3.255 \(\int \frac{\sin (a+\frac{b}{(c+d x)^{2/3}})}{(c e+d e x)^{5/3}} \, dx\)

Optimal. Leaf size=47 \[ \frac{3 (c+d x)^{2/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.066342, antiderivative size = 47, 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} \[ \frac{3 (c+d x)^{2/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(5/3),x]

[Out]

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

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+\frac{b}{(c+d x)^{2/3}}\right )}{(c e+d e x)^{5/3}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{b}{x^{2/3}}\right )}{(e x)^{5/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x)^{2/3} \operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{b}{x^{2/3}}\right )}{x^{5/3}} \, dx,x,c+d x\right )}{d e (e (c+d x))^{2/3}}\\ &=-\frac{\left (3 (c+d x)^{2/3}\right ) \operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d e (e (c+d x))^{2/3}}\\ &=\frac{3 (c+d x)^{2/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e (e (c+d x))^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0819698, size = 44, normalized size = 0.94 \[ \frac{3 (c+d x)^{5/3} \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d (e (c+d x))^{5/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(5/3),x]

[Out]

(3*(c + d*x)^(5/3)*Cos[a + b/(c + d*x)^(2/3)])/(2*b*d*(e*(c + d*x))^(5/3))

________________________________________________________________________________________

Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+{b \left ( dx+c \right ) ^{-{\frac{2}{3}}}} \right ) \left ( dex+ce \right ) ^{-{\frac{5}{3}}}}\, 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)^(5/3),x)

[Out]

int(sin(a+b/(d*x+c)^(2/3))/(d*e*x+c*e)^(5/3),x)

________________________________________________________________________________________

Maxima [A]  time = 1.03258, size = 42, normalized size = 0.89 \begin{align*} \frac{3 \, \cos \left (\frac{{\left (d x + c\right )}^{\frac{2}{3}} a + b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right )}{2 \, b d e^{\frac{5}{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)^(5/3),x, algorithm="maxima")

[Out]

3/2*cos(((d*x + c)^(2/3)*a + b)/(d*x + c)^(2/3))/(b*d*e^(5/3))

________________________________________________________________________________________

Fricas [A]  time = 1.46317, size = 157, normalized size = 3.34 \begin{align*} \frac{3 \,{\left (d e x + c e\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}} \cos \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\right )}{2 \,{\left (b d^{2} e^{2} x + b c d e^{2}\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)^(5/3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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)**(2/3))/(d*e*x+c*e)**(5/3),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac{5}{3}}}\,{d x} \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)^(5/3),x, algorithm="giac")

[Out]

integrate(sin(a + b/(d*x + c)^(2/3))/(d*e*x + c*e)^(5/3), x)