Integrand size = 22, antiderivative size = 434 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{e+f x} \, dx=-\frac {3 \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{f}+\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right ) \sin \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )}{f}+\frac {\operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right ) \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )}{f}+\frac {\operatorname {CosIntegral}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right ) \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right )}{f}-\frac {3 \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{f}-\frac {\cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {\cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{f} \]
-3*cos(a)*Si(b/(d*x+c)^(1/3))/f-cos(a+(-1)^(1/3)*b*f^(1/3)/(-c*f+d*e)^(1/3 ))*Si((-1)^(1/3)*b*f^(1/3)/(-c*f+d*e)^(1/3)-b/(d*x+c)^(1/3))/f+cos(a-b*f^( 1/3)/(-c*f+d*e)^(1/3))*Si(b*f^(1/3)/(-c*f+d*e)^(1/3)+b/(d*x+c)^(1/3))/f+co s(a-(-1)^(2/3)*b*f^(1/3)/(-c*f+d*e)^(1/3))*Si((-1)^(2/3)*b*f^(1/3)/(-c*f+d *e)^(1/3)+b/(d*x+c)^(1/3))/f-3*Ci(b/(d*x+c)^(1/3))*sin(a)/f+Ci(b*f^(1/3)/( -c*f+d*e)^(1/3)+b/(d*x+c)^(1/3))*sin(a-b*f^(1/3)/(-c*f+d*e)^(1/3))/f+Ci((- 1)^(1/3)*b*f^(1/3)/(-c*f+d*e)^(1/3)-b/(d*x+c)^(1/3))*sin(a+(-1)^(1/3)*b*f^ (1/3)/(-c*f+d*e)^(1/3))/f+Ci((-1)^(2/3)*b*f^(1/3)/(-c*f+d*e)^(1/3)+b/(d*x+ c)^(1/3))*sin(a-(-1)^(2/3)*b*f^(1/3)/(-c*f+d*e)^(1/3))/f
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 25.31 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.39 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{e+f x} \, dx=\frac {i \left (\left (-3 \operatorname {ExpIntegralEi}\left (-\frac {i b}{\sqrt [3]{c+d x}}\right )+\text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,e^{-\frac {i b}{\text {$\#$1}}} \operatorname {ExpIntegralEi}\left (-i b \left (\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{\text {$\#$1}}\right )\right )\&\right ]\right ) (\cos (a)-i \sin (a))+\left (3 \operatorname {ExpIntegralEi}\left (\frac {i b}{\sqrt [3]{c+d x}}\right )-\text {RootSum}\left [d e-c f+f \text {$\#$1}^3\&,e^{\frac {i b}{\text {$\#$1}}} \operatorname {ExpIntegralEi}\left (i b \left (\frac {1}{\sqrt [3]{c+d x}}-\frac {1}{\text {$\#$1}}\right )\right )\&\right ]\right ) (\cos (a)+i \sin (a))\right )}{2 f} \]
((I/2)*((-3*ExpIntegralEi[((-I)*b)/(c + d*x)^(1/3)] + RootSum[d*e - c*f + f*#1^3 & , ExpIntegralEi[(-I)*b*((c + d*x)^(-1/3) - #1^(-1))]/E^((I*b)/#1) & ])*(Cos[a] - I*Sin[a]) + (3*ExpIntegralEi[(I*b)/(c + d*x)^(1/3)] - Root Sum[d*e - c*f + f*#1^3 & , E^((I*b)/#1)*ExpIntegralEi[I*b*((c + d*x)^(-1/3 ) - #1^(-1))] & ])*(Cos[a] + I*Sin[a])))/f
Time = 1.70 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{e+f x} \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle -\frac {3 \int \left (\frac {d \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{f}-\frac {d (d e-c f) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{f (c+d x)^{2/3} \left (f+\frac {d e-c f}{c+d x}\right )}\right )d\frac {1}{\sqrt [3]{c+d x}}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \left (-\frac {d \sin \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f}-\frac {d \sin \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f}-\frac {d \sin \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \operatorname {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f}+\frac {d \sin (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{f}+\frac {d \cos \left (a+\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}-\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f}-\frac {d \cos \left (a-\frac {b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {\sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f}-\frac {d \cos \left (a-\frac {(-1)^{2/3} b \sqrt [3]{f}}{\sqrt [3]{d e-c f}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{f} b}{\sqrt [3]{d e-c f}}+\frac {b}{\sqrt [3]{c+d x}}\right )}{3 f}+\frac {d \cos (a) \text {Si}\left (\frac {b}{\sqrt [3]{c+d x}}\right )}{f}\right )}{d}\) |
(-3*((d*CosIntegral[b/(c + d*x)^(1/3)]*Sin[a])/f - (d*CosIntegral[(b*f^(1/ 3))/(d*e - c*f)^(1/3) + b/(c + d*x)^(1/3)]*Sin[a - (b*f^(1/3))/(d*e - c*f) ^(1/3)])/(3*f) - (d*CosIntegral[((-1)^(1/3)*b*f^(1/3))/(d*e - c*f)^(1/3) - b/(c + d*x)^(1/3)]*Sin[a + ((-1)^(1/3)*b*f^(1/3))/(d*e - c*f)^(1/3)])/(3* f) - (d*CosIntegral[((-1)^(2/3)*b*f^(1/3))/(d*e - c*f)^(1/3) + b/(c + d*x) ^(1/3)]*Sin[a - ((-1)^(2/3)*b*f^(1/3))/(d*e - c*f)^(1/3)])/(3*f) + (d*Cos[ a]*SinIntegral[b/(c + d*x)^(1/3)])/f + (d*Cos[a + ((-1)^(1/3)*b*f^(1/3))/( d*e - c*f)^(1/3)]*SinIntegral[((-1)^(1/3)*b*f^(1/3))/(d*e - c*f)^(1/3) - b /(c + d*x)^(1/3)])/(3*f) - (d*Cos[a - (b*f^(1/3))/(d*e - c*f)^(1/3)]*SinIn tegral[(b*f^(1/3))/(d*e - c*f)^(1/3) + b/(c + d*x)^(1/3)])/(3*f) - (d*Cos[ a - ((-1)^(2/3)*b*f^(1/3))/(d*e - c*f)^(1/3)]*SinIntegral[((-1)^(2/3)*b*f^ (1/3))/(d*e - c*f)^(1/3) + b/(c + d*x)^(1/3)])/(3*f)))/d
3.3.20.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.99 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.36
| method | result | size |
| derivativedivides | \(-3 b^{3} \left (\frac {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )+\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{f \,b^{3}}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (c f -d e \right ) \textit {\_Z}^{3}+\left (-3 a c f +3 a d e \right ) \textit {\_Z}^{2}+\left (3 a^{2} c f -3 a^{2} d e \right ) \textit {\_Z} -a^{3} c f +a^{3} d e -f \,b^{3}\right )}{\sum }\left (-\operatorname {Si}\left (-\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )\right )}{3 f \,b^{3}}\right )\) | \(156\) |
| default | \(-3 b^{3} \left (\frac {\operatorname {Si}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \cos \left (a \right )+\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right ) \sin \left (a \right )}{f \,b^{3}}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (c f -d e \right ) \textit {\_Z}^{3}+\left (-3 a c f +3 a d e \right ) \textit {\_Z}^{2}+\left (3 a^{2} c f -3 a^{2} d e \right ) \textit {\_Z} -a^{3} c f +a^{3} d e -f \,b^{3}\right )}{\sum }\left (-\operatorname {Si}\left (-\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}+\textit {\_R1} -a \right ) \cos \left (\textit {\_R1} \right )+\operatorname {Ci}\left (\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}-\textit {\_R1} +a \right ) \sin \left (\textit {\_R1} \right )\right )}{3 f \,b^{3}}\right )\) | \(156\) |
-3*b^3*(1/f/b^3*(Si(b/(d*x+c)^(1/3))*cos(a)+Ci(b/(d*x+c)^(1/3))*sin(a))-1/ 3/f/b^3*sum(-Si(-b/(d*x+c)^(1/3)+_R1-a)*cos(_R1)+Ci(b/(d*x+c)^(1/3)-_R1+a) *sin(_R1),_R1=RootOf((c*f-d*e)*_Z^3+(-3*a*c*f+3*a*d*e)*_Z^2+(3*a^2*c*f-3*a ^2*d*e)*_Z-a^3*c*f+a^3*d*e-f*b^3)))
