sqrt{\pi } \csc ^2\left (a+b x^2\right ) C\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{16 b^{3/2}}-\frac{x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \sin \left (2 a+2 b x^2\right )}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.287334, size = 113, normalized size = 0.58 \[ \frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (3 \sqrt{\pi } \sin (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right )+3 \sqrt{\pi } \cos (2 a) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right )+2 \sqrt{b} x \left (4 b x^2-3 \sin \left (2 \left (a+b x^2\right )\right )\right )\right )}{48 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(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)*(3*Sqrt[Pi]*Cos[2*a]*FresnelS[(2*Sqrt[b]*x)/Sqrt[Pi]] + 3*Sqrt[Pi
]*FresnelC[(2*Sqrt[b]*x)/Sqrt[Pi]]*Sin[2*a] + 2*Sqrt[b]*x*(4*b*x^2 - 3*Sin[2*(a + b*x^2)])))/(48*b^(3/2))

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Maple [C]  time = 0.112, size = 309, normalized size = 1.6 \begin{align*}{\frac{{\frac{i}{16}}x}{ \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}{64}}{{\rm e}^{2\,ib{x}^{2}}}\sqrt{\pi }\sqrt{2}}{ \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}}}{\it Erf} \left ( \sqrt{2}\sqrt{ib}x \right ){\frac{1}{\sqrt{ib}}}}+{\frac{1}{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}}} \left ({\frac{-{\frac{i}{4}}x{{\rm e}^{4\,i \left ( b{x}^{2}+a \right ) }}}{b}}+{\frac{{\frac{i}{8}}\sqrt{\pi }{{\rm e}^{2\,i \left ( b{x}^{2}+2\,a \right ) }}}{b}{\it Erf} \left ( \sqrt{-2\,ib}x \right ){\frac{1}{\sqrt{-2\,ib}}}} \right ) }-{\frac{{x}^{3}{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{6\, \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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*x-1/64*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*exp(2*I*b*x^2)/b*Pi^(1/2)*2^(1/2)/(I*b
)^(1/2)*erf(2^(1/2)*(I*b)^(1/2)*x)+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*(-1/4*I/b*x*exp(4*I*(b*x^2+a))+1/8*I/b*Pi^(1/2)/(-2*I*b)^(1/2)*erf((-2*I*b)^(1/2)*x)*exp(2*I*(b*x^2
+2*a)))-1/6*(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^3*exp(2*I*(b*x
^2+a))

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Maxima [C]  time = 1.76104, size = 539, normalized size = 2.76 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768*(sqrt(2)*sqrt(pi)*((((3*sqrt(3) - 3*I)*cos(1/4*pi + 1/2*arctan2(0, b)) - (3*sqrt(3) + 3*I)*cos(-1/4*pi +
1/2*arctan2(0, b)) - 3*(I*sqrt(3) + 1)*sin(1/4*pi + 1/2*arctan2(0, b)) - 3*(I*sqrt(3) - 1)*sin(-1/4*pi + 1/2*
arctan2(0, b)))*cos(2*a) - (3*(I*sqrt(3) + 1)*cos(1/4*pi + 1/2*arctan2(0, b)) + 3*(-I*sqrt(3) + 1)*cos(-1/4*pi
+ 1/2*arctan2(0, b)) + (3*sqrt(3) - 3*I)*sin(1/4*pi + 1/2*arctan2(0, b)) + (3*sqrt(3) + 3*I)*sin(-1/4*pi + 1/
2*arctan2(0, b)))*sin(2*a))*erf(sqrt(2*I*b)*x) + (((3*sqrt(3) + 3*I)*cos(1/4*pi + 1/2*arctan2(0, b)) - (3*sqrt
(3) - 3*I)*cos(-1/4*pi + 1/2*arctan2(0, b)) - 3*(-I*sqrt(3) + 1)*sin(1/4*pi + 1/2*arctan2(0, b)) - 3*(-I*sqrt(
3) - 1)*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(2*a) - (3*(-I*sqrt(3) + 1)*cos(1/4*pi + 1/2*arctan2(0, b)) + 3*(
I*sqrt(3) + 1)*cos(-1/4*pi + 1/2*arctan2(0, b)) + (3*sqrt(3) + 3*I)*sin(1/4*pi + 1/2*arctan2(0, b)) + (3*sqrt(
3) - 3*I)*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(2*a))*erf(sqrt(-2*I*b)*x))*c^(2/3)*sqrt(abs(b)) - 16*(4*b*x^3*
abs(b) - 3*x*abs(b)*sin(2*b*x^2 + 2*a))*c^(2/3))/(b*abs(b))

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Fricas [A]  time = 1.85172, size = 387, normalized size = 1.98 \begin{align*} -\frac{4^{\frac{2}{3}}{\left (8 \cdot 4^{\frac{1}{3}} b^{2} x^{3} - 12 \cdot 4^{\frac{1}{3}} b x \cos \left (b x^{2} + a\right ) \sin \left (b x^{2} + a\right ) + 3 \cdot 4^{\frac{1}{3}} \pi \sqrt{\frac{b}{\pi }} \cos \left (2 \, a\right ) \operatorname{S}\left (2 \, x \sqrt{\frac{b}{\pi }}\right ) + 3 \cdot 4^{\frac{1}{3}} \pi \sqrt{\frac{b}{\pi }} \operatorname{C}\left (2 \, x \sqrt{\frac{b}{\pi }}\right ) \sin \left (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}}}{192 \,{\left (b^{2} \cos \left (b x^{2} + a\right )^{2} - b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*4^(2/3)*(8*4^(1/3)*b^2*x^3 - 12*4^(1/3)*b*x*cos(b*x^2 + a)*sin(b*x^2 + a) + 3*4^(1/3)*pi*sqrt(b/pi)*cos
(2*a)*fresnel_sin(2*x*sqrt(b/pi)) + 3*4^(1/3)*pi*sqrt(b/pi)*fresnel_cos(2*x*sqrt(b/pi))*sin(2*a))*(-(c*cos(b*x
^2 + a)^2 - c)*sin(b*x^2 + a))^(2/3)/(b^2*cos(b*x^2 + a)^2 - b^2)

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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