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

Optimal. Leaf size=42 \[ -\frac{3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d (e (c+d x))^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.0515908, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3431, 15, 2638} \[ -\frac{3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(c + d*x)^(2/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0678411, size = 42, normalized size = 1. \[ -\frac{3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+b\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 [A]  time = 1.0494, size = 31, normalized size = 0.74 \begin{align*} -\frac{3 \, \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{b d e^{\frac{2}{3}}} \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]

-3*cos((d*x + c)^(1/3)*b + a)/(b*d*e^(2/3))

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Fricas [A]  time = 1.66474, size = 120, normalized size = 2.86 \begin{align*} -\frac{3 \,{\left (d e x + c e\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \cos \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{b d^{2} e x + b c d e} \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]

-3*(d*e*x + c*e)^(1/3)*(d*x + c)^(2/3)*cos((d*x + c)^(1/3)*b + a)/(b*d^2*e*x + b*c*d*e)

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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{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 [A]  time = 1.17757, size = 47, normalized size = 1.12 \begin{align*} -\frac{3 \, \cos \left ({\left ({\left (d x e + c e\right )}^{\frac{1}{3}} b e^{\frac{2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\left (-\frac{2}{3}\right )}}{b d} \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]

-3*cos(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))*e^(-2/3)/(b*d)