3.315 \(\int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x} \, dx\)

Optimal. Leaf size=55 \[ \sin (a) \text {Ci}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}+\cos (a) \text {Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.17, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6720, 3303, 3299, 3302} \[ \sin (a) \text {CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}+\cos (a) \text {Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x]^3)^(1/3)/x,x]

[Out]

CosIntegral[b*x]*Csc[a + b*x]*Sin[a]*(c*Sin[a + b*x]^3)^(1/3) + Cos[a]*Csc[a + b*x]*(c*Sin[a + b*x]^3)^(1/3)*S
inIntegral[b*x]

Rule 3299

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 3302

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 3303

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 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 {\sqrt [3]{c \sin ^3(a+b x)}}{x} \, dx &=\left (\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\sin (a+b x)}{x} \, dx\\ &=\left (\cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\sin (b x)}{x} \, dx+\left (\csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\cos (b x)}{x} \, dx\\ &=\text {Ci}(b x) \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}+\cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \text {Si}(b x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 36, normalized size = 0.65 \[ \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} (\sin (a) \text {Ci}(b x)+\cos (a) \text {Si}(b x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x]^3)^(1/3)/x,x]

[Out]

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

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fricas [A]  time = 0.79, size = 80, normalized size = 1.45 \[ -\frac {4^{\frac {1}{3}} {\left (2 \cdot 4^{\frac {2}{3}} \cos \relax (a) \operatorname {Si}\left (b x\right ) + {\left (4^{\frac {2}{3}} \operatorname {Ci}\left (b x\right ) + 4^{\frac {2}{3}} \operatorname {Ci}\left (-b x\right )\right )} \sin \relax (a)\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {1}{3}} \sin \left (b x + a\right )}{8 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a)^3)^(1/3)/x,x, algorithm="fricas")

[Out]

-1/8*4^(1/3)*(2*4^(2/3)*cos(a)*sin_integral(b*x) + (4^(2/3)*cos_integral(b*x) + 4^(2/3)*cos_integral(-b*x))*si
n(a))*(-(c*cos(b*x + a)^2 - c)*sin(b*x + a))^(1/3)*sin(b*x + a)/(cos(b*x + a)^2 - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sin \left (b x + a\right )^{3}\right )^{\frac {1}{3}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a)^3)^(1/3)/x,x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a)^3)^(1/3)/x, x)

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maple [C]  time = 0.19, size = 228, normalized size = 4.15 \[ -\frac {\Ei \left (1, -i b x \right ) \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i \left (b x +2 a \right )}}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b x} \pi \,\mathrm {csgn}\left (b x \right )}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b x} \Si \left (b x \right )}{{\mathrm e}^{2 i \left (b x +a \right )}-1}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b x} \Ei \left (1, -i b x \right )}{2 \,{\mathrm e}^{2 i \left (b x +a \right )}-2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(b*x+a)^3)^(1/3)/x,x)

[Out]

-1/2*Ei(1,-I*b*x)/(exp(2*I*(b*x+a))-1)*(I*c*(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(1/3)*exp(I*(b*x+2*a))-1
/2*I*(I*c*(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(1/3)/(exp(2*I*(b*x+a))-1)*exp(I*b*x)*Pi*csgn(b*x)+I*(I*c*
(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(1/3)/(exp(2*I*(b*x+a))-1)*exp(I*b*x)*Si(b*x)+1/2*(I*c*(exp(2*I*(b*x
+a))-1)^3*exp(-3*I*(b*x+a)))^(1/3)/(exp(2*I*(b*x+a))-1)*exp(I*b*x)*Ei(1,-I*b*x)

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maxima [C]  time = 1.21, size = 42, normalized size = 0.76 \[ \frac {1}{4} \, {\left ({\left (i \, E_{1}\left (i \, b x\right ) - i \, E_{1}\left (-i \, b x\right )\right )} \cos \relax (a) + {\left (E_{1}\left (i \, b x\right ) + E_{1}\left (-i \, b x\right )\right )} \sin \relax (a)\right )} c^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a)^3)^(1/3)/x,x, algorithm="maxima")

[Out]

1/4*((I*exp_integral_e(1, I*b*x) - I*exp_integral_e(1, -I*b*x))*cos(a) + (exp_integral_e(1, I*b*x) + exp_integ
ral_e(1, -I*b*x))*sin(a))*c^(1/3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{1/3}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x)^3)^(1/3)/x,x)

[Out]

int((c*sin(a + b*x)^3)^(1/3)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a)**3)**(1/3)/x,x)

[Out]

Integral((c*sin(a + b*x)**3)**(1/3)/x, x)

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