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

Optimal. Leaf size=148 \[ -\frac{\sqrt{\pi } \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{\sqrt{\pi } \sin (2 a) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{1}{2} x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]

[Out]

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

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Rubi [A]  time = 0.0583402, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {6720, 3357, 3354, 3352, 3351} \[ -\frac{\sqrt{\pi } \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{\sqrt{\pi } \sin (2 a) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{1}{2} x \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^2]^3)^(2/3),x]

[Out]

(x*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/2 - (Sqrt[Pi]*Cos[2*a]*Csc[a + b*x^2]^2*FresnelC[(2*Sqrt[b]*x)
/Sqrt[Pi]]*(c*Sin[a + b*x^2]^3)^(2/3))/(4*Sqrt[b]) + (Sqrt[Pi]*Csc[a + b*x^2]^2*FresnelS[(2*Sqrt[b]*x)/Sqrt[Pi
]]*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3))/(4*Sqrt[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 3357

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

Rule 3354

Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[
Sin[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 \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 \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{1}{2}-\frac{1}{2} \cos \left (2 a+2 b x^2\right )\right ) \, dx\\ &=\frac{1}{2} x \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 \cos \left (2 a+2 b x^2\right ) \, dx\\ &=\frac{1}{2} x \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 (\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 \cos \left (2 b x^2\right ) \, dx+\frac{1}{2} \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 \sin \left (2 b x^2\right ) \, dx\\ &=\frac{1}{2} x \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 ) C\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 \sqrt{b}}+\frac{\sqrt{\pi } \csc ^2\left (a+b x^2\right ) S\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}}{4 \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.102306, size = 93, normalized size = 0.63 \[ \frac{\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (-\sqrt{\pi } \cos (2 a) \text{FresnelC}\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right )+\sqrt{\pi } \sin (2 a) S\left (\frac{2 \sqrt{b} x}{\sqrt{\pi }}\right )+2 \sqrt{b} x\right )}{4 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x^2]^2*(2*Sqrt[b]*x - Sqrt[Pi]*Cos[2*a]*FresnelC[(2*Sqrt[b]*x)/Sqrt[Pi]] + Sqrt[Pi]*FresnelS[(2*Sqr
t[b]*x)/Sqrt[Pi]]*Sin[2*a])*(c*Sin[a + b*x^2]^3)^(2/3))/(4*Sqrt[b])

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Maple [C]  time = 0.087, size = 224, normalized size = 1.5 \begin{align*}{\frac{{{\rm e}^{2\,ib{x}^{2}}}\sqrt{\pi }\sqrt{2}}{16\, \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}}}{\it Erf} \left ( \sqrt{2}\sqrt{ib}x \right ){\frac{1}{\sqrt{ib}}}}+{\frac{\sqrt{\pi }{{\rm e}^{2\,i \left ( b{x}^{2}+2\,a \right ) }}}{8\, \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}}}{\it Erf} \left ( \sqrt{-2\,ib}x \right ){\frac{1}{\sqrt{-2\,ib}}}}-{\frac{x{{\rm e}^{2\,i \left ( b{x}^{2}+a \right ) }}}{2\, \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((c*sin(b*x^2+a)^3)^(2/3),x)

[Out]

1/16*(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)*Pi^(1/2)
*2^(1/2)/(I*b)^(1/2)*erf(2^(1/2)*(I*b)^(1/2)*x)+1/8*(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*Pi^(1/2)/(-2*I*b)^(1/2)*erf((-2*I*b)^(1/2)*x)*exp(2*I*(b*x^2+2*a))-1/2*(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*exp(2*I*(b*x^2+a))

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Maxima [C]  time = 1.72626, size = 483, normalized size = 3.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(sqrt(2)*sqrt(pi)*((((-I*sqrt(3) - 1)*cos(1/4*pi + 1/2*arctan2(0, b)) + (I*sqrt(3) - 1)*cos(-1/4*pi + 1/
2*arctan2(0, b)) - (sqrt(3) - I)*sin(1/4*pi + 1/2*arctan2(0, b)) - (sqrt(3) + I)*sin(-1/4*pi + 1/2*arctan2(0,
b)))*cos(2*a) - ((sqrt(3) - I)*cos(1/4*pi + 1/2*arctan2(0, b)) - (sqrt(3) + I)*cos(-1/4*pi + 1/2*arctan2(0, b)
) - (I*sqrt(3) + 1)*sin(1/4*pi + 1/2*arctan2(0, b)) - (I*sqrt(3) - 1)*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(2*
a))*erf(sqrt(2*I*b)*x) + (((I*sqrt(3) - 1)*cos(1/4*pi + 1/2*arctan2(0, b)) + (-I*sqrt(3) - 1)*cos(-1/4*pi + 1/
2*arctan2(0, b)) - (sqrt(3) + I)*sin(1/4*pi + 1/2*arctan2(0, b)) - (sqrt(3) - I)*sin(-1/4*pi + 1/2*arctan2(0,
b)))*cos(2*a) - ((sqrt(3) + I)*cos(1/4*pi + 1/2*arctan2(0, b)) - (sqrt(3) - I)*cos(-1/4*pi + 1/2*arctan2(0, b)
) - (-I*sqrt(3) + 1)*sin(1/4*pi + 1/2*arctan2(0, b)) - (-I*sqrt(3) - 1)*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*c^(2/3)*x*abs(b))/abs(b)

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Fricas [A]  time = 1.70179, size = 305, normalized size = 2.06 \begin{align*} \frac{4^{\frac{2}{3}}{\left (4^{\frac{1}{3}} \pi \sqrt{\frac{b}{\pi }} \cos \left (2 \, a\right ) \operatorname{C}\left (2 \, x \sqrt{\frac{b}{\pi }}\right ) - 4^{\frac{1}{3}} \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (2 \, x \sqrt{\frac{b}{\pi }}\right ) \sin \left (2 \, a\right ) - 2 \cdot 4^{\frac{1}{3}} b x\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}}}{16 \,{\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((c*sin(b*x^2+a)^3)^(2/3),x, algorithm="fricas")

[Out]

1/16*4^(2/3)*(4^(1/3)*pi*sqrt(b/pi)*cos(2*a)*fresnel_cos(2*x*sqrt(b/pi)) - 4^(1/3)*pi*sqrt(b/pi)*fresnel_sin(2
*x*sqrt(b/pi))*sin(2*a) - 2*4^(1/3)*b*x)*(-(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 \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((c*sin(b*x**2+a)**3)**(2/3),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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