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

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

[Out]

1/2*b^3*(-c*f+d*e)*Ci(b/(d*x+c)^(1/3))*cos(a)/d^2+1/240*b^5*f*(d*x+c)^(1/3)*cos(a+b/(d*x+c)^(1/3))/d^2+1/2*b*(
-c*f+d*e)*(d*x+c)^(2/3)*cos(a+b/(d*x+c)^(1/3))/d^2-1/120*b^3*f*(d*x+c)*cos(a+b/(d*x+c)^(1/3))/d^2+1/10*b*f*(d*
x+c)^(5/3)*cos(a+b/(d*x+c)^(1/3))/d^2+1/240*b^6*f*cos(a)*Si(b/(d*x+c)^(1/3))/d^2+1/240*b^6*f*Ci(b/(d*x+c)^(1/3
))*sin(a)/d^2-1/2*b^3*(-c*f+d*e)*Si(b/(d*x+c)^(1/3))*sin(a)/d^2-1/2*b^2*(-c*f+d*e)*(d*x+c)^(1/3)*sin(a+b/(d*x+
c)^(1/3))/d^2+1/240*b^4*f*(d*x+c)^(2/3)*sin(a+b/(d*x+c)^(1/3))/d^2+(-c*f+d*e)*(d*x+c)*sin(a+b/(d*x+c)^(1/3))/d
^2-1/40*b^2*f*(d*x+c)^(4/3)*sin(a+b/(d*x+c)^(1/3))/d^2+1/2*f*(d*x+c)^2*sin(a+b/(d*x+c)^(1/3))/d^2

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Rubi [A]  time = 0.50, antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.89, size = 540, normalized size = 1.29 \[ \frac {b^3 f \left (b^3 \sin (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+b^3 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )-120 c \cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+120 c \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{240 d^2}+\frac {b^3 e \left (\cos (a) \text {Ci}\left (\frac {b}{\sqrt [3]{c+d x}}\right )-\sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{2 d}+\frac {e \sqrt [3]{c+d x} \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right ) \left (b^2 (-\sin (a))+b \cos (a) \sqrt [3]{c+d x}+2 \sin (a) (c+d x)^{2/3}\right )}{2 d}+\frac {e \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right ) \left (b^2 (-\cos (a))-b \sin (a) \sqrt [3]{c+d x}+2 \cos (a) (c+d x)^{2/3}\right )}{2 d}+\frac {f \sqrt [3]{c+d x} \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right ) \left (b^5 \cos (a)+b^4 \sin (a) \sqrt [3]{c+d x}-2 b^3 \cos (a) (c+d x)^{2/3}-6 b^2 \sin (a) (c+d x)+120 b^2 c \sin (a)+24 b \cos (a) (c+d x)^{4/3}-120 b c \cos (a) \sqrt [3]{c+d x}+120 \sin (a) (c+d x)^{5/3}-240 c \sin (a) (c+d x)^{2/3}\right )}{240 d^2}+\frac {f \sqrt [3]{c+d x} \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right ) \left (b^5 (-\sin (a))+b^4 \cos (a) \sqrt [3]{c+d x}+2 b^3 \sin (a) (c+d x)^{2/3}-6 b^2 \cos (a) (c+d x)+120 b^2 c \cos (a)-24 b \sin (a) (c+d x)^{4/3}+120 b c \sin (a) \sqrt [3]{c+d x}+120 \cos (a) (c+d x)^{5/3}-240 c \cos (a) (c+d x)^{2/3}\right )}{240 d^2} \]

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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fricas [A]  time = 0.78, size = 299, normalized size = 0.71 \[ \frac {2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{5} f - 2 \, b^{3} d f x - 2 \, b^{3} c f + 24 \, {\left (b d f x + 5 \, b d e - 4 \, b c f\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) + {\left (b^{6} f \sin \relax (a) + 120 \, {\left (b^{3} d e - b^{3} c f\right )} \cos \relax (a)\right )} \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\left (b^{6} f \sin \relax (a) + 120 \, {\left (b^{3} d e - b^{3} c f\right )} \cos \relax (a)\right )} \operatorname {Ci}\left (-\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + 2 \, {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{4} f + 120 \, d^{2} f x^{2} + 240 \, d^{2} e x + 240 \, c d e - 120 \, c^{2} f - 6 \, {\left (b^{2} d f x + 20 \, b^{2} d e - 19 \, b^{2} c f\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) + 2 \, {\left (b^{6} f \cos \relax (a) - 120 \, {\left (b^{3} d e - b^{3} c f\right )} \sin \relax (a)\right )} \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{480 \, d^{2}} \]

