3.232 \(\int \frac{\sin (a+b \sqrt [3]{c+d x})}{(c e+d e x)^{4/3}} \, dx\)

Optimal. Leaf size=120 \[ \frac{3 b \cos (a) \sqrt [3]{c+d x} \text{CosIntegral}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 b \sin (a) \sqrt [3]{c+d x} \text{Si}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.146167, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3431, 15, 3297, 3303, 3299, 3302} \[ \frac{3 b \cos (a) \sqrt [3]{c+d x} \text{CosIntegral}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 b \sin (a) \sqrt [3]{c+d x} \text{Si}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(4/3),x]

[Out]

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

Rule 3431

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - (e*h)/f + (h*x^(1/n))/f)^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 3297

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 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 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]

Rubi steps

\begin{align*} \int \frac{\sin \left (a+b \sqrt [3]{c+d x}\right )}{(c e+d e x)^{4/3}} \, dx &=\frac{3 \operatorname{Subst}\left (\int \frac{x^2 \sin (a+b x)}{\left (e x^3\right )^{4/3}} \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac{\left (3 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}+\frac{\left (3 b \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}+\frac{\left (3 b \sqrt [3]{c+d x} \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{\left (3 b \sqrt [3]{c+d x} \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=\frac{3 b \sqrt [3]{c+d x} \cos (a) \text{Ci}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac{3 b \sqrt [3]{c+d x} \sin (a) \text{Si}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.14493, size = 85, normalized size = 0.71 \[ -\frac{3 \left (-b \cos (a) \sqrt [3]{c+d x} \text{CosIntegral}\left (b \sqrt [3]{c+d x}\right )+b \sin (a) \sqrt [3]{c+d x} \text{Si}\left (b \sqrt [3]{c+d x}\right )+\sin \left (a+b \sqrt [3]{c+d x}\right )\right )}{d e \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(4/3),x]

[Out]

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

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+b\sqrt [3]{dx+c} \right ) \left ( dex+ce \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(4/3),x)

[Out]

int(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(4/3),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(4/3),x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d e x + c e\right )}^{\frac{2}{3}} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(4/3),x, algorithm="fricas")

[Out]

integral((d*e*x + c*e)^(2/3)*sin((d*x + c)^(1/3)*b + a)/(d^2*e^2*x^2 + 2*c*d*e^2*x + c^2*e^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (a + b \sqrt [3]{c + d x} \right )}}{\left (e \left (c + d x\right )\right )^{\frac{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*(d*x+c)**(1/3))/(d*e*x+c*e)**(4/3),x)

[Out]

Integral(sin(a + b*(c + d*x)**(1/3))/(e*(c + d*x))**(4/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(4/3),x, algorithm="giac")

[Out]

integrate(sin((d*x + c)^(1/3)*b + a)/(d*e*x + c*e)^(4/3), x)