Integrand size = 20, antiderivative size = 419 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\frac {b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac {b^3 f (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac {b^3 (d e-c f) \cos (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^6 f \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{240 d^2}-\frac {b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^4 f (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac {b^6 f \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac {b^3 (d e-c f) \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d^2} \]
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
Time = 0.70 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.29 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\frac {e \sqrt [3]{c+d x} \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right ) \left (b \sqrt [3]{c+d x} \cos (a)-b^2 \sin (a)+2 (c+d x)^{2/3} \sin (a)\right )}{2 d}+\frac {f \sqrt [3]{c+d x} \cos \left (\frac {b}{\sqrt [3]{c+d x}}\right ) \left (b^5 \cos (a)-120 b c \sqrt [3]{c+d x} \cos (a)-2 b^3 (c+d x)^{2/3} \cos (a)+24 b (c+d x)^{4/3} \cos (a)+120 b^2 c \sin (a)+b^4 \sqrt [3]{c+d x} \sin (a)-240 c (c+d x)^{2/3} \sin (a)-6 b^2 (c+d x) \sin (a)+120 (c+d x)^{5/3} \sin (a)\right )}{240 d^2}+\frac {e \sqrt [3]{c+d x} \left (-b^2 \cos (a)+2 (c+d x)^{2/3} \cos (a)-b \sqrt [3]{c+d x} \sin (a)\right ) \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )}{2 d}+\frac {f \sqrt [3]{c+d x} \left (120 b^2 c \cos (a)+b^4 \sqrt [3]{c+d x} \cos (a)-240 c (c+d x)^{2/3} \cos (a)-6 b^2 (c+d x) \cos (a)+120 (c+d x)^{5/3} \cos (a)-b^5 \sin (a)+120 b c \sqrt [3]{c+d x} \sin (a)+2 b^3 (c+d x)^{2/3} \sin (a)-24 b (c+d x)^{4/3} \sin (a)\right ) \sin \left (\frac {b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac {b^3 e \left (\cos (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )-\sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{2 d}+\frac {b^3 f \left (-120 c \cos (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+b^3 \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)+b^3 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )+120 c \sin (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{240 d^2} \]
(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*Cos[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] + 120*(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]*SinI ntegral[b/(c + d*x)^(1/3)]))/(2*d) + (b^3*f*(-120*c*Cos[a]*CosIntegral[b/( c + d*x)^(1/3)] + b^3*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a] + b^3*Cos[a]*S inIntegral[b/(c + d*x)^(1/3)] + 120*c*Sin[a]*SinIntegral[b/(c + d*x)^(1/3) ]))/(240*d^2)
Time = 0.62 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3912, 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.
| \(\displaystyle \int (e+f x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle -\frac {3 \int \left (\frac {f \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) (c+d x)^{7/3}}{d}+\frac {(d e-c f) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) (c+d x)^{4/3}}{d}\right )d\frac {1}{\sqrt [3]{c+d x}}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \left (-\frac {b^6 f \sin (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{720 d}-\frac {b^6 f \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{720 d}-\frac {b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{720 d}-\frac {b^4 f (c+d x)^{2/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{720 d}-\frac {b^3 \cos (a) (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{6 d}+\frac {b^3 \sin (a) (d e-c f) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{6 d}+\frac {b^3 f (c+d x) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{360 d}+\frac {b^2 \sqrt [3]{c+d x} (d e-c f) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{6 d}+\frac {b^2 f (c+d x)^{4/3} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{120 d}-\frac {(c+d x) (d e-c f) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 d}-\frac {b (c+d x)^{2/3} (d e-c f) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{6 d}-\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{6 d}-\frac {b f (c+d x)^{5/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{30 d}\right )}{d}\) |
(-3*(-1/720*(b^5*f*(c + d*x)^(1/3)*Cos[a + b/(c + d*x)^(1/3)])/d - (b*(d*e - c*f)*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(6*d) + (b^3*f*(c + d* x)*Cos[a + b/(c + d*x)^(1/3)])/(360*d) - (b*f*(c + d*x)^(5/3)*Cos[a + b/(c + d*x)^(1/3)])/(30*d) - (b^3*(d*e - c*f)*Cos[a]*CosIntegral[b/(c + d*x)^( 1/3)])/(6*d) - (b^6*f*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a])/(720*d) + (b^ 2*(d*e - c*f)*(c + d*x)^(1/3)*Sin[a + b/(c + d*x)^(1/3)])/(6*d) - (b^4*f*( c + d*x)^(2/3)*Sin[a + b/(c + d*x)^(1/3)])/(720*d) - ((d*e - c*f)*(c + d*x )*Sin[a + b/(c + d*x)^(1/3)])/(3*d) + (b^2*f*(c + d*x)^(4/3)*Sin[a + b/(c + d*x)^(1/3)])/(120*d) - (f*(c + d*x)^2*Sin[a + b/(c + d*x)^(1/3)])/(6*d) - (b^6*f*Cos[a]*SinIntegral[b/(c + d*x)^(1/3)])/(720*d) + (b^3*(d*e - c*f) *Sin[a]*SinIntegral[b/(c + d*x)^(1/3)])/(6*d)))/d