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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giac [B]  time = 3.10, size = 3728, normalized size = 8.90 \[ \text {result too large to display} \]

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]

1/240*(120*(a^3*b^4*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) + a^3*b^4*sin(a)*sin_int
egral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 3*((d*x + c)^(1/3)*a + b)*a^2*b^4*cos(a)*cos_integral(-a
+ ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) - 3*((d*x + c)^(1/3)*a + b)*a^2*b^4*sin(a)*sin_inte
gral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) + 3*((d*x + c)^(1/3)*a + b)^2*a*b^4*cos(a)*c
os_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) + 3*((d*x + c)^(1/3)*a + b)^2*a*b^4*
sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) - ((d*x + c)^(1/3)*a + b)^3*b
^4*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c) - ((d*x + c)^(1/3)*a + b)^3*b^4
*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c) + a^2*b^4*sin(((d*x + c)^(1/3)*a +
b)/(d*x + c)^(1/3)) - 2*((d*x + c)^(1/3)*a + b)*a*b^4*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^
(1/3) + ((d*x + c)^(1/3)*a + b)^2*b^4*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) + a*b^4*cos
(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - ((d*x + c)^(1/3)*a + b)*b^4*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^
(1/3))/(d*x + c)^(1/3) - 2*b^4*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)))*e/((a^3 - 3*((d*x + c)^(1/3)*a +
b)*a^2/(d*x + c)^(1/3) + 3*((d*x + c)^(1/3)*a + b)^2*a/(d*x + c)^(2/3) - ((d*x + c)^(1/3)*a + b)^3/(d*x + c))*
b) + (a^6*b^7*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))*sin(a) - a^6*b^7*cos(a)*sin_integral(
a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 6*((d*x + c)^(1/3)*a + b)*a^5*b^7*cos_integral(-a + ((d*x + c)^
(1/3)*a + b)/(d*x + c)^(1/3))*sin(a)/(d*x + c)^(1/3) + 6*((d*x + c)^(1/3)*a + b)*a^5*b^7*cos(a)*sin_integral(a
- ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) + 15*((d*x + c)^(1/3)*a + b)^2*a^4*b^7*cos_integra
l(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))*sin(a)/(d*x + c)^(2/3) - 15*((d*x + c)^(1/3)*a + b)^2*a^4*b^7*
cos(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) - 20*((d*x + c)^(1/3)*a + b)^
3*a^3*b^7*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))*sin(a)/(d*x + c) + 20*((d*x + c)^(1/3)*a
+ b)^3*a^3*b^7*cos(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c) + 15*((d*x + c)^(1/3
)*a + b)^4*a^2*b^7*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))*sin(a)/(d*x + c)^(4/3) - 15*((d*
x + c)^(1/3)*a + b)^4*a^2*b^7*cos(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(4/3)
- 6*((d*x + c)^(1/3)*a + b)^5*a*b^7*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))*sin(a)/(d*x +
c)^(5/3) + 6*((d*x + c)^(1/3)*a + b)^5*a*b^7*cos(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/
(d*x + c)^(5/3) - a^5*b^7*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 120*a^6*b^4*c*cos(a)*cos_integral(-a
+ ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) + ((d*x + c)^(1/3)*a + b)^6*b^7*cos_integral(-a + ((d*x + c)^(1/3)*
a + b)/(d*x + c)^(1/3))*sin(a)/(d*x + c)^2 - ((d*x + c)^(1/3)*a + b)^6*b^7*cos(a)*sin_integral(a - ((d*x + c)^
(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^2 - 120*a^6*b^4*c*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x
+ c)^(1/3)) + 5*((d*x + c)^(1/3)*a + b)*a^4*b^7*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3)
+ 720*((d*x + c)^(1/3)*a + b)*a^5*b^4*c*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x
+ c)^(1/3) + 720*((d*x + c)^(1/3)*a + b)*a^5*b^4*c*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^
(1/3))/(d*x + c)^(1/3) - 10*((d*x + c)^(1/3)*a + b)^2*a^3*b^7*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*
x + c)^(2/3) - 1800*((d*x + c)^(1/3)*a + b)^2*a^4*b^4*c*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x
+ c)^(1/3))/(d*x + c)^(2/3) - 1800*((d*x + c)^(1/3)*a + b)^2*a^4*b^4*c*sin(a)*sin_integral(a - ((d*x + c)^(1/3
)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) + 10*((d*x + c)^(1/3)*a + b)^3*a^2*b^7*cos(((d*x + c)^(1/3)*a + b)/(
