3.256 \(\int \frac{\sin (a+\frac{b}{(c+d x)^{2/3}})}{(c e+d e x)^{7/3}} \, dx\)

Optimal. Leaf size=95 \[ \frac{3 \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac{3 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b^2 d e^2 \sqrt [3]{e (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0955332, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3435, 3381, 3379, 3296, 2637} \[ \frac{3 \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac{3 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b^2 d e^2 \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(7/3),x]

[Out]

(3*Cos[a + b/(c + d*x)^(2/3)])/(2*b*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)) - (3*(c + d*x)^(1/3)*Sin[a + b/
(c + d*x)^(2/3)])/(2*b^2*d*e^2*(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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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)^{7/3}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{b}{x^{2/3}}\right )}{(e x)^{7/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\sqrt [3]{c+d x} \operatorname{Subst}\left (\int \frac{\sin \left (a+\frac{b}{x^{2/3}}\right )}{x^{7/3}} \, dx,x,c+d x\right )}{d e^2 \sqrt [3]{e (c+d x)}}\\ &=-\frac{\left (3 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int x \sin (a+b x) \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac{3 \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac{\left (3 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \cos (a+b x) \, dx,x,\frac{1}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac{3 \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac{3 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )}{2 b^2 d e^2 \sqrt [3]{e (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.108754, size = 72, normalized size = 0.76 \[ -\frac{3 (c+d x)^{5/3} \left ((c+d x)^{2/3} \sin \left (a+\frac{b}{(c+d x)^{2/3}}\right )-b \cos \left (a+\frac{b}{(c+d x)^{2/3}}\right )\right )}{2 b^2 d (e (c+d x))^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/(c + d*x)^(2/3)]/(c*e + d*e*x)^(7/3),x]

[Out]

(-3*(c + d*x)^(5/3)*(-(b*Cos[a + b/(c + d*x)^(2/3)]) + (c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(2/3)]))/(2*b^2*d*(
e*(c + d*x))^(7/3))

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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{7}{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)^(7/3),x)

[Out]

int(sin(a+b/(d*x+c)^(2/3))/(d*e*x+c*e)^(7/3),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \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)^(7/3),x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [A]  time = 6.57708, size = 317, normalized size = 3.34 \begin{align*} \frac{3 \,{\left ({\left (d e x + c e\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{2}{3}} b \cos \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\right ) -{\left (d e x + c e\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{4}{3}} \sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{1}{3}} b}{d x + c}\right )\right )}}{2 \,{\left (b^{2} d^{3} e^{3} x^{2} + 2 \, b^{2} c d^{2} e^{3} x + b^{2} c^{2} d e^{3}\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)^(7/3),x, algorithm="fricas")

[Out]

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

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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)**(2/3))/(d*e*x+c*e)**(7/3),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{2}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac{7}{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)^(7/3),x, algorithm="giac")

[Out]

integrate(sin(a + b/(d*x + c)^(2/3))/(d*e*x + c*e)^(7/3), x)