Integrand size = 17, antiderivative size = 218 \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=-\frac {2 a \cos (c+d x)}{b^2 d^3}+\frac {a^3 \cos (c+d x)}{b^4 d}+\frac {6 x \cos (c+d x)}{b d^3}-\frac {a^2 x \cos (c+d x)}{b^3 d}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {x^3 \cos (c+d x)}{b d}+\frac {a^4 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {6 \sin (c+d x)}{b d^4}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {3 x^2 \sin (c+d x)}{b d^2}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5} \]
-2*a*cos(d*x+c)/b^2/d^3+a^3*cos(d*x+c)/b^4/d+6*x*cos(d*x+c)/b/d^3-a^2*x*co s(d*x+c)/b^3/d+a*x^2*cos(d*x+c)/b^2/d-x^3*cos(d*x+c)/b/d+a^4*cos(-c+a*d/b) *Si(a*d/b+d*x)/b^5-a^4*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^5-6*sin(d*x+c)/b/d^4+ a^2*sin(d*x+c)/b^3/d^2-2*a*x*sin(d*x+c)/b^2/d^2+3*x^2*sin(d*x+c)/b/d^2
Time = 0.41 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.72 \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\frac {a^4 d^4 \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+b \left (d \left (a^3 d^2-a^2 b d^2 x+b^3 x \left (6-d^2 x^2\right )+a b^2 \left (-2+d^2 x^2\right )\right ) \cos (c+d x)+b \left (a^2 d^2-2 a b d^2 x+3 b^2 \left (-2+d^2 x^2\right )\right ) \sin (c+d x)\right )+a^4 d^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^5 d^4} \]
(a^4*d^4*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + b*(d*(a^3*d^2 - a^2*b *d^2*x + b^3*x*(6 - d^2*x^2) + a*b^2*(-2 + d^2*x^2))*Cos[c + d*x] + b*(a^2 *d^2 - 2*a*b*d^2*x + 3*b^2*(-2 + d^2*x^2))*Sin[c + d*x]) + a^4*d^4*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(b^5*d^4)
Time = 0.66 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {7293, 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.
| \(\displaystyle \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {a^4 \sin (c+d x)}{b^4 (a+b x)}-\frac {a^3 \sin (c+d x)}{b^4}+\frac {a^2 x \sin (c+d x)}{b^3}-\frac {a x^2 \sin (c+d x)}{b^2}+\frac {x^3 \sin (c+d x)}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^4 \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^4 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {a^3 \cos (c+d x)}{b^4 d}+\frac {a^2 \sin (c+d x)}{b^3 d^2}-\frac {a^2 x \cos (c+d x)}{b^3 d}-\frac {2 a \cos (c+d x)}{b^2 d^3}-\frac {2 a x \sin (c+d x)}{b^2 d^2}+\frac {a x^2 \cos (c+d x)}{b^2 d}-\frac {6 \sin (c+d x)}{b d^4}+\frac {6 x \cos (c+d x)}{b d^3}+\frac {3 x^2 \sin (c+d x)}{b d^2}-\frac {x^3 \cos (c+d x)}{b d}\) |
(-2*a*Cos[c + d*x])/(b^2*d^3) + (a^3*Cos[c + d*x])/(b^4*d) + (6*x*Cos[c + d*x])/(b*d^3) - (a^2*x*Cos[c + d*x])/(b^3*d) + (a*x^2*Cos[c + d*x])/(b^2*d ) - (x^3*Cos[c + d*x])/(b*d) + (a^4*CosIntegral[(a*d)/b + d*x]*Sin[c - (a* d)/b])/b^5 - (6*Sin[c + d*x])/(b*d^4) + (a^2*Sin[c + d*x])/(b^3*d^2) - (2* a*x*Sin[c + d*x])/(b^2*d^2) + (3*x^2*Sin[c + d*x])/(b*d^2) + (a^4*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5
3.1.18.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.42 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.50
| method | result | size |
| risch | \(-\frac {i \pi \,\operatorname {csgn}\left (\frac {d \left (b x +a \right )}{b}\right ) \sin \left (\frac {d a -c b}{b}\right ) a^{4}}{2 b^{5}}-\frac {x^{3} \cos \left (d x +c \right )}{b d}-\frac {\pi \,\operatorname {csgn}\left (\frac {d \left (b x +a \right )}{b}\right ) \cos \left (\frac {d a -c b}{b}\right ) a^{4}}{2 b^{5}}+\frac {i \operatorname {Si}\left (\frac {d \left (b x +a \right )}{b}\right ) \sin \left (\frac {d a -c b}{b}\right ) a^{4}}{b^{5}}+\frac {a \,x^{2} \cos \left (d x +c \right )}{b^{2} d}+\frac {\operatorname {Si}\left (\frac {d \left (b x +a \right )}{b}\right ) \cos \left (\frac {d a -c b}{b}\right ) a^{4}}{b^{5}}+\frac {\operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right ) \sin \left (\frac {d a -c b}{b}\right ) a^{4}}{b^{5}}+\frac {3 x^{2} \sin \left (d x +c \right )}{b \,d^{2}}-\frac {a^{2} x \cos \left (d x +c \right )}{b^{3} d}-\frac {2 a x \sin \left (d x +c \right )}{b^{2} d^{2}}+\frac {a^{3} \cos \left (d x +c \right )}{b^{4} d}+\frac {a^{2} \sin \left (d x +c \right )}{b^{3} d^{2}}+\frac {6 x \cos \left (d x +c \right )}{b \,d^{3}}-\frac {2 a \cos \left (d x +c \right )}{b^{2} d^{3}}-\frac {6 \sin \left (d x +c \right )}{b \,d^{4}}\) | \(328\) |
