3.217 \(\int (e+f x)^2 \sin (a+\frac{b}{\sqrt [3]{c+d x}}) \, dx\)

Optimal. Leaf size=855 \[ \text{result too large to display} \]

[Out]

(b^5*f*(d*e - c*f)*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(120*d^3) - (b^7*f^2*(c + d*x)^(2/3)*Cos[a + b/
(c + d*x)^(1/3)])/(120960*d^3) + (b*(d*e - c*f)^2*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(2*d^3) - (b^3*f
*(d*e - c*f)*(c + d*x)*Cos[a + b/(c + d*x)^(1/3)])/(60*d^3) + (b^5*f^2*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(1/
3)])/(20160*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(5/3)*Cos[a + b/(c + d*x)^(1/3)])/(5*d^3) - (b^3*f^2*(c + d*x)^2
*Cos[a + b/(c + d*x)^(1/3)])/(1008*d^3) + (b*f^2*(c + d*x)^(8/3)*Cos[a + b/(c + d*x)^(1/3)])/(24*d^3) - (b^9*f
^2*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)])/(120960*d^3) + (b^3*(d*e - c*f)^2*Cos[a]*CosIntegral[b/(c + d*x)^(1/
3)])/(2*d^3) + (b^6*f*(d*e - c*f)*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a])/(120*d^3) + (b^8*f^2*(c + d*x)^(1/3)*
Sin[a + b/(c + d*x)^(1/3)])/(120960*d^3) - (b^2*(d*e - c*f)^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(2*d
^3) + (b^4*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)])/(120*d^3) - (b^6*f^2*(c + d*x)*Sin[a + b/
(c + d*x)^(1/3)])/(60480*d^3) + ((d*e - c*f)^2*(c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/d^3 - (b^2*f*(d*e - c*f)*
(c + d*x)^(4/3)*Sin[a + b/(c + d*x)^(1/3)])/(20*d^3) + (b^4*f^2*(c + d*x)^(5/3)*Sin[a + b/(c + d*x)^(1/3)])/(5
040*d^3) + (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/(c + d*x)^(1/3)])/d^3 - (b^2*f^2*(c + d*x)^(7/3)*Sin[a + b/(c
+ d*x)^(1/3)])/(168*d^3) + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(1/3)])/(3*d^3) + (b^6*f*(d*e - c*f)*Cos[a]*Si
nIntegral[b/(c + d*x)^(1/3)])/(120*d^3) + (b^9*f^2*Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(120960*d^3) - (b^3*
(d*e - c*f)^2*Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(2*d^3)

________________________________________________________________________________________

Rubi [A]  time = 1.05102, antiderivative size = 855, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {3431, 3297, 3303, 3299, 3302} \[ -\frac{f^2 \cos (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) b^9}{120960 d^3}+\frac{f^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) b^9}{120960 d^3}+\frac{f^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^8}{120960 d^3}-\frac{f^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^7}{120960 d^3}+\frac{f (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) \sin (a) b^6}{120 d^3}-\frac{f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^6}{60480 d^3}+\frac{f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) b^6}{120 d^3}+\frac{f^2 (c+d x)^{4/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^5}{20160 d^3}+\frac{f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^5}{120 d^3}+\frac{f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^4}{5040 d^3}+\frac{f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^4}{120 d^3}-\frac{f^2 (c+d x)^2 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^3}{1008 d^3}-\frac{f (d e-c f) (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^3}{60 d^3}+\frac{(d e-c f)^2 \cos (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) b^3}{2 d^3}-\frac{(d e-c f)^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) b^3}{2 d^3}-\frac{f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^2}{168 d^3}-\frac{f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^2}{20 d^3}-\frac{(d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b^2}{2 d^3}+\frac{f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b}{24 d^3}+\frac{f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b}{5 d^3}+\frac{(d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) b}{2 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3} \]

Antiderivative was successfully verified.

