3.218 \(\int (e+f x) \sin (a+\frac{b}{\sqrt [3]{c+d x}}) \, dx\)

Optimal. Leaf size=419 \[ \frac{b^3 \cos (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 \sin (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 \sqrt [3]{c+d x} (d e-c f) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{(c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac{b (c+d x)^{2/3} (d e-c f) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2} \]

[Out]

(b^5*f*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(240*d^2) + (b*(d*e - c*f)*(c + d*x)^(2/3)*Cos[a + b/(c + d
*x)^(1/3)])/(2*d^2) - (b^3*f*(c + d*x)*Cos[a + b/(c + d*x)^(1/3)])/(120*d^2) + (b*f*(c + d*x)^(5/3)*Cos[a + b/
(c + d*x)^(1/3)])/(10*d^2) + (b^3*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)])/(2*d^2) + (b^6*f*CosInteg
ral[b/(c + d*x)^(1/3)]*Sin[a])/(240*d^2) - (b^2*(d*e - c*f)*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(2*d^2
) + (b^4*f*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)])/(240*d^2) + ((d*e - c*f)*(c + d*x)*Sin[a + b/(c + d*x)^
(1/3)])/d^2 - (b^2*f*(c + d*x)^(4/3)*Sin[a + b/(c + d*x)^(1/3)])/(40*d^2) + (f*(c + d*x)^2*Sin[a + b/(c + d*x)
^(1/3)])/(2*d^2) + (b^6*f*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])/(240*d^2) - (b^3*(d*e - c*f)*Sin[a]*SinIntegr
al[b/(c + d*x)^(1/3)])/(2*d^2)

________________________________________________________________________________________

Rubi [A]  time = 0.504016, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3431, 3297, 3303, 3299, 3302} \[ \frac{b^3 \cos (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 \sin (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 \sqrt [3]{c+d x} (d e-c f) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{(c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac{b (c+d x)^{2/3} (d e-c f) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^5*f*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(240*d^2) + (b*(d*e - c*f)*(c + d*x)^(2/3)*Cos[a + b/(c + d
*x)^(1/3)])/(2*d^2) - (b^3*f*(c + d*x)*Cos[a + b/(c + d*x)^(1/3)])/(120*d^2) + (b*f*(c + d*x)^(5/3)*Cos[a + b/
(c + d*x)^(1/3)])/(10*d^2) + (b^3*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)])/(2*d^2) + (b^6*f*CosInteg
ral[b/(c + d*x)^(1/3)]*Sin[a])/(240*d^2) - (b^2*(d*e - c*f)*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(2*d^2
) + (b^4*f*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)])/(240*d^2) + ((d*e - c*f)*(c + d*x)*Sin[a + b/(c + d*x)^
(1/3)])/d^2 - (b^2*f*(c + d*x)^(4/3)*Sin[a + b/(c + d*x)^(1/3)])/(40*d^2) + (f*(c + d*x)^2*Sin[a + b/(c + d*x)
^(1/3)])/(2*d^2) + (b^6*f*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])/(240*d^2) - (b^3*(d*e - c*f)*Sin[a]*SinIntegr
al[b/(c + d*x)^(1/3)])/(2*d^2)

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 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 (e+f x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) \, dx &=-\frac{3 \operatorname{Subst}\left (\int \left (\frac{f \sin (a+b x)}{d x^7}+\frac{(d e-c f) \sin (a+b x)}{d x^4}\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac{(3 f) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^7} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{(3 (d e-c f)) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^2}\\ &=\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{(b f) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^6} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{(b (d e-c f)) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^2}\\ &=\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{\left (b^2 f\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{\left (b^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ &=\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{\left (b^3 f\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{\left (b^3 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ &=\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{\left (b^4 f\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{\left (b^3 (d e-c f) \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{\left (b^3 (d e-c f) \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ &=\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{b^3 (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 (d e-c f) \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{\left (b^5 f\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{240 d^2}\\ &=\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{b^3 (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 (d e-c f) \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{\left (b^6 f\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{240 d^2}\\ &=\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{b^3 (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 (d e-c f) \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{\left (b^6 f \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{\left (b^6 f \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{240 d^2}\\ &=\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{b^3 (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{240 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 (d e-c f) \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ \end{align*}