3.3.18.3.1 Defintions of rubi rules used
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]
Time = 1.18 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.93
| method | result | size |
| derivativedivides | \(-\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 {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{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 {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{6}\right )+f \,b^{3} \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 {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{720}-\frac {\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{720}\right )\right )}{d^{2}}\) | \(391\) |
| default | \(-\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 {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{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 {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{6}-\frac {\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{6}\right )+f \,b^{3} \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 {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )}{720}-\frac {\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{720}\right )\right )}{d^{2}}\) | \(391\) |
| parts | \(\text {Expression too large to display}\) | \(956\) |
-3/d^2*b^3*(-c*f*(-1/3*sin(a+b/(d*x+c)^(1/3))/b^3*(d*x+c)-1/6*cos(a+b/(d*x +c)^(1/3))/b^2*(d*x+c)^(2/3)+1/6*sin(a+b/(d*x+c)^(1/3))/b*(d*x+c)^(1/3)+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*sin (a+b/(d*x+c)^(1/3))/b^3*(d*x+c)-1/6*cos(a+b/(d*x+c)^(1/3))/b^2*(d*x+c)^(2/ 3)+1/6*sin(a+b/(d*x+c)^(1/3))/b*(d*x+c)^(1/3)+1/6*Si(b/(d*x+c)^(1/3))*sin( a)-1/6*Ci(b/(d*x+c)^(1/3))*cos(a))+f*b^3*(-1/6*sin(a+b/(d*x+c)^(1/3))/b^6* (d*x+c)^2-1/30*cos(a+b/(d*x+c)^(1/3))/b^5*(d*x+c)^(5/3)+1/120*sin(a+b/(d*x +c)^(1/3))/b^4*(d*x+c)^(4/3)+1/360*cos(a+b/(d*x+c)^(1/3))/b^3*(d*x+c)-1/72 0*sin(a+b/(d*x+c)^(1/3))/b^2*(d*x+c)^(2/3)-1/720*cos(a+b/(d*x+c)^(1/3))/b* (d*x+c)^(1/3)-1/720*Si(b/(d*x+c)^(1/3))*cos(a)-1/720*Ci(b/(d*x+c)^(1/3))*s in(a)))
Time = 0.35 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.62 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\frac {{\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{5} f - 2 \, b^{3} d f x - 2 \, b^{3} c f + 24 \, {\left (b d f x + 5 \, b d e - 4 \, b c f\right )} {\left (d x + c\right )}^{\frac {2}{3}}\right )} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) + {\left (b^{6} f \sin \left (a\right ) + 120 \, {\left (b^{3} d e - b^{3} c f\right )} \cos \left (a\right )\right )} \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{4} f + 120 \, d^{2} f x^{2} + 240 \, d^{2} e x + 240 \, c d e - 120 \, c^{2} f - 6 \, {\left (b^{2} d f x + 20 \, b^{2} d e - 19 \, b^{2} c f\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) + {\left (b^{6} f \cos \left (a\right ) - 120 \, {\left (b^{3} d e - b^{3} c f\right )} \sin \left (a\right )\right )} \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{240 \, d^{2}} \]
1/240*(((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_integral(b/ (d*x + c)^(1/3)) + ((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)) + (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
\[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\int \left (e + f x\right ) \sin {\left (a + \frac {b}{\sqrt [3]{c + d x}} \right )}\, dx \]
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.09 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\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 \left (a\right ) + {\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 \left (a\right )\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 \left (a\right ) + {\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 \left (a\right )\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 \left (a\right ) + {\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 \left (a\right )\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} \]
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)))*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)))*f/d )/d
Leaf count of result is larger than twice the leaf count of optimal. 3727 vs. \(2 (357) = 714\).
Time = 0.66 (sec) , antiderivative size = 3727, normalized size of antiderivative = 8.89 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\text {Too large to display} \]
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_integral(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_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)^2*a*b^4*cos(a)*cos_i ntegral(-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...
Timed out. \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right ) \, dx=\int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )\,\left (e+f\,x\right ) \,d x \]