Integrand size = 22, antiderivative size = 555 \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(e+f x)^2} \, dx=-\frac {\sqrt [3]{-1} b d \cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac {b d \cos \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}+\frac {(-1)^{2/3} b d \cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{f (e+f x)}-\frac {\sqrt [3]{-1} b d \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {b d \sin \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}}-\frac {(-1)^{2/3} b d \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}+b \sqrt [3]{c+d x}\right )}{3 f^{4/3} (d e-c f)^{2/3}} \]
1/3*b*d*Ci(b*(-c*f+d*e)^(1/3)/f^(1/3)+b*(d*x+c)^(1/3))*cos(a-b*(-c*f+d*e)^ (1/3)/f^(1/3))/f^(4/3)/(-c*f+d*e)^(2/3)-1/3*(-1)^(1/3)*b*d*Ci((-1)^(1/3)*b *(-c*f+d*e)^(1/3)/f^(1/3)-b*(d*x+c)^(1/3))*cos(a+(-1)^(1/3)*b*(-c*f+d*e)^( 1/3)/f^(1/3))/f^(4/3)/(-c*f+d*e)^(2/3)+1/3*(-1)^(2/3)*b*d*Ci((-1)^(2/3)*b* (-c*f+d*e)^(1/3)/f^(1/3)+b*(d*x+c)^(1/3))*cos(a-(-1)^(2/3)*b*(-c*f+d*e)^(1 /3)/f^(1/3))/f^(4/3)/(-c*f+d*e)^(2/3)-1/3*b*d*Si(b*(-c*f+d*e)^(1/3)/f^(1/3 )+b*(d*x+c)^(1/3))*sin(a-b*(-c*f+d*e)^(1/3)/f^(1/3))/f^(4/3)/(-c*f+d*e)^(2 /3)+1/3*(-1)^(1/3)*b*d*Si(-(-1)^(1/3)*b*(-c*f+d*e)^(1/3)/f^(1/3)+b*(d*x+c) ^(1/3))*sin(a+(-1)^(1/3)*b*(-c*f+d*e)^(1/3)/f^(1/3))/f^(4/3)/(-c*f+d*e)^(2 /3)-1/3*(-1)^(2/3)*b*d*Si((-1)^(2/3)*b*(-c*f+d*e)^(1/3)/f^(1/3)+b*(d*x+c)^ (1/3))*sin(a-(-1)^(2/3)*b*(-c*f+d*e)^(1/3)/f^(1/3))/f^(4/3)/(-c*f+d*e)^(2/ 3)-sin(a+b*(d*x+c)^(1/3))/f/(f*x+e)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 0.87 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.32 \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(e+f x)^2} \, dx=\frac {\frac {3 i e^{-i \left (a+b \sqrt [3]{c+d x}\right )} \left (-1+e^{2 i \left (a+b \sqrt [3]{c+d x}\right )}\right ) f}{e+f x}+b d \text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,\frac {e^{-i a-i b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (-i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]+b d \text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,\frac {e^{i a+i b \text {$\#$1}} \operatorname {ExpIntegralEi}\left (i b \left (\sqrt [3]{c+d x}-\text {$\#$1}\right )\right )}{\text {$\#$1}^2}\&\right ]}{6 f^2} \]
(((3*I)*(-1 + E^((2*I)*(a + b*(c + d*x)^(1/3))))*f)/(E^(I*(a + b*(c + d*x) ^(1/3)))*(e + f*x)) + b*d*RootSum[d*e - c*f + f*#1^3 & , (E^((-I)*a - I*b* #1)*ExpIntegralEi[(-I)*b*((c + d*x)^(1/3) - #1)])/#1^2 & ] + b*d*RootSum[d *e - c*f + f*#1^3 & , (E^(I*a + I*b*#1)*ExpIntegralEi[I*b*((c + d*x)^(1/3) - #1)])/#1^2 & ])/(6*f^2)
Time = 1.64 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3912, 27, 3822, 3815, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(e+f x)^2} \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle \frac {3 \int \frac {d^2 (c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2}d\sqrt [3]{c+d x}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 d \int \frac {(c+d x)^{2/3} \sin \left (a+b \sqrt [3]{c+d x}\right )}{(d e-c f+f (c+d x))^2}d\sqrt [3]{c+d x}\) |
| \(\Big \downarrow \) 3822 |
| \(\displaystyle 3 d \left (\frac {b \int \frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{d e-c f+f (c+d x)}d\sqrt [3]{c+d x}}{3 f}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{3 f (f (c+d x)-c f+d e)}\right )\) |
