3.4.49 \(\int \frac {(c \sin ^3(a+b x^2))^{2/3}}{x^3} \, dx\) [349]

Optimal. Leaf size=161 \[ -\frac {\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac {\cos \left (2 \left (a+b x^2\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac {1}{2} b \text {Ci}\left (2 b x^2\right ) \csc ^2\left (a+b x^2\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac {1}{2} b \cos (2 a) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \text {Si}\left (2 b x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6852, 3484, 3461, 3378, 3384, 3380, 3383} \begin {gather*} \frac {1}{2} b \sin (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}{2} b \cos (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 {\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2}+\frac {\cos \left (2 \left (a+b x^2\right )\right ) \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^2]^3)^(2/3)/x^3,x]

[Out]

-1/4*(Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/x^2 + (Cos[2*(a + b*x^2)]*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2
]^3)^(2/3))/(4*x^2) + (b*CosIntegral[2*b*x^2]*Csc[a + b*x^2]^2*Sin[2*a]*(c*Sin[a + b*x^2]^3)^(2/3))/2 + (b*Cos
[2*a]*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3)*SinIntegral[2*b*x^2])/2

Rule 3378

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

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3461

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3484

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

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Mathematica [A]
time = 0.10, size = 79, normalized size = 0.49 \begin {gather*} \frac {\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (-1+\cos \left (2 \left (a+b x^2\right )\right )+2 b x^2 \text {Ci}\left (2 b x^2\right ) \sin (2 a)+2 b x^2 \cos (2 a) \text {Si}\left (2 b x^2\right )\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3)*(-1 + Cos[2*(a + b*x^2)] + 2*b*x^2*CosIntegral[2*b*x^2]*Sin[2*a]
+ 2*b*x^2*Cos[2*a]*SinIntegral[2*b*x^2]))/(4*x^2)

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Maple [C] Result contains complex when optimal does not.
time = 0.17, size = 277, normalized size = 1.72

method	result	size
risch	\(-\frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}}}{8 x^{2} \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i x^{2} b} b \expIntegral \left (1, 2 i x^{2} b \right )}{4 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} \left (-\frac {{\mathrm e}^{4 i \left (b \,x^{2}+a \right )}}{2 x^{2}}-i b \expIntegral \left (1, -2 i x^{2} b \right ) {\mathrm e}^{2 i \left (b \,x^{2}+2 a \right )}\right )}{4 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b \,x^{2}+a \right )}}{4 x^{2} \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}\)	\(277\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/x^2/(exp(2*I*(b*x^2+a))-1)^2*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(2/3)+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)*b*Ei(1,2*I*b*x^2)+1/4/(exp
(2*I*(b*x^2+a))-1)^2*(I*c*(exp(2*I*(b*x^2+a))-1)^3*exp(-3*I*(b*x^2+a)))^(2/3)*(-1/2/x^2*exp(4*I*(b*x^2+a))-I*b
*Ei(1,-2*I*b*x^2)*exp(2*I*(b*x^2+2*a)))+1/4/x^2/(exp(2*I*(b*x^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))

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Maxima [C] Result contains complex when optimal does not.
time = 0.61, size = 64, normalized size = 0.40 \begin {gather*} -\frac {{\left ({\left ({\left (i \, \Gamma \left (-1, 2 i \, b x^{2}\right ) - i \, \Gamma \left (-1, -2 i \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) + {\left (\Gamma \left (-1, 2 i \, b x^{2}\right ) + \Gamma \left (-1, -2 i \, b x^{2}\right )\right )} \sin \left (2 \, a\right )\right )} b x^{2} - 1\right )} c^{\frac {2}{3}}}{8 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(((I*gamma(-1, 2*I*b*x^2) - I*gamma(-1, -2*I*b*x^2))*cos(2*a) + (gamma(-1, 2*I*b*x^2) + gamma(-1, -2*I*b*
x^2))*sin(2*a))*b*x^2 - 1)*c^(2/3)/x^2

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Fricas [A]
time = 0.38, size = 132, normalized size = 0.82 \begin {gather*} -\frac {4^{\frac {2}{3}} {\left (2 \cdot 4^{\frac {1}{3}} b x^{2} \cos \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{2}\right ) + 2 \cdot 4^{\frac {1}{3}} \cos \left (b x^{2} + a\right )^{2} + {\left (4^{\frac {1}{3}} b x^{2} \operatorname {Ci}\left (2 \, b x^{2}\right ) + 4^{\frac {1}{3}} b x^{2} \operatorname {Ci}\left (-2 \, b x^{2}\right )\right )} \sin \left (2 \, a\right ) - 2 \cdot 4^{\frac {1}{3}}\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 (x^{2} \cos \left (b x^{2} + a\right )^{2} - x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*4^(2/3)*(2*4^(1/3)*b*x^2*cos(2*a)*sin_integral(2*b*x^2) + 2*4^(1/3)*cos(b*x^2 + a)^2 + (4^(1/3)*b*x^2*co
s_integral(2*b*x^2) + 4^(1/3)*b*x^2*cos_integral(-2*b*x^2))*sin(2*a) - 2*4^(1/3))*(-(c*cos(b*x^2 + a)^2 - c)*s
in(b*x^2 + a))^(2/3)/(x^2*cos(b*x^2 + a)^2 - x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \sin ^{3}{\left (a + b x^{2} \right )}\right )^{\frac {2}{3}}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{2/3}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x^2)^3)^(2/3)/x^3,x)

[Out]

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

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