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

Optimal. Leaf size=115 \[ -\frac{1}{4} \cos (2 a) \text{CosIntegral}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{1}{4} \sin (2 a) \text{Si}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{1}{2} \log (x) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]

[Out]

-(Cos[2*a]*CosIntegral[2*b*x^2]*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/4 + (Csc[a + b*x^2]^2*Log[x]*(c*S
in[a + b*x^2]^3)^(2/3))/2 + (Csc[a + b*x^2]^2*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3)*SinIntegral[2*b*x^2])/4

________________________________________________________________________________________

Rubi [A]  time = 0.133211, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6720, 3403, 3378, 3376, 3375} \[ -\frac{1}{4} \cos (2 a) \text{CosIntegral}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{1}{4} \sin (2 a) \text{Si}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{1}{2} \log (x) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^2]^3)^(2/3)/x,x]

[Out]

-(Cos[2*a]*CosIntegral[2*b*x^2]*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/4 + (Csc[a + b*x^2]^2*Log[x]*(c*S
in[a + b*x^2]^3)^(2/3))/2 + (Csc[a + b*x^2]^2*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3)*SinIntegral[2*b*x^2])/4

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 3378

Int[Cos[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Cos[c], Int[Cos[d*x^n]/x, x], x] - Dist[Sin[c], Int[Si
n[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rule 3376

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3375

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rubi steps

\begin{align*} \int \frac{\left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{x} \, 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 \frac{\sin ^2\left (a+b x^2\right )}{x} \, 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 x}-\frac{\cos \left (2 a+2 b x^2\right )}{2 x}\right ) \, dx\\ &=\frac{1}{2} \csc ^2\left (a+b x^2\right ) \log (x) \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 \frac{\cos \left (2 a+2 b x^2\right )}{x} \, dx\\ &=\frac{1}{2} \csc ^2\left (a+b x^2\right ) \log (x) \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 \frac{\cos \left (2 b x^2\right )}{x} \, 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 \frac{\sin \left (2 b x^2\right )}{x} \, dx\\ &=-\frac{1}{4} \cos (2 a) \text{Ci}\left (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}+\frac{1}{2} \csc ^2\left (a+b x^2\right ) \log (x) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac{1}{4} \csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \text{Si}\left (2 b x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.096372, size = 60, normalized size = 0.52 \[ \frac{1}{4} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (-\cos (2 a) \text{CosIntegral}\left (2 b x^2\right )+\sin (2 a) \text{Si}\left (2 b x^2\right )+2 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3)*(-(Cos[2*a]*CosIntegral[2*b*x^2]) + 2*Log[x] + Sin[2*a]*SinIntegr
al[2*b*x^2]))/4

________________________________________________________________________________________

Maple [C]  time = 0.085, size = 331, normalized size = 2.9 \begin{align*}{\frac{{\frac{i}{8}}{{\rm e}^{2\,ib{x}^{2}}}\pi \,{\it csgn} \left ( b{x}^{2} \right ) }{ \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}{4}}{{\rm e}^{2\,ib{x}^{2}}}{\it Si} \left ( 2\,b{x}^{2} \right ) }{ \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{{{\rm e}^{2\,ib{x}^{2}}}{\it Ei} \left ( 1,-2\,ib{x}^{2} \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}}}}-{\frac{{\it Ei} \left ( 1,-2\,ib{x}^{2} \right ){{\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}}}}-{\frac{\ln \left ( x \right ){{\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,x)

[Out]

1/8*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)*Pi*csgn
(b*x^2)-1/4*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)
*Si(2*b*x^2)-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*exp(2*I*b*x
^2)*Ei(1,-2*I*b*x^2)-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*Ei(
1,-2*I*b*x^2)*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*ln(x)*exp(2*I*(b*x^2+a))

________________________________________________________________________________________

Maxima [C]  time = 1.68878, size = 74, normalized size = 0.64 \begin{align*} \frac{1}{16} \,{\left ({\left ({\rm Ei}\left (2 i \, b x^{2}\right ) +{\rm Ei}\left (-2 i \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) -{\left (-i \,{\rm Ei}\left (2 i \, b x^{2}\right ) + i \,{\rm Ei}\left (-2 i \, b x^{2}\right )\right )} \sin \left (2 \, a\right ) - 4 \, \log \left (x\right )\right )} c^{\frac{2}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*((Ei(2*I*b*x^2) + Ei(-2*I*b*x^2))*cos(2*a) - (-I*Ei(2*I*b*x^2) + I*Ei(-2*I*b*x^2))*sin(2*a) - 4*log(x))*c
^(2/3)

________________________________________________________________________________________

Fricas [A]  time = 1.61394, size = 304, normalized size = 2.64 \begin{align*} -\frac{4^{\frac{2}{3}}{\left (2 \cdot 4^{\frac{1}{3}} \sin \left (2 \, a\right ) \operatorname{Si}\left (2 \, b x^{2}\right ) -{\left (4^{\frac{1}{3}} \operatorname{Ci}\left (2 \, b x^{2}\right ) + 4^{\frac{1}{3}} \operatorname{Ci}\left (-2 \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) + 4 \cdot 4^{\frac{1}{3}} \log \left (x\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}}}{32 \,{\left (\cos \left (b x^{2} + a\right )^{2} - 1\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,x, algorithm="fricas")

[Out]

-1/32*4^(2/3)*(2*4^(1/3)*sin(2*a)*sin_integral(2*b*x^2) - (4^(1/3)*cos_integral(2*b*x^2) + 4^(1/3)*cos_integra
l(-2*b*x^2))*cos(2*a) + 4*4^(1/3)*log(x))*(-(c*cos(b*x^2 + a)^2 - c)*sin(b*x^2 + a))^(2/3)/(cos(b*x^2 + a)^2 -
1)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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