| \(\Big \downarrow \) 3815 |
| \(\displaystyle 3 d \left (\frac {b \int \left (-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (d e-c f)^{2/3} \left (-\sqrt [3]{d e-c f}-\sqrt [3]{f} \sqrt [3]{c+d x}\right )}-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (d e-c f)^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{f} \sqrt [3]{c+d x}-\sqrt [3]{d e-c f}\right )}-\frac {\cos \left (a+b \sqrt [3]{c+d x}\right )}{3 (d e-c f)^{2/3} \left (-\sqrt [3]{d e-c f}-(-1)^{2/3} \sqrt [3]{f} \sqrt [3]{c+d x}\right )}\right )d\sqrt [3]{c+d x}}{3 f}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{3 f (f (c+d x)-c f+d e)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 d \left (\frac {b \left (-\frac {\sqrt [3]{-1} \cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 \sqrt [3]{f} (d e-c f)^{2/3}}+\frac {\cos \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 \sqrt [3]{f} (d e-c f)^{2/3}}+\frac {(-1)^{2/3} \cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 \sqrt [3]{f} (d e-c f)^{2/3}}-\frac {\sqrt [3]{-1} \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}-b \sqrt [3]{c+d x}\right )}{3 \sqrt [3]{f} (d e-c f)^{2/3}}-\frac {\sin \left (a-\frac {b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {\sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 \sqrt [3]{f} (d e-c f)^{2/3}}-\frac {(-1)^{2/3} \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{d e-c f}}{\sqrt [3]{f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{d e-c f} b}{\sqrt [3]{f}}+\sqrt [3]{c+d x} b\right )}{3 \sqrt [3]{f} (d e-c f)^{2/3}}\right )}{3 f}-\frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{3 f (f (c+d x)-c f+d e)}\right )\) |
3*d*(-1/3*Sin[a + b*(c + d*x)^(1/3)]/(f*(d*e - c*f + f*(c + d*x))) + (b*(- 1/3*((-1)^(1/3)*Cos[a + ((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*CosInteg ral[((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3) - b*(c + d*x)^(1/3)])/(f^(1/3 )*(d*e - c*f)^(2/3)) + (Cos[a - (b*(d*e - c*f)^(1/3))/f^(1/3)]*CosIntegral [(b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(1/3)*(d*e - c*f )^(2/3)) + ((-1)^(2/3)*Cos[a - ((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3)]*C osIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x)^(1/3)])/ (3*f^(1/3)*(d*e - c*f)^(2/3)) - ((-1)^(1/3)*Sin[a + ((-1)^(1/3)*b*(d*e - c *f)^(1/3))/f^(1/3)]*SinIntegral[((-1)^(1/3)*b*(d*e - c*f)^(1/3))/f^(1/3) - b*(c + d*x)^(1/3)])/(3*f^(1/3)*(d*e - c*f)^(2/3)) - (Sin[a - (b*(d*e - c* f)^(1/3))/f^(1/3)]*SinIntegral[(b*(d*e - c*f)^(1/3))/f^(1/3) + b*(c + d*x) ^(1/3)])/(3*f^(1/3)*(d*e - c*f)^(2/3)) - ((-1)^(2/3)*Sin[a - ((-1)^(2/3)*b *(d*e - c*f)^(1/3))/f^(1/3)]*SinIntegral[((-1)^(2/3)*b*(d*e - c*f)^(1/3))/ f^(1/3) + b*(c + d*x)^(1/3)])/(3*f^(1/3)*(d*e - c*f)^(2/3))))/(3*f))
3.3.11.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int [ExpandIntegrand[Cos[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_) ], x_Symbol] :> Simp[e^m*(a + b*x^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))), x] - Simp[d*(e^m/(b*n*(p + 1))) Int[(a + b*x^n)^(p + 1)*Cos[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && ( IntegerQ[n] || GtQ[e, 0])