d*x + c)^(1/3))/(d*x + c) + 2400*((d*x + c)^(1/3)*a + b)^3*a^3*b^4*c*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)
*a + b)/(d*x + c)^(1/3))/(d*x + c) + a^4*b^7*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) + 2400*((d*x + c)^(1
/3)*a + b)^3*a^3*b^4*c*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c) - 5*((d*x +
c)^(1/3)*a + b)^4*a*b^7*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(4/3) - 1800*((d*x + c)^(1/3)*a
+ b)^4*a^2*b^4*c*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(4/3) - 4*((d*x
+ c)^(1/3)*a + b)*a^3*b^7*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) - 1800*((d*x + c)^(1/3)
*a + b)^4*a^2*b^4*c*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(4/3) + ((d*x +
c)^(1/3)*a + b)^5*b^7*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(5/3) + 720*((d*x + c)^(1/3)*a +
b)^5*a*b^4*c*cos(a)*cos_integral(-a + ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(5/3) + 6*((d*x + c)
^(1/3)*a + b)^2*a^2*b^7*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) + 720*((d*x + c)^(1/3)*a
+ b)^5*a*b^4*c*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(5/3) + 2*a^3*b^7*co
s(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 120*((d*x + c)^(1/3)*a + b)^6*b^4*c*cos(a)*cos_integral(-a + ((d*
x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^2 - 4*((d*x + c)^(1/3)*a + b)^3*a*b^7*sin(((d*x + c)^(1/3)*a +
b)/(d*x + c)^(1/3))/(d*x + c) - 120*a^5*b^4*c*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 120*((d*x + c)^(1
/3)*a + b)^6*b^4*c*sin(a)*sin_integral(a - ((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^2 - 6*((d*x + c)
^(1/3)*a + b)*a^2*b^7*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) + ((d*x + c)^(1/3)*a + b)^4
*b^7*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(4/3) + 600*((d*x + c)^(1/3)*a + b)*a^4*b^4*c*sin(
((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) + 6*((d*x + c)^(1/3)*a + b)^2*a*b^7*cos(((d*x + c)^(1
/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) - 1200*((d*x + c)^(1/3)*a + b)^2*a^3*b^4*c*sin(((d*x + c)^(1/3)*a
+ b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) - 2*((d*x + c)^(1/3)*a + b)^3*b^7*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^
(1/3))/(d*x + c) - 120*a^4*b^4*c*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 6*a^2*b^7*sin(((d*x + c)^(1/3)
*a + b)/(d*x + c)^(1/3)) + 1200*((d*x + c)^(1/3)*a + b)^3*a^2*b^4*c*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3
))/(d*x + c) + 480*((d*x + c)^(1/3)*a + b)*a^3*b^4*c*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1
/3) + 12*((d*x + c)^(1/3)*a + b)*a*b^7*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(1/3) - 600*((d*
x + c)^(1/3)*a + b)^4*a*b^4*c*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(4/3) - 720*((d*x + c)^(1
/3)*a + b)^2*a^2*b^4*c*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) - 6*((d*x + c)^(1/3)*a + b
)^2*b^7*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(2/3) + 120*((d*x + c)^(1/3)*a + b)^5*b^4*c*sin
(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(5/3) - 24*a*b^7*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/
3)) + 480*((d*x + c)^(1/3)*a + b)^3*a*b^4*c*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c) + 240*a^3*b
^4*c*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) + 24*((d*x + c)^(1/3)*a + b)*b^7*cos(((d*x + c)^(1/3)*a + b)
/(d*x + c)^(1/3))/(d*x + c)^(1/3) - 120*((d*x + c)^(1/3)*a + b)^4*b^4*c*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^
(1/3))/(d*x + c)^(4/3) - 720*((d*x + c)^(1/3)*a + b)*a^2*b^4*c*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d
*x + c)^(1/3) + 720*((d*x + c)^(1/3)*a + b)^2*a*b^4*c*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c)^(
2/3) + 120*b^7*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - 240*((d*x + c)^(1/3)*a + b)^3*b^4*c*sin(((d*x +
c)^(1/3)*a + b)/(d*x + c)^(1/3))/(d*x + c))*f/((a^6 - 6*((d*x + c)^(1/3)*a + b)*a^5/(d*x + c)^(1/3) + 15*((d*x
+ c)^(1/3)*a + b)^2*a^4/(d*x + c)^(2/3) - 20*((d*x + c)^(1/3)*a + b)^3*a^3/(d*x + c) + 15*((d*x + c)^(1/3)*a
+ b)^4*a^2/(d*x + c)^(4/3) - 6*((d*x + c)^(1/3)*a + b)^5*a/(d*x + c)^(5/3) + ((d*x + c)^(1/3)*a + b)^6/(d*x +
c)^2)*b*d))/d