[In]

Int[(e + f*x)^2*Sin[a + b/(c + d*x)^(1/3)],x]

[Out]

(b^5*f*(d*e - c*f)*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/(120*d^3) - (b^7*f^2*(c + d*x)^(2/3)*Cos[a + b/
(c + d*x)^(1/3)])/(120960*d^3) + (b*(d*e - c*f)^2*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(2*d^3) - (b^3*f
*(d*e - c*f)*(c + d*x)*Cos[a + b/(c + d*x)^(1/3)])/(60*d^3) + (b^5*f^2*(c + d*x)^(4/3)*Cos[a + b/(c + d*x)^(1/
3)])/(20160*d^3) + (b*f*(d*e - c*f)*(c + d*x)^(5/3)*Cos[a + b/(c + d*x)^(1/3)])/(5*d^3) - (b^3*f^2*(c + d*x)^2
*Cos[a + b/(c + d*x)^(1/3)])/(1008*d^3) + (b*f^2*(c + d*x)^(8/3)*Cos[a + b/(c + d*x)^(1/3)])/(24*d^3) - (b^9*f
^2*Cos[a]*CosIntegral[b/(c + d*x)^(1/3)])/(120960*d^3) + (b^3*(d*e - c*f)^2*Cos[a]*CosIntegral[b/(c + d*x)^(1/
3)])/(2*d^3) + (b^6*f*(d*e - c*f)*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a])/(120*d^3) + (b^8*f^2*(c + d*x)^(1/3)*
Sin[a + b/(c + d*x)^(1/3)])/(120960*d^3) - (b^2*(d*e - c*f)^2*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(2*d
^3) + (b^4*f*(d*e - c*f)*(c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)])/(120*d^3) - (b^6*f^2*(c + d*x)*Sin[a + b/
(c + d*x)^(1/3)])/(60480*d^3) + ((d*e - c*f)^2*(c + d*x)*Sin[a + b/(c + d*x)^(1/3)])/d^3 - (b^2*f*(d*e - c*f)*
(c + d*x)^(4/3)*Sin[a + b/(c + d*x)^(1/3)])/(20*d^3) + (b^4*f^2*(c + d*x)^(5/3)*Sin[a + b/(c + d*x)^(1/3)])/(5
040*d^3) + (f*(d*e - c*f)*(c + d*x)^2*Sin[a + b/(c + d*x)^(1/3)])/d^3 - (b^2*f^2*(c + d*x)^(7/3)*Sin[a + b/(c
+ d*x)^(1/3)])/(168*d^3) + (f^2*(c + d*x)^3*Sin[a + b/(c + d*x)^(1/3)])/(3*d^3) + (b^6*f*(d*e - c*f)*Cos[a]*Si
nIntegral[b/(c + d*x)^(1/3)])/(120*d^3) + (b^9*f^2*Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(120960*d^3) - (b^3*
(d*e - c*f)^2*Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(2*d^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 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)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) \, dx &=-\frac{3 \operatorname{Subst}\left (\int \left (\frac{f^2 \sin (a+b x)}{d^2 x^{10}}+\frac{2 f (d e-c f) \sin (a+b x)}{d^2 x^7}+\frac{(d e-c f)^2 \sin (a+b x)}{d^2 x^4}\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac{\left (3 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^{10}} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{(6 f (d e-c f)) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^7} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{\left (3 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}-\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^9} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{3 d^3}-\frac{(b f (d e-c f)) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^6} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{\left (b (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^3}\\ &=\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^8} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{24 d^3}+\frac{\left (b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{5 d^3}+\frac{\left (b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^3}\\ &=\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}-\frac{b^2 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}+\frac{\left (b^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^7} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{\left (b^3 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{\left (b^3 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^3}\\ &=\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^3 f (d e-c f) (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}-\frac{b^3 f^2 (c+d x)^2 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{1008 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}-\frac{b^2 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}-\frac{\left (b^4 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^6} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{1008 d^3}-\frac{\left (b^4 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{60 d^3}+\frac{\left (b^3 (d e-c f)^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{\left (b^3 (d e-c f)^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^3}\\ &=\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^3 f (d e-c f) (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}-\frac{b^3 f^2 (c+d x)^2 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{1008 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}+\frac{b^3 (d e-c f)^2 \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^2 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^4 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{b^4 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5040 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}-\frac{b^3 (d e-c f)^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{\left (b^5 f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{5040 d^3}-\frac{\left (b^5 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120 d^3}\\ &=\frac{b^5 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}+\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^3 f (d e-c f) (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 (c+d x)^{4/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20160 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}-\frac{b^3 f^2 (c+d x)^2 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{1008 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}+\frac{b^3 (d e-c f)^2 \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^2 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^4 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{b^4 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5040 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}-\frac{b^3 (d e-c f)^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{\left (b^6 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{20160 d^3}+\frac{\left (b^6 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120 d^3}\\ &=\frac{b^5 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}+\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^3 f (d e-c f) (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 (c+d x)^{4/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20160 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}-\frac{b^3 f^2 (c+d x)^2 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{1008 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}+\frac{b^3 (d e-c f)^2 \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^2 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^4 