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 548, normalized size of antiderivative = 1.26 \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{e+f x} \, dx=\frac {i \, {\rm Ei}\left (\frac {-2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (-i \, d x - i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, a\right )} - i \, {\rm Ei}\left (\frac {2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (-\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (-i \, d x - i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, a\right )} + i \, {\rm Ei}\left (\frac {-2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (i \, d x + i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, a\right )} - i \, {\rm Ei}\left (\frac {2 i \, {\left (d x + c\right )}^{\frac {2}{3}} b - \left (-\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (d x - \sqrt {3} {\left (i \, d x + i \, c\right )} + c\right )}}{2 \, {\left (d x + c\right )}}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, a\right )} - i \, {\rm Ei}\left (\frac {i \, {\left (d x + c\right )}^{\frac {2}{3}} b + \left (-\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (d x + c\right )}}{d x + c}\right ) e^{\left (i \, a - \left (-\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}}\right )} + i \, {\rm Ei}\left (\frac {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b + \left (\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}} {\left (d x + c\right )}}{d x + c}\right ) e^{\left (-i \, a - \left (\frac {i \, b^{3} f}{d e - c f}\right )^{\frac {1}{3}}\right )} - 6 \, \operatorname {Ci}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right ) \sin \left (a\right ) - 6 \, \cos \left (a\right ) \operatorname {Si}\left (\frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{2 \, f} \]
1/2*(I*Ei(1/2*(-2*I*(d*x + c)^(2/3)*b - (I*b^3*f/(d*e - c*f))^(1/3)*(d*x - sqrt(3)*(-I*d*x - I*c) + c))/(d*x + c))*e^(1/2*(I*b^3*f/(d*e - c*f))^(1/3 )*(I*sqrt(3) + 1) - I*a) - I*Ei(1/2*(2*I*(d*x + c)^(2/3)*b - (-I*b^3*f/(d* e - c*f))^(1/3)*(d*x - sqrt(3)*(-I*d*x - I*c) + c))/(d*x + c))*e^(1/2*(-I* b^3*f/(d*e - c*f))^(1/3)*(I*sqrt(3) + 1) + I*a) + I*Ei(1/2*(-2*I*(d*x + c) ^(2/3)*b - (I*b^3*f/(d*e - c*f))^(1/3)*(d*x - sqrt(3)*(I*d*x + I*c) + c))/ (d*x + c))*e^(1/2*(I*b^3*f/(d*e - c*f))^(1/3)*(-I*sqrt(3) + 1) - I*a) - I* Ei(1/2*(2*I*(d*x + c)^(2/3)*b - (-I*b^3*f/(d*e - c*f))^(1/3)*(d*x - sqrt(3 )*(I*d*x + I*c) + c))/(d*x + c))*e^(1/2*(-I*b^3*f/(d*e - c*f))^(1/3)*(-I*s qrt(3) + 1) + I*a) - I*Ei((I*(d*x + c)^(2/3)*b + (-I*b^3*f/(d*e - c*f))^(1 /3)*(d*x + c))/(d*x + c))*e^(I*a - (-I*b^3*f/(d*e - c*f))^(1/3)) + I*Ei((- I*(d*x + c)^(2/3)*b + (I*b^3*f/(d*e - c*f))^(1/3)*(d*x + c))/(d*x + c))*e^ (-I*a - (I*b^3*f/(d*e - c*f))^(1/3)) - 6*cos_integral(b/(d*x + c)^(1/3))*s in(a) - 6*cos(a)*sin_integral(b/(d*x + c)^(1/3)))/f
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{e+f x} \, dx=\int \frac {\sin {\left (a + \frac {b}{\sqrt [3]{c + d x}} \right )}}{e + f x}\, dx \]
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{e+f x} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{f x + e} \,d x } \]
\[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{e+f x} \, dx=\int { \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{f x + e} \,d x } \]
Timed out. \[ \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{e+f x} \, dx=\int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{e+f\,x} \,d x \]