Mathematica [A]  time = 0.855827, size = 540, normalized size = 1.29 \[ \frac{b^3 f \left (b^3 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )+b^3 \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )-120 c \cos (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )+120 c \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )\right )}{240 d^2}+\frac{b^3 e \left (\cos (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )-\sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )\right )}{2 d}+\frac{f \sqrt [3]{c+d x} \cos \left (\frac{b}{\sqrt [3]{c+d x}}\right ) \left (b^4 \sin (a) \sqrt [3]{c+d x}-6 b^2 \sin (a) (c+d x)-2 b^3 \cos (a) (c+d x)^{2/3}+120 b^2 c \sin (a)+b^5 \cos (a)+24 b \cos (a) (c+d x)^{4/3}-120 b c \cos (a) \sqrt [3]{c+d x}+120 \sin (a) (c+d x)^{5/3}-240 c \sin (a) (c+d x)^{2/3}\right )}{240 d^2}+\frac{f \sqrt [3]{c+d x} \sin \left (\frac{b}{\sqrt [3]{c+d x}}\right ) \left (2 b^3 \sin (a) (c+d x)^{2/3}+b^4 \cos (a) \sqrt [3]{c+d x}-6 b^2 \cos (a) (c+d x)+120 b^2 c \cos (a)+b^5 (-\sin (a))-24 b \sin (a) (c+d x)^{4/3}+120 b c \sin (a) \sqrt [3]{c+d x}+120 \cos (a) (c+d x)^{5/3}-240 c \cos (a) (c+d x)^{2/3}\right )}{240 d^2}+\frac{e \sqrt [3]{c+d x} \cos \left (\frac{b}{\sqrt [3]{c+d x}}\right ) \left (b^2 (-\sin (a))+b \cos (a) \sqrt [3]{c+d x}+2 \sin (a) (c+d x)^{2/3}\right )}{2 d}+\frac{e \sqrt [3]{c+d x} \sin \left (\frac{b}{\sqrt [3]{c+d x}}\right ) \left (b^2 (-\cos (a))-b \sin (a) \sqrt [3]{c+d x}+2 \cos (a) (c+d x)^{2/3}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(c + d*x)^(1/3)*Cos[b/(c + d*x)^(1/3)]*(b*(c + d*x)^(1/3)*Cos[a] - b^2*Sin[a] + 2*(c + d*x)^(2/3)*Sin[a]))/
(2*d) + (f*(c + d*x)^(1/3)*Cos[b/(c + d*x)^(1/3)]*(b^5*Cos[a] - 120*b*c*(c + d*x)^(1/3)*Cos[a] - 2*b^3*(c + d*
x)^(2/3)*Cos[a] + 24*b*(c + d*x)^(4/3)*Cos[a] + 120*b^2*c*Sin[a] + b^4*(c + d*x)^(1/3)*Sin[a] - 240*c*(c + d*x
)^(2/3)*Sin[a] - 6*b^2*(c + d*x)*Sin[a] + 120*(c + d*x)^(5/3)*Sin[a]))/(240*d^2) + (e*(c + d*x)^(1/3)*(-(b^2*C
os[a]) + 2*(c + d*x)^(2/3)*Cos[a] - b*(c + d*x)^(1/3)*Sin[a])*Sin[b/(c + d*x)^(1/3)])/(2*d) + (f*(c + d*x)^(1/
3)*(120*b^2*c*Cos[a] + b^4*(c + d*x)^(1/3)*Cos[a] - 240*c*(c + d*x)^(2/3)*Cos[a] - 6*b^2*(c + d*x)*Cos[a] + 12
0*(c + d*x)^(5/3)*Cos[a] - b^5*Sin[a] + 120*b*c*(c + d*x)^(1/3)*Sin[a] + 2*b^3*(c + d*x)^(2/3)*Sin[a] - 24*b*(
c + d*x)^(4/3)*Sin[a])*Sin[b/(c + d*x)^(1/3)])/(240*d^2) + (b^3*e*(Cos[a]*CosIntegral[b/(c + d*x)^(1/3)] - Sin
[a]*SinIntegral[b/(c + d*x)^(1/3)]))/(2*d) + (b^3*f*(-120*c*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)] + b^3*CosInt
egral[b/(c + d*x)^(1/3)]*Sin[a] + b^3*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)] + 120*c*Sin[a]*SinIntegral[b/(c +
d*x)^(1/3)]))/(240*d^2)

