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

Optimal. Leaf size=126 \[ \frac{3 b \cos (a) (c+d x)^{2/3} \text{CosIntegral}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac{3 b \sin (a) (c+d x)^{2/3} \text{Si}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac{3 \sin \left (a+b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.153494, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {3435, 3381, 3379, 3297, 3303, 3299, 3302} \[ \frac{3 b \cos (a) (c+d x)^{2/3} \text{CosIntegral}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac{3 b \sin (a) (c+d x)^{2/3} \text{Si}\left (b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}}-\frac{3 \sin \left (a+b (c+d x)^{2/3}\right )}{2 d e (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*(c + d*x)^(2/3)*Cos[a]*CosIntegral[b*(c + d*x)^(2/3)])/(2*d*e*(e*(c + d*x))^(2/3)) - (3*Sin[a + b*(c + d*
x)^(2/3)])/(2*d*e*(e*(c + d*x))^(2/3)) - (3*b*(c + d*x)^(2/3)*Sin[a]*SinIntegral[b*(c + d*x)^(2/3)])/(2*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 3297

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

Rule 3303

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

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.15564, size = 87, normalized size = 0.69 \[ -\frac{3 \left (-b \cos (a) (c+d x)^{2/3} \text{CosIntegral}\left (b (c+d x)^{2/3}\right )+b \sin (a) (c+d x)^{2/3} \text{Si}\left (b (c+d x)^{2/3}\right )+\sin \left (a+b (c+d x)^{2/3}\right )\right )}{2 d e (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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 ) \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 [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)^(5/3),x, algorithm="maxima")

[Out]

Exception raised: IndexError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d e x + c e\right )}^{\frac{1}{3}} \sin \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\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]

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

________________________________________________________________________________________

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 ({\left (d x + c\right )}^{\frac{2}{3}} b + a\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((d*x + c)^(2/3)*b + a)/(d*e*x + c*e)^(5/3), x)