3.3.31 \(\int \frac {\sin (a+b \sqrt [3]{c+d x})}{(c e+d e x)^{2/3}} \, dx\) [231]

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

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Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 = {3512, 15, 2718} \begin {gather*} -\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 {gather*}

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 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 2718

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

Rule 3512

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]

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 \text {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 ) \text {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*}

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Mathematica [A]
time = 0.05, size = 42, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{\left (d e x +c e \right )^{\frac {2}{3}}}\, dx\]

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 = 0.32, size = 22, normalized size = 0.52 \begin {gather*} -\frac {3 \, \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e^{\left (-\frac {2}{3}\right )}}{b d} \end {gather*}

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)*e^(-2/3)/(b*d)

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Fricas [A]
time = 0.35, size = 34, normalized size = 0.81 \begin {gather*} -\frac {3 \, {\left (d x + c\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e^{\left (-\frac {2}{3}\right )}}{b d^{2} x + b c d} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 6.51, size = 35, normalized size = 0.83 \begin {gather*} -\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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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