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/(n*f) Subst[Int[ExpandIntegra nd[(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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.36 (sec) , antiderivative size = 1176, normalized size of antiderivative = 2.12
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1176\) |
| default | \(\text {Expression too large to display}\) | \(1176\) |
3*d/b^3*(b^6*a^2*(sin(a+b*(d*x+c)^(1/3))*(1/3/b^3/(c*f-d*e)*(a+b*(d*x+c)^( 1/3))-1/3*a/b^3/(c*f-d*e))/(b^3*c*f-b^3*d*e+a^3*f-3*a^2*f*(a+b*(d*x+c)^(1/ 3))+3*a*f*(a+b*(d*x+c)^(1/3))^2-f*(a+b*(d*x+c)^(1/3))^3)-2/9/b^3/f*sum(1/( c*f-d*e)/(_R1^2-2*_R1*a+a^2)*(-Si(-b*(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b*(d *x+c)^(1/3)-_R1+a)*sin(_R1)),_R1=RootOf(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f +3*_Z*a^2*f-a^3*f))-1/9/b^3/f*sum(1/(-_RR1+a)/(c*f-d*e)*(Si(-b*(d*x+c)^(1/ 3)+_RR1-a)*sin(_RR1)+Ci(b*(d*x+c)^(1/3)-_RR1+a)*cos(_RR1)),_RR1=RootOf(-b^ 3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f)))+sin(a+b*(d*x+c)^(1/3)) *(-2/3*a*b^3/(c*f-d*e)*(a+b*(d*x+c)^(1/3))^2+2/3*a^2*b^3/(c*f-d*e)*(a+b*(d *x+c)^(1/3)))/(b^3*c*f-b^3*d*e+a^3*f-3*a^2*f*(a+b*(d*x+c)^(1/3))+3*a*f*(a+ b*(d*x+c)^(1/3))^2-f*(a+b*(d*x+c)^(1/3))^3)+2/9*a*b^3/f*sum((_R1+a)/(c*f-d *e)/(_R1^2-2*_R1*a+a^2)*(-Si(-b*(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b*(d*x+c) ^(1/3)-_R1+a)*sin(_R1)),_R1=RootOf(-b^3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z *a^2*f-a^3*f))+2/9*a*b^3/f*sum(_RR1/(-_RR1+a)/(c*f-d*e)*(Si(-b*(d*x+c)^(1/ 3)+_RR1-a)*sin(_RR1)+Ci(b*(d*x+c)^(1/3)-_RR1+a)*cos(_RR1)),_RR1=RootOf(-b^ 3*c*f+b^3*d*e+_Z^3*f-3*_Z^2*a*f+3*_Z*a^2*f-a^3*f))+sin(a+b*(d*x+c)^(1/3))* (2/3*a*b^3/(c*f-d*e)*(a+b*(d*x+c)^(1/3))^2-a^2*b^3/(c*f-d*e)*(a+b*(d*x+c)^ (1/3))+1/3*b^3*(b^3*c*f-b^3*d*e+a^3*f)/f/(c*f-d*e))/(b^3*c*f-b^3*d*e+a^3*f -3*a^2*f*(a+b*(d*x+c)^(1/3))+3*a*f*(a+b*(d*x+c)^(1/3))^2-f*(a+b*(d*x+c)^(1 /3))^3)-2/9*a*b^3/f*sum(_R1/(c*f-d*e)/(_R1^2-2*_R1*a+a^2)*(-Si(-b*(d*x+...
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 730, normalized size of antiderivative = 1.32 \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(e+f x)^2} \, dx=-\frac {{\left (i \, d f x + i \, d e - \sqrt {3} {\left (d f x + d e\right )}\right )} \left (\frac {i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\right )} \left (\frac {i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (i \, \sqrt {3} + 1\right )} \left (\frac {i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac {1}{3}} - i \, a\right )} + {\left (i \, d f x + i \, d e + \sqrt {3} {\left (d f x + d e\right )}\right )} \left (\frac {i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, {\left (i \, \sqrt {3} - 1\right )} \left (\frac {i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (-i \, \sqrt {3} + 1\right )} \left (\frac {i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac {1}{3}} - i \, a\right )} + {\left (-i \, d f x - i \, d e + \sqrt {3} {\left (d f x + d e\right )}\right )} \left (\frac {-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, {\left (-i \, \sqrt {3} - 1\right )} \left (\frac {-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (i \, \sqrt {3} + 1\right )} \left (\frac {-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac {1}{3}} + i \, a\right )} + {\left (-i \, d f x - i \, d