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^6 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60480 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{b^4 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5040 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}-\frac{b^3 (d e-c f)^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{\left (b^7 f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{60480 d^3}+\frac{\left (b^6 f (d e-c f) \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120 d^3}+\frac{\left (b^6 f (d e-c f) \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120 d^3}\\ &=\frac{b^5 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^7 f^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120960 d^3}+\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^3 f (d e-c f) (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 (c+d x)^{4/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20160 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}-\frac{b^3 f^2 (c+d x)^2 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{1008 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}+\frac{b^3 (d e-c f)^2 \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^6 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{120 d^3}-\frac{b^2 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^4 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^6 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60480 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{b^4 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5040 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}+\frac{b^6 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^3 (d e-c f)^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{\left (b^8 f^2\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120960 d^3}\\ &=\frac{b^5 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^7 f^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120960 d^3}+\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^3 f (d e-c f) (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 (c+d x)^{4/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20160 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}-\frac{b^3 f^2 (c+d x)^2 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{1008 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}+\frac{b^3 (d e-c f)^2 \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^6 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{120 d^3}+\frac{b^8 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120960 d^3}-\frac{b^2 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^4 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^6 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60480 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{b^4 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5040 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}+\frac{b^6 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^3 (d e-c f)^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{\left (b^9 f^2\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120960 d^3}\\ &=\frac{b^5 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^7 f^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120960 d^3}+\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^3 f (d e-c f) (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 (c+d x)^{4/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20160 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}-\frac{b^3 f^2 (c+d x)^2 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{1008 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}+\frac{b^3 (d e-c f)^2 \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^6 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{120 d^3}+\frac{b^8 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120960 d^3}-\frac{b^2 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^4 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^6 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60480 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{b^4 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5040 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}+\frac{b^6 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^3 (d e-c f)^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{\left (b^9 f^2 \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120960 d^3}+\frac{\left (b^9 f^2 \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120960 d^3}\\ &=\frac{b^5 f (d e-c f) \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^7 f^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120960 d^3}+\frac{b (d e-c f)^2 (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}-\frac{b^3 f (d e-c f) (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60 d^3}+\frac{b^5 f^2 (c+d x)^{4/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20160 d^3}+\frac{b f (d e-c f) (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5 d^3}-\frac{b^3 f^2 (c+d x)^2 \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{1008 d^3}+\frac{b f^2 (c+d x)^{8/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{24 d^3}-\frac{b^9 f^2 \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{120960 d^3}+\frac{b^3 (d e-c f)^2 \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^6 f (d e-c f) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{120 d^3}+\frac{b^8 f^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120960 d^3}-\frac{b^2 (d e-c f)^2 \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}+\frac{b^4 f (d e-c f) (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}-\frac{b^6 f^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{60480 d^3}+\frac{(d e-c f)^2 (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f (d e-c f) (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{20 d^3}+\frac{b^4 f^2 (c+d x)^{5/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{5040 d^3}+\frac{f (d e-c f) (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^3}-\frac{b^2 f^2 (c+d x)^{7/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{168 d^3}+\frac{f^2 (c+d x)^3 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{3 d^3}+\frac{b^6 f (d e-c f) \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^3}+\frac{b^9 f^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{120960 d^3}-\frac{b^3 (d e-c f)^2 \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^3}\\ \end{align*}