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Maple [A]  time = 0.03, size = 391, normalized size = 0.9 \begin{align*} -3\,{\frac{{b}^{3}}{{d}^{2}} \left ( -cf \left ( -1/3\,{\frac{dx+c}{{b}^{3}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-1/6\,{\frac{ \left ( dx+c \right ) ^{2/3}}{{b}^{2}}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+1/6\,{\frac{\sqrt [3]{dx+c}}{b}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+1/6\,{\it Si} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \sin \left ( a \right ) -1/6\,{\it Ci} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \cos \left ( a \right ) \right ) +de \left ( -1/3\,{\frac{dx+c}{{b}^{3}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-1/6\,{\frac{ \left ( dx+c \right ) ^{2/3}}{{b}^{2}}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+1/6\,{\frac{\sqrt [3]{dx+c}}{b}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+1/6\,{\it Si} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \sin \left ( a \right ) -1/6\,{\it Ci} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \cos \left ( a \right ) \right ) +{b}^{3}f \left ( -1/6\,{\frac{ \left ( dx+c \right ) ^{2}}{{b}^{6}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-1/30\,{\frac{ \left ( dx+c \right ) ^{5/3}}{{b}^{5}}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+{\frac{ \left ( dx+c \right ) ^{4/3}}{120\,{b}^{4}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+{\frac{dx+c}{360\,{b}^{3}}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-{\frac{ \left ( dx+c \right ) ^{2/3}}{720\,{b}^{2}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-{\frac{\sqrt [3]{dx+c}}{720\,b}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-{\frac{\cos \left ( a \right ) }{720}{\it Si} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) }-{\frac{\sin \left ( a \right ) }{720}{\it Ci} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) } \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/d^2*b^3*(-c*f*(-1/3*sin(a+b/(d*x+c)^(1/3))*(d*x+c)/b^3-1/6*cos(a+b/(d*x+c)^(1/3))*(d*x+c)^(2/3)/b^2+1/6*sin
(a+b/(d*x+c)^(1/3))*(d*x+c)^(1/3)/b+1/6*Si(b/(d*x+c)^(1/3))*sin(a)-1/6*Ci(b/(d*x+c)^(1/3))*cos(a))+d*e*(-1/3*s
in(a+b/(d*x+c)^(1/3))*(d*x+c)/b^3-1/6*cos(a+b/(d*x+c)^(1/3))*(d*x+c)^(2/3)/b^2+1/6*sin(a+b/(d*x+c)^(1/3))*(d*x
+c)^(1/3)/b+1/6*Si(b/(d*x+c)^(1/3))*sin(a)-1/6*Ci(b/(d*x+c)^(1/3))*cos(a))+b^3*f*(-1/6*sin(a+b/(d*x+c)^(1/3))*
(d*x+c)^2/b^6-1/30*cos(a+b/(d*x+c)^(1/3))*(d*x+c)^(5/3)/b^5+1/120*sin(a+b/(d*x+c)^(1/3))*(d*x+c)^(4/3)/b^4+1/3
60*cos(a+b/(d*x+c)^(1/3))*(d*x+c)/b^3-1/720*sin(a+b/(d*x+c)^(1/3))*(d*x+c)^(2/3)/b^2-1/720*cos(a+b/(d*x+c)^(1/
3))*(d*x+c)^(1/3)/b-1/720*Si(b/(d*x+c)^(1/3))*cos(a)-1/720*Ci(b/(d*x+c)^(1/3))*sin(a)))

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Maxima [C]  time = 1.57241, size = 618, normalized size = 1.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b/(d*x+c)^(1/3)),x, algorithm="maxima")

[Out]