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maple [A]  time = 0.05, size = 391, normalized size = 0.93 \[ -\frac {3 b^{3} \left (-c f \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{6 b}+\frac {\Si \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \relax (a )}{6}-\frac {\Ci \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \relax (a )}{6}\right )+d e \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{3 b^{3}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{6 b^{2}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{6 b}+\frac {\Si \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \relax (a )}{6}-\frac {\Ci \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \relax (a )}{6}\right )+b^{3} f \left (-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{2}}{6 b^{6}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {5}{3}}}{30 b^{5}}+\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {4}{3}}}{120 b^{4}}+\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )}{360 b^{3}}-\frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {2}{3}}}{720 b^{2}}-\frac {\cos \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \left (d x +c \right )^{\frac {1}{3}}}{720 b}-\frac {\Si \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \relax (a )}{720}-\frac {\Ci \left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \relax (a )}{720}\right )\right )}{d^{2}} \]

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 = 0.74, size = 458, normalized size = 1.09 \[ \frac {120 \, {\left ({\left ({\left ({\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \relax (a) + {\left (i \, {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \relax (a)\right )} b^{3} + 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b \cos \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{2} - 2 \, d x - 2 \, c\right )} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} e - \frac {120 \, {\left ({\left ({\left ({\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \relax (a) + {\left (i \, {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \relax (a)\right )} b^{3} + 2 \, {\left (d x + c\right )}^{\frac {2}{3}} b \cos \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{2} - 2 \, d x - 2 \, c\right )} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} c f}{d} + \frac {{\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \cos \relax (a) + {\left ({\rm Ei}\left (\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\rm Ei}\left (-\frac {i \, b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} \sin \relax (a)\right )} b^{6} + 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{5} - 2 \, {\left (d x + c\right )} b^{3} + 24 \, {\left (d x + c\right )}^{\frac {5}{3}} b\right )} \cos \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + 2 \, {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{4} - 6 \, {\left (d x + c\right )}^{\frac {4}{3}} b^{2} + 120 \, {\left (d x + c\right )}^{2}\right )} \sin \left (\frac {{\left (d x + c\right )}^{\frac {1}{3}} a + b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )\right )} f}{d}}{480 \, d} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )\,\left (e+f\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e + f x\right ) \sin {\left (a + \frac {b}{\sqrt [3]{c + d x}} \right )}\, dx \]

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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