e - \sqrt {3} {\left (d f x + d e\right )}\right )} \left (\frac {-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \frac {1}{2} \, {\left (i \, \sqrt {3} - 1\right )} \left (\frac {-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac {1}{3}}\right ) e^{\left (\frac {1}{2} \, {\left (-i \, \sqrt {3} + 1\right )} \left (\frac {-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac {1}{3}} + i \, a\right )} - 2 \, {\left (-i \, d f x - i \, d e\right )} \left (\frac {-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (\frac {-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac {1}{3}}\right ) e^{\left (i \, a - \left (\frac {-i \, b^{3} d e + i \, b^{3} c f}{f}\right )^{\frac {1}{3}}\right )} - 2 \, {\left (i \, d f x + i \, d e\right )} \left (\frac {i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, {\left (d x + c\right )}^{\frac {1}{3}} b + \left (\frac {i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, a - \left (\frac {i \, b^{3} d e - i \, b^{3} c f}{f}\right )^{\frac {1}{3}}\right )} + 12 \, {\left (d e - c f\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{12 \, {\left (d e^{2} f - c e f^{2} + {\left (d e f^{2} - c f^{3}\right )} x\right )}} \]
-1/12*((I*d*f*x + I*d*e - sqrt(3)*(d*f*x + d*e))*((I*b^3*d*e - I*b^3*c*f)/ f)^(1/3)*Ei(-I*(d*x + c)^(1/3)*b + 1/2*(-I*sqrt(3) - 1)*((I*b^3*d*e - I*b^ 3*c*f)/f)^(1/3))*e^(1/2*(I*sqrt(3) + 1)*((I*b^3*d*e - I*b^3*c*f)/f)^(1/3) - I*a) + (I*d*f*x + I*d*e + sqrt(3)*(d*f*x + d*e))*((I*b^3*d*e - I*b^3*c*f )/f)^(1/3)*Ei(-I*(d*x + c)^(1/3)*b + 1/2*(I*sqrt(3) - 1)*((I*b^3*d*e - I*b ^3*c*f)/f)^(1/3))*e^(1/2*(-I*sqrt(3) + 1)*((I*b^3*d*e - I*b^3*c*f)/f)^(1/3 ) - I*a) + (-I*d*f*x - I*d*e + sqrt(3)*(d*f*x + d*e))*((-I*b^3*d*e + I*b^3 *c*f)/f)^(1/3)*Ei(I*(d*x + c)^(1/3)*b + 1/2*(-I*sqrt(3) - 1)*((-I*b^3*d*e + I*b^3*c*f)/f)^(1/3))*e^(1/2*(I*sqrt(3) + 1)*((-I*b^3*d*e + I*b^3*c*f)/f) ^(1/3) + I*a) + (-I*d*f*x - I*d*e - sqrt(3)*(d*f*x + d*e))*((-I*b^3*d*e + I*b^3*c*f)/f)^(1/3)*Ei(I*(d*x + c)^(1/3)*b + 1/2*(I*sqrt(3) - 1)*((-I*b^3* d*e + I*b^3*c*f)/f)^(1/3))*e^(1/2*(-I*sqrt(3) + 1)*((-I*b^3*d*e + I*b^3*c* f)/f)^(1/3) + I*a) - 2*(-I*d*f*x - I*d*e)*((-I*b^3*d*e + I*b^3*c*f)/f)^(1/ 3)*Ei(I*(d*x + c)^(1/3)*b + ((-I*b^3*d*e + I*b^3*c*f)/f)^(1/3))*e^(I*a - ( (-I*b^3*d*e + I*b^3*c*f)/f)^(1/3)) - 2*(I*d*f*x + I*d*e)*((I*b^3*d*e - I*b ^3*c*f)/f)^(1/3)*Ei(-I*(d*x + c)^(1/3)*b + ((I*b^3*d*e - I*b^3*c*f)/f)^(1/ 3))*e^(-I*a - ((I*b^3*d*e - I*b^3*c*f)/f)^(1/3)) + 12*(d*e - c*f)*sin((d*x + c)^(1/3)*b + a))/(d*e^2*f - c*e*f^2 + (d*e*f^2 - c*f^3)*x)
\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(e+f x)^2} \, dx=\int \frac {\sin {\left (a + b \sqrt [3]{c + d x} \right )}}{\left (e + f x\right )^{2}}\, dx \]
\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(e+f x)^2} \, dx=\int { \frac {\sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{{\left (f x + e\right )}^{2}} \,d x } \]
\[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(e+f x)^2} \, dx=\int { \frac {\sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{{\left (f x + e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sin \left (a+b \sqrt [3]{c+d x}\right )}{(e+f x)^2} \, dx=\int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )}{{\left (e+f\,x\right )}^2} \,d x \]