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

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

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 3375

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

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*}

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Mathematica [A]  time = 0.10, size = 60, normalized size = 0.52 \[ \frac {1}{4} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \left (-\cos (2 a) \text {Ci}\left (2 b x^2\right )+\sin (2 a) \text {Si}\left (2 b x^2\right )+2 \log (x)\right ) \]

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

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fricas [A]  time = 0.68, size = 100, normalized size = 0.87 \[ -\frac {4^{\frac {2}{3}} {\left (2 \cdot 4^{\frac {1}{3}} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{2}\right ) - {\left (4^{\frac {1}{3}} \operatorname {Ci}\left (2 \, b x^{2}\right ) + 4^{\frac {1}{3}} \operatorname {Ci}\left (-2 \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) + 4 \cdot 4^{\frac {1}{3}} \log \relax (x)\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {2}{3}}}{32 \, {\left (\cos \left (b x^{2} + a\right )^{2} - 1\right )}} \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x^{2} + a\right )^{3}\right )^{\frac {2}{3}}}{x}\,{d x} \]

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)

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maple [C]  time = 0.22, size = 331, normalized size = 2.88 \[ \frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{2}} \pi \,\mathrm {csgn}\left (b \,x^{2}\right )}{8 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{2}} \Si \left (2 b \,x^{2}\right )}{4 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {\left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{2}} \Ei \left (1, -2 i b \,x^{2}\right )}{8 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {\Ei \left (1, -2 i b \,x^{2}\right ) \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b \,x^{2}+2 a \right )}}{8 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}}-\frac {\ln \relax (x ) \left (i c \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b \,x^{2}+a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b \,x^{2}+a \right )}}{2 \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{2}} \]

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*Ei(1,-2*I*b*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+2*a))-1/2*ln(x)/(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]  time = 1.26, size = 55, normalized size = 0.48 \[ \frac {1}{16} \, {\left ({\left ({\rm Ei}\left (2 i \, b x^{2}\right ) + {\rm Ei}\left (-2 i \, b x^{2}\right )\right )} \cos \left (2 \, a\right ) - {\left (-i \, {\rm Ei}\left (2 i \, b x^{2}\right ) + i \, {\rm Ei}\left (-2 i \, b x^{2}\right )\right )} \sin \left (2 \, a\right ) - 4 \, \log \relax (x)\right )} c^{\frac {2}{3}} \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x^2)^3)^(2/3)/x,x)

[Out]

int((c*sin(a + b*x^2)^3)^(2/3)/x, x)

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

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)

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