Mathematica [C]  time = 4.58093, size = 929, normalized size = 1.09 \[ -\frac{i \left ((\cos (a)+i \sin (a)) \left (-i f^2 \text{Ei}\left (\frac{i b}{\sqrt [3]{c+d x}}\right ) b^9-1008 c f^2 \text{Ei}\left (\frac{i b}{\sqrt [3]{c+d x}}\right ) b^6+1008 d e f \text{Ei}\left (\frac{i b}{\sqrt [3]{c+d x}}\right ) b^6+60480 i d^2 e^2 \text{Ei}\left (\frac{i b}{\sqrt [3]{c+d x}}\right ) b^3+60480 i c^2 f^2 \text{Ei}\left (\frac{i b}{\sqrt [3]{c+d x}}\right ) b^3-120960 i c d e f \text{Ei}\left (\frac{i b}{\sqrt [3]{c+d x}}\right ) b^3+\sqrt [3]{c+d x} \left (f^2 b^8-i f^2 \sqrt [3]{c+d x} b^7-2 f^2 (c+d x)^{2/3} b^6+6 i f (168 d e-167 c f+d f x) b^5+24 f \sqrt [3]{c+d x} (42 d e-41 c f+d f x) b^4+24 i f (c+d x)^{2/3} (-84 d e+79 c f-5 d f x) b^3-144 \left (\left (420 e^2+42 f x e+5 f^2 x^2\right ) d^2-2 c f (399 e+16 f x) d+383 c^2 f^2\right ) b^2+1008 i \sqrt [3]{c+d x} \left (\left (60 e^2+24 f x e+5 f^2 x^2\right ) d^2-2 c f (48 e+7 f x) d+41 c^2 f^2\right ) b+40320 (c+d x)^{2/3} \left (\left (3 e^2+3 f x e+f^2 x^2\right ) d^2-c f (3 e+f x) d+c^2 f^2\right )\right ) \left (\cos \left (\frac{b}{\sqrt [3]{c+d x}}\right )+i \sin \left (\frac{b}{\sqrt [3]{c+d x}}\right )\right )\right )-\left (i \left (-60480 d^2 e^2+1008 \left (120 c-i b^3\right ) d f e+\left (b^6+1008 i c b^3-60480 c^2\right ) f^2\right ) \text{Ei}\left (-\frac{i b}{\sqrt [3]{c+d x}}\right ) \left (\cos \left (\frac{b}{\sqrt [3]{c+d x}}\right )+i \sin \left (\frac{b}{\sqrt [3]{c+d x}}\right )\right ) b^3+\sqrt [3]{c+d x} \left (f^2 b^8+i f^2 \sqrt [3]{c+d x} b^7-2 f^2 (c+d x)^{2/3} b^6-6 i f (168 d e-167 c f+d f x) b^5+24 f \sqrt [3]{c+d x} (42 d e-41 c f+d f x) b^4+24 i f (c+d x)^{2/3} (84 d e-79 c f+5 d f x) b^3-144 \left (\left (420 e^2+42 f x e+5 f^2 x^2\right ) d^2-2 c f (399 e+16 f x) d+383 c^2 f^2\right ) b^2-1008 i \sqrt [3]{c+d x} \left (\left (60 e^2+24 f x e+5 f^2 x^2\right ) d^2-2 c f (48 e+7 f x) d+41 c^2 f^2\right ) b+40320 (c+d x)^{2/3} \left (\left (3 e^2+3 f x e+f^2 x^2\right ) d^2-c f (3 e+f x) d+c^2 f^2\right )\right )\right ) \left (\cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )-i \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )\right )\right )}{241920 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(e + f*x)^2*Sin[a + b/(c + d*x)^(1/3)],x]

