3.3.32 \(\int \frac {\sin (a+b \sqrt [3]{c+d x})}{(c e+d e x)^{4/3}} \, dx\) [232]

Optimal. Leaf size=120 \[ \frac {3 b \sqrt [3]{c+d x} \cos (a) \text {Ci}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 b \sqrt [3]{c+d x} \sin (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3512, 15, 3378, 3384, 3380, 3383} \begin {gather*} \frac {3 b \cos (a) \sqrt [3]{c+d x} \text {CosIntegral}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 b \sin (a) \sqrt [3]{c+d x} \text {Si}\left (b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sin \left (a+b \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(4/3),x]

[Out]

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

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 3378

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 3380

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 3383

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]

Rule 3384

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

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 85, normalized size = 0.71 \begin {gather*} -\frac {3 \left (-b \sqrt [3]{c+d x} \cos (a) \text {Ci}\left (b \sqrt [3]{c+d x}\right )+\sin \left (a+b \sqrt [3]{c+d x}\right )+b \sqrt [3]{c+d x} \sin (a) \text {Si}\left (b \sqrt [3]{c+d x}\right )\right )}{d e \sqrt [3]{e (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*(c + d*x)^(1/3)]/(c*e + d*e*x)^(4/3),x]

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{\left (d e x +c e \right )^{\frac {4}{3}}}\, dx\]

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)^(4/3),x)

[Out]

int(sin(a+b*(d*x+c)^(1/3))/(d*e*x+c*e)^(4/3),x)

________________________________________________________________________________________

Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.49, size = 125, normalized size = 1.04 \begin {gather*} \frac {3 \, {\left ({\left (\Gamma \left (-1, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-1, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + \Gamma \left (-1, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + \Gamma \left (-1, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) + {\left (-i \, \Gamma \left (-1, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, \Gamma \left (-1, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, \Gamma \left (-1, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + i \, \Gamma \left (-1, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} b e^{\left (-\frac {4}{3}\right )}}{4 \, 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)^(4/3),x, algorithm="maxima")

[Out]

3/4*((gamma(-1, I*b*conjugate((d*x + c)^(1/3))) + gamma(-1, -I*b*conjugate((d*x + c)^(1/3))) + gamma(-1, I*(d*
x + c)^(1/3)*b) + gamma(-1, -I*(d*x + c)^(1/3)*b))*cos(a) + (-I*gamma(-1, I*b*conjugate((d*x + c)^(1/3))) + I*
gamma(-1, -I*b*conjugate((d*x + c)^(1/3))) - I*gamma(-1, I*(d*x + c)^(1/3)*b) + I*gamma(-1, -I*(d*x + c)^(1/3)
*b))*sin(a))*b*e^(-4/3)/d

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)^(4/3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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 {4}{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)**(4/3),x)

[Out]

Integral(sin(a + b*(c + d*x)**(1/3))/(e*(c + d*x))**(4/3), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)^(4/3),x, algorithm="giac")

[Out]

integrate(sin((d*x + c)^(1/3)*b + a)/(d*x*e + c*e)^(4/3), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{4/3}} \,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)^(4/3),x)

[Out]

int(sin(a + b*(c + d*x)^(1/3))/(c*e + d*e*x)^(4/3), x)

________________________________________________________________________________________