3.244 \(\int \frac{\sin (a+\frac{b}{\sqrt [3]{c+d x}})}{(c e+d e x)^{2/3}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{3 b \cos (a) (c+d x)^{2/3} \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{3 b \sin (a) (c+d x)^{2/3} \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{3 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.129129, antiderivative size = 116, 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} \[ -\frac{3 b \cos (a) (c+d x)^{2/3} \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{3 b \sin (a) (c+d x)^{2/3} \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}}+\frac{3 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(2/3),x]

[Out]

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

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

Mathematica [A]  time = 0.181369, size = 88, normalized size = 0.76 \[ \frac{3 \left (-b \cos (a) (c+d x)^{2/3} \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )+b \sin (a) (c+d x)^{2/3} \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )+(c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )\right )}{d (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(2/3),x]

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+{b{\frac{1}{\sqrt [3]{dx+c}}}} \right ) \left ( dex+ce \right ) ^{-{\frac{2}{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)^(2/3),x)

[Out]

int(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(2/3),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)^(2/3),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{\sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right )}{{\left (d e x + c e\right )}^{\frac{2}{3}}}, 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)^(2/3),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac{2}{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)^(2/3),x, algorithm="giac")

[Out]

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