[Out]

((-I/241920)*((Cos[a] + I*Sin[a])*((60480*I)*b^3*d^2*e^2*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] + 1008*b^6*d*e*f
*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] - (120960*I)*b^3*c*d*e*f*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] - I*b^9*f^
2*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] - 1008*b^6*c*f^2*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] + (60480*I)*b^3*c
^2*f^2*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] + (c + d*x)^(1/3)*(b^8*f^2 - I*b^7*f^2*(c + d*x)^(1/3) - 2*b^6*f^2
*(c + d*x)^(2/3) + (24*I)*b^3*f*(c + d*x)^(2/3)*(-84*d*e + 79*c*f - 5*d*f*x) + (6*I)*b^5*f*(168*d*e - 167*c*f
+ d*f*x) + 24*b^4*f*(c + d*x)^(1/3)*(42*d*e - 41*c*f + d*f*x) + 40320*(c + d*x)^(2/3)*(c^2*f^2 - c*d*f*(3*e +
f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2)) + (1008*I)*b*(c + d*x)^(1/3)*(41*c^2*f^2 - 2*c*d*f*(48*e + 7*f*x) + d^
2*(60*e^2 + 24*e*f*x + 5*f^2*x^2)) - 144*b^2*(383*c^2*f^2 - 2*c*d*f*(399*e + 16*f*x) + d^2*(420*e^2 + 42*e*f*x
+ 5*f^2*x^2)))*(Cos[b/(c + d*x)^(1/3)] + I*Sin[b/(c + d*x)^(1/3)])) - ((c + d*x)^(1/3)*(b^8*f^2 + I*b^7*f^2*(
c + d*x)^(1/3) - 2*b^6*f^2*(c + d*x)^(2/3) - (6*I)*b^5*f*(168*d*e - 167*c*f + d*f*x) + 24*b^4*f*(c + d*x)^(1/3
)*(42*d*e - 41*c*f + d*f*x) + (24*I)*b^3*f*(c + d*x)^(2/3)*(84*d*e - 79*c*f + 5*d*f*x) + 40320*(c + d*x)^(2/3)
*(c^2*f^2 - c*d*f*(3*e + f*x) + d^2*(3*e^2 + 3*e*f*x + f^2*x^2)) - (1008*I)*b*(c + d*x)^(1/3)*(41*c^2*f^2 - 2*
c*d*f*(48*e + 7*f*x) + d^2*(60*e^2 + 24*e*f*x + 5*f^2*x^2)) - 144*b^2*(383*c^2*f^2 - 2*c*d*f*(399*e + 16*f*x)
+ d^2*(420*e^2 + 42*e*f*x + 5*f^2*x^2))) + I*b^3*(-60480*d^2*e^2 + 1008*((-I)*b^3 + 120*c)*d*e*f + (b^6 + (100
8*I)*b^3*c - 60480*c^2)*f^2)*ExpIntegralEi[((-I)*b)/(c + d*x)^(1/3)]*(Cos[b/(c + d*x)^(1/3)] + I*Sin[b/(c + d*
x)^(1/3)]))*(Cos[a + b/(c + d*x)^(1/3)] - I*Sin[a + b/(c + d*x)^(1/3)])))/d^3

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Maple [A]  time = 0.078, size = 936, normalized size = 1.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/d^3*b^3*(d^2*e^2*(-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))+c^2*f^2*
(-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))+b^6*f^2*(-1/9*sin(a+b/(d*x+c
)^(1/3))*(d*x+c)^3/b^9-1/72*cos(a+b/(d*x+c)^(1/3))*(d*x+c)^(8/3)/b^8+1/504*sin(a+b/(d*x+c)^(1/3))*(d*x+c)^(7/3
)/b^7+1/3024*cos(a+b/(d*x+c)^(1/3))*(d*x+c)^2/b^6-1/15120*sin(a+b/(d*x+c)^(1/3))*(d*x+c)^(5/3)/b^5-1/60480*cos
(a+b/(d*x+c)^(1/3))*(d*x+c)^(4/3)/b^4+1/181440*sin(a+b/(d*x+c)^(1/3))*(d*x+c)/b^3+1/362880*cos(a+b/(d*x+c)^(1/
3))*(d*x+c)^(2/3)/b^2-1/362880*sin(a+b/(d*x+c)^(1/3))*(d*x+c)^(1/3)/b-1/362880*Si(b/(d*x+c)^(1/3))*sin(a)+1/36
2880*Ci(b/(d*x+c)^(1/3))*cos(a))-2*c*d*e*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))+2*b^3*d*e*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+1/360*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))-2*b^3*c*f^2*(-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+1/360*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 = 2.10109, size = 1354, normalized size = 1.58 \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)^2*sin(a+b/(d*x+c)^(1/3)),x, algorithm="maxima")