1/480*(120*(((Ei(I*b/(d*x + c)^(1/3)) + Ei(-I*b/(d*x + c)^(1/3)))*cos(a) + (I*Ei(I*b/(d*x + c)^(1/3)) - I*Ei(-
I*b/(d*x + c)^(1/3)))*sin(a))*b^3 + 2*(d*x + c)^(2/3)*b*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 2*((d*x
+ c)^(1/3)*b^2 - 2*d*x - 2*c)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)))*e - 120*(((Ei(I*b/(d*x + c)^(1/3)
) + Ei(-I*b/(d*x + c)^(1/3)))*cos(a) + (I*Ei(I*b/(d*x + c)^(1/3)) - I*Ei(-I*b/(d*x + c)^(1/3)))*sin(a))*b^3 +
2*(d*x + c)^(2/3)*b*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 2*((d*x + c)^(1/3)*b^2 - 2*d*x - 2*c)*sin((
(d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)))*c*f/d + (((-I*Ei(I*b/(d*x + c)^(1/3)) + I*Ei(-I*b/(d*x + c)^(1/3)))*c
os(a) + (Ei(I*b/(d*x + c)^(1/3)) + Ei(-I*b/(d*x + c)^(1/3)))*sin(a))*b^6 + 2*((d*x + c)^(1/3)*b^5 - 2*(d*x + c
)*b^3 + 24*(d*x + c)^(5/3)*b)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) + 2*((d*x + c)^(2/3)*b^4 - 6*(d*x +
c)^(4/3)*b^2 + 120*(d*x + c)^2)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)))*f/d)/d

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Fricas [A]  time = 2.03869, size = 814, normalized size = 1.94 \begin{align*} \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b^{5} f - 2 \, b^{3} d f x - 2 \, b^{3} c f + 24 \,{\left (b d f x + 5 \, b d e - 4 \, b c f\right )}{\left (d x + c\right )}^{\frac{2}{3}}\right )} \cos \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right ) +{\left (b^{6} f \sin \left (a\right ) + 120 \,{\left (b^{3} d e - b^{3} c f\right )} \cos \left (a\right )\right )} \operatorname{Ci}\left (\frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right ) +{\left (b^{6} f \sin \left (a\right ) + 120 \,{\left (b^{3} d e - b^{3} c f\right )} \cos \left (a\right )\right )} \operatorname{Ci}\left (-\frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right ) + 2 \,{\left ({\left (d x + c\right )}^{\frac{2}{3}} b^{4} f + 120 \, d^{2} f x^{2} + 240 \, d^{2} e x + 240 \, c d e - 120 \, c^{2} f - 6 \,{\left (b^{2} d f x + 20 \, b^{2} d e - 19 \, b^{2} c f\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )} \sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right ) + 2 \,{\left (b^{6} f \cos \left (a\right ) - 120 \,{\left (b^{3} d e - b^{3} c f\right )} \sin \left (a\right )\right )} \operatorname{Si}\left (\frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right )}{480 \, d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b/(d*x+c)^(1/3)),x, algorithm="fricas")

[Out]

1/480*(2*((d*x + c)^(1/3)*b^5*f - 2*b^3*d*f*x - 2*b^3*c*f + 24*(b*d*f*x + 5*b*d*e - 4*b*c*f)*(d*x + c)^(2/3))*
cos((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x + c)) + (b^6*f*sin(a) + 120*(b^3*d*e - b^3*c*f)*cos(a))*cos_integra
l(b/(d*x + c)^(1/3)) + (b^6*f*sin(a) + 120*(b^3*d*e - b^3*c*f)*cos(a))*cos_integral(-b/(d*x + c)^(1/3)) + 2*((
d*x + c)^(2/3)*b^4*f + 120*d^2*f*x^2 + 240*d^2*e*x + 240*c*d*e - 120*c^2*f - 6*(b^2*d*f*x + 20*b^2*d*e - 19*b^
2*c*f)*(d*x + c)^(1/3))*sin((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x + c)) + 2*(b^6*f*cos(a) - 120*(b^3*d*e - b^
3*c*f)*sin(a))*sin_integral(b/(d*x + c)^(1/3)))/d^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e + f x\right ) \sin{\left (a + \frac{b}{\sqrt [3]{c + d x}} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sin(a+b/(d*x+c)^(1/3)),x, algorithm="giac")

[Out]

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