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

Optimal. Leaf size=44 \[ -\frac{3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0628817, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3435, 3381, 3379, 2638} \[ -\frac{3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3435

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

Rule 3381

Int[((e_)*(x_))^(m_)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[(e^IntPart[m]*(e*x)
^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] &&
IntegerQ[Simplify[(m + 1)/n]]

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

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

Mathematica [A]  time = 0.074525, size = 44, normalized size = 1. \[ -\frac{3 \sqrt [3]{c+d x} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+b \left ( dx+c \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{dex+ce}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04933, size = 31, normalized size = 0.7 \begin{align*} -\frac{3 \, \cos \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )}{2 \, b d e^{\frac{1}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.71172, size = 123, normalized size = 2.8 \begin{align*} -\frac{3 \,{\left (d e x + c e\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}} \cos \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )}{2 \,{\left (b d^{2} e x + b c d e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(d*e*x + c*e)^(2/3)*(d*x + c)^(1/3)*cos((d*x + c)^(2/3)*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 \left (c + d x\right )^{\frac{2}{3}} \right )}}{\sqrt [3]{e \left (c + d x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.22065, size = 70, normalized size = 1.59 \begin{align*} -\frac{3 \,{\left (\cos \left ({\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} + a\right ) + \cos \left (-{\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} - a\right )\right )} e^{\left (-\frac{1}{3}\right )}}{4 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(cos((d*x*e + c*e)^(2/3)*b*e^(-2/3) + a) + cos(-(d*x*e + c*e)^(2/3)*b*e^(-2/3) - a))*e^(-1/3)/(b*d)