[Out]

1/241920*(60480*(((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^2 - 120960*(((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*e*f/d + 60480*(((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^2*f^2/d^2 + 1008*(((-I*Ei(I*b/(d*x + c)^(1/3)) + I*Ei(-I*b/(d*x + c)^(1/3)))*cos(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)))*e*f/d - 1008*(((-I*Ei(I*b/(d*x + c)^(1
/3)) + I*Ei(-I*b/(d*x + c)^(1/3)))*cos(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)))*c*f^2/d^2 - (((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^9 + 2*((d*x + c)^(2/3)*b^7 - 6*(d*x + c)^(4/3)*b^5 + 120*(d*x + c)^2*b^
3 - 5040*(d*x + c)^(8/3)*b)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 2*((d*x + c)^(1/3)*b^8 - 2*(d*x + c
)*b^6 + 24*(d*x + c)^(5/3)*b^4 - 720*(d*x + c)^(7/3)*b^2 + 40320*(d*x + c)^3)*sin(((d*x + c)^(1/3)*a + b)/(d*x
+ c)^(1/3)))*f^2/d^2)/d

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Fricas [A]  time = 2.17636, size = 1670, normalized size = 1.95 \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)^2*sin(a+b/(d*x+c)^(1/3)),x, algorithm="fricas")

[Out]

-1/241920*(2*(120*b^3*d^2*f^2*x^2 + 2016*b^3*c*d*e*f - 1896*b^3*c^2*f^2 + 48*(42*b^3*d^2*e*f - 37*b^3*c*d*f^2)
*x - (5040*b*d^2*f^2*x^2 + 60480*b*d^2*e^2 - 96768*b*c*d*e*f - (b^7 - 41328*b*c^2)*f^2 + 2016*(12*b*d^2*e*f -
7*b*c*d*f^2)*x)*(d*x + c)^(2/3) - 6*(b^5*d*f^2*x + 168*b^5*d*e*f - 167*b^5*c*f^2)*(d*x + c)^(1/3))*cos((a*d*x
+ a*c + (d*x + c)^(2/3)*b)/(d*x + c)) - ((60480*b^3*d^2*e^2 - 120960*b^3*c*d*e*f - (b^9 - 60480*b^3*c^2)*f^2)*
cos(a) + 1008*(b^6*d*e*f - b^6*c*f^2)*sin(a))*cos_integral(b/(d*x + c)^(1/3)) - ((60480*b^3*d^2*e^2 - 120960*b
^3*c*d*e*f - (b^9 - 60480*b^3*c^2)*f^2)*cos(a) + 1008*(b^6*d*e*f - b^6*c*f^2)*sin(a))*cos_integral(-b/(d*x + c
)^(1/3)) - 2*(40320*d^3*f^2*x^3 + 120960*d^3*e*f*x^2 + 120960*c*d^2*e^2 - 120960*c^2*d*e*f - 2*(b^6*c - 20160*
c^3)*f^2 - 2*(b^6*d*f^2 - 60480*d^3*e^2)*x + 24*(b^4*d*f^2*x + 42*b^4*d*e*f - 41*b^4*c*f^2)*(d*x + c)^(2/3) -
(720*b^2*d^2*f^2*x^2 + 60480*b^2*d^2*e^2 - 114912*b^2*c*d*e*f - (b^8 - 55152*b^2*c^2)*f^2 + 288*(21*b^2*d^2*e*
f - 16*b^2*c*d*f^2)*x)*(d*x + c)^(1/3))*sin((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x + c)) - 2*(1008*(b^6*d*e*f
- b^6*c*f^2)*cos(a) - (60480*b^3*d^2*e^2 - 120960*b^3*c*d*e*f - (b^9 - 60480*b^3*c^2)*f^2)*sin(a))*sin_integra
l(b/(d*x + c)^(1/3)))/d^3

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

[Out]

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

[Out]

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