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

Optimal. Leaf size=45 \[ \frac{3 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e \sqrt [3]{e (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0474572, antiderivative size = 45, 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} \[ \frac{3 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d e \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0664153, size = 42, normalized size = 0.93 \[ \frac{3 (c+d x)^{4/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{b d (e (c+d x))^{4/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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{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 [A]  time = 1.03975, size = 42, normalized size = 0.93 \begin{align*} \frac{3 \, \cos \left (\frac{{\left (d x + c\right )}^{\frac{1}{3}} a + b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right )}{b d e^{\frac{4}{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)^(4/3),x, algorithm="maxima")

[Out]

3*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(b*d*e^(4/3))

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Fricas [A]  time = 2.07689, size = 154, normalized size = 3.42 \begin{align*} \frac{3 \,{\left (d e x + c e\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} \cos \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right )}{b d^{2} e^{2} x + b c d e^{2}} \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]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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]

Timed out

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