3.234 \(\int \frac{\sin (a+b \sqrt [3]{c+d x})}{(c e+d e x)^{7/3}} \, dx\)

Optimal. Leaf size=267 \[ \frac{b^4 \sin (a) \sqrt [3]{c+d x} \text{CosIntegral}\left (b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}+\frac{b^4 \cos (a) \sqrt [3]{c+d x} \text{Si}\left (b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}+\frac{b^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac{b^3 \cos \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x) \sqrt [3]{e (c+d x)}}-\frac{b \cos \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}} \]

[Out]

(b^3*Cos[a + b*(c + d*x)^(1/3)])/(8*d*e^2*(e*(c + d*x))^(1/3)) - (b*Cos[a + b*(c + d*x)^(1/3)])/(4*d*e^2*(c +
d*x)^(2/3)*(e*(c + d*x))^(1/3)) + (b^4*(c + d*x)^(1/3)*CosIntegral[b*(c + d*x)^(1/3)]*Sin[a])/(8*d*e^2*(e*(c +
d*x))^(1/3)) - (3*Sin[a + b*(c + d*x)^(1/3)])/(4*d*e^2*(c + d*x)*(e*(c + d*x))^(1/3)) + (b^2*Sin[a + b*(c + d
*x)^(1/3)])/(8*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)) + (b^4*(c + d*x)^(1/3)*Cos[a]*SinIntegral[b*(c + d*x
)^(1/3)])/(8*d*e^2*(e*(c + d*x))^(1/3))

________________________________________________________________________________________

Rubi [A]  time = 0.262629, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3431, 15, 3297, 3303, 3299, 3302} \[ \frac{b^4 \sin (a) \sqrt [3]{c+d x} \text{CosIntegral}\left (b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}+\frac{b^4 \cos (a) \sqrt [3]{c+d x} \text{Si}\left (b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}+\frac{b^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac{b^3 \cos \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x) \sqrt [3]{e (c+d x)}}-\frac{b \cos \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(7/3),x]

[Out]

(b^3*Cos[a + b*(c + d*x)^(1/3)])/(8*d*e^2*(e*(c + d*x))^(1/3)) - (b*Cos[a + b*(c + d*x)^(1/3)])/(4*d*e^2*(c +
d*x)^(2/3)*(e*(c + d*x))^(1/3)) + (b^4*(c + d*x)^(1/3)*CosIntegral[b*(c + d*x)^(1/3)]*Sin[a])/(8*d*e^2*(e*(c +
d*x))^(1/3)) - (3*Sin[a + b*(c + d*x)^(1/3)])/(4*d*e^2*(c + d*x)*(e*(c + d*x))^(1/3)) + (b^2*Sin[a + b*(c + d
*x)^(1/3)])/(8*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(1/3)) + (b^4*(c + d*x)^(1/3)*Cos[a]*SinIntegral[b*(c + d*x
)^(1/3)])/(8*d*e^2*(e*(c + d*x))^(1/3))

Rule 3431

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]

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 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 \sqrt [3]{c+d x}\right )}{(c e+d e x)^{7/3}} \, dx &=\frac{3 \operatorname{Subst}\left (\int \frac{x^2 \sin (a+b x)}{\left (e x^3\right )^{7/3}} \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac{\left (3 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\sqrt [3]{c+d x}\right )}{d e^2 \sqrt [3]{e (c+d x)}}\\ &=-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x) \sqrt [3]{e (c+d x)}}+\frac{\left (3 b \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\sqrt [3]{c+d x}\right )}{4 d e^2 \sqrt [3]{e (c+d x)}}\\ &=-\frac{b \cos \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x) \sqrt [3]{e (c+d x)}}-\frac{\left (b^2 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\sqrt [3]{c+d x}\right )}{4 d e^2 \sqrt [3]{e (c+d x)}}\\ &=-\frac{b \cos \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x) \sqrt [3]{e (c+d x)}}+\frac{b^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}-\frac{\left (b^3 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac{b^3 \cos \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}-\frac{b \cos \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x) \sqrt [3]{e (c+d x)}}+\frac{b^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac{\left (b^4 \sqrt [3]{c+d x}\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac{b^3 \cos \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}-\frac{b \cos \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x) \sqrt [3]{e (c+d x)}}+\frac{b^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac{\left (b^4 \sqrt [3]{c+d x} \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}+\frac{\left (b^4 \sqrt [3]{c+d x} \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}\\ &=\frac{b^3 \cos \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}-\frac{b \cos \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x)^{2/3} \sqrt [3]{e (c+d x)}}+\frac{b^4 \sqrt [3]{c+d x} \text{Ci}\left (b \sqrt [3]{c+d x}\right ) \sin (a)}{8 d e^2 \sqrt [3]{e (c+d x)}}-\frac{3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{4 d e^2 (c+d x) \sqrt [3]{e (c+d x)}}+\frac{b^2 \sin \left (a+b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)}}+\frac{b^4 \sqrt [3]{c+d x} \cos (a) \text{Si}\left (b \sqrt [3]{c+d x}\right )}{8 d e^2 \sqrt [3]{e (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.355332, size = 184, normalized size = 0.69 \[ \frac{b^4 \sin (a) (c+d x)^{4/3} \text{CosIntegral}\left (b \sqrt [3]{c+d x}\right )+b^4 \cos (a) (c+d x)^{4/3} \text{Si}\left (b \sqrt [3]{c+d x}\right )+b^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )+b^3 c \cos \left (a+b \sqrt [3]{c+d x}\right )+b^3 d x \cos \left (a+b \sqrt [3]{c+d x}\right )-6 \sin \left (a+b \sqrt [3]{c+d x}\right )-2 b \sqrt [3]{c+d x} \cos \left (a+b \sqrt [3]{c+d x}\right )}{8 d e (e (c+d x))^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(7/3),x]

[Out]

(b^3*c*Cos[a + b*(c + d*x)^(1/3)] + b^3*d*x*Cos[a + b*(c + d*x)^(1/3)] - 2*b*(c + d*x)^(1/3)*Cos[a + b*(c + d*
x)^(1/3)] + b^4*(c + d*x)^(4/3)*CosIntegral[b*(c + d*x)^(1/3)]*Sin[a] - 6*Sin[a + b*(c + d*x)^(1/3)] + b^2*(c
+ d*x)^(2/3)*Sin[a + b*(c + d*x)^(1/3)] + b^4*(c + d*x)^(4/3)*Cos[a]*SinIntegral[b*(c + d*x)^(1/3)])/(8*d*e*(e
*(c + d*x))^(4/3))

________________________________________________________________________________________

Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+b\sqrt [3]{dx+c} \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)^(1/3))/(d*e*x+c*e)^(7/3),x)

[Out]

int(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(7/3),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)^(1/3))/(d*e*x+c*e)^(7/3),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{2}{3}} \sin \left ({\left (d x + c\right )}^{\frac{1}{3}} b + a\right )}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \end{align*}

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)^(7/3),x, algorithm="fricas")

[Out]

integral((d*e*x + c*e)^(2/3)*sin((d*x + c)^(1/3)*b + a)/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e
^3), 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)**(1/3))/(d*e*x+c*e)**(7/3),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{1}{3}} b + a\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)^(1/3))/(d*e*x+c*e)^(7/3),x, algorithm="giac")

[Out]

integrate(sin((d*x + c)^(1/3)*b + a)/(d*e*x + c*e)^(7/3), x)