| derivativedivides | \(\frac {d \,c^{4} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )+\frac {4 \left (d a -c b \right ) d \,c^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}+\frac {4 d \,c^{3} \cos \left (d x +c \right )}{b}+\frac {6 \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) d \,c^{2} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}-\frac {6 d \,c^{2} \left (d a -c b -b \right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{2}}+\frac {4 \left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d c \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{3}}-\frac {4 d c \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}-a b d +b^{2} c +b^{2}\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{b^{3}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{4}}-\frac {d \left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}-d^{2} a^{2} b +2 a \,b^{2} c d -b^{3} c^{2}+a \,b^{2} d -b^{3} c -b^{3}\right ) \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{4}}}{d^{5}}\) | \(784\) |
| default | \(\frac {d \,c^{4} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )+\frac {4 \left (d a -c b \right ) d \,c^{3} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b}+\frac {4 d \,c^{3} \cos \left (d x +c \right )}{b}+\frac {6 \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}\right ) d \,c^{2} \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{2}}-\frac {6 d \,c^{2} \left (d a -c b -b \right ) \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{2}}+\frac {4 \left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) d c \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{3}}-\frac {4 d c \left (d^{2} a^{2}-2 a b c d +b^{2} c^{2}-a b d +b^{2} c +b^{2}\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{b^{3}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{b^{4}}-\frac {d \left (d^{3} a^{3}-3 c \,d^{2} a^{2} b +3 a \,b^{2} c^{2} d -b^{3} c^{3}-d^{2} a^{2} b +2 a \,b^{2} c d -b^{3} c^{2}+a \,b^{2} d -b^{3} c -b^{3}\right ) \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \cos \left (d x +c \right ) \left (d x +c \right )\right )}{b^{4}}}{d^{5}}\) | \(784\) |
-1/2*I/b^5*Pi*csgn(d*(b*x+a)/b)*sin((a*d-b*c)/b)*a^4-x^3*cos(d*x+c)/b/d-1/ 2/b^5*Pi*csgn(d*(b*x+a)/b)*cos((a*d-b*c)/b)*a^4+I/b^5*Si(d*(b*x+a)/b)*sin( (a*d-b*c)/b)*a^4+a*x^2*cos(d*x+c)/b^2/d+1/b^5*Si(d*(b*x+a)/b)*cos((a*d-b*c )/b)*a^4+1/b^5*Ei(1,-I*d*(b*x+a)/b)*sin((a*d-b*c)/b)*a^4+3*x^2*sin(d*x+c)/ b/d^2-a^2*x*cos(d*x+c)/b^3/d-2*a*x*sin(d*x+c)/b^2/d^2+a^3*cos(d*x+c)/b^4/d +a^2*sin(d*x+c)/b^3/d^2+6*x*cos(d*x+c)/b/d^3-2*a*cos(d*x+c)/b^2/d^3-6*sin( d*x+c)/b/d^4
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.86 \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=-\frac {a^{4} d^{4} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - a^{4} d^{4} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + {\left (b^{4} d^{3} x^{3} - a b^{3} d^{3} x^{2} - a^{3} b d^{3} + 2 \, a b^{3} d + {\left (a^{2} b^{2} d^{3} - 6 \, b^{4} d\right )} x\right )} \cos \left (d x + c\right ) - {\left (3 \, b^{4} d^{2} x^{2} - 2 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2} - 6 \, b^{4}\right )} \sin \left (d x + c\right )}{b^{5} d^{4}} \]
-(a^4*d^4*cos_integral((b*d*x + a*d)/b)*sin(-(b*c - a*d)/b) - a^4*d^4*cos( -(b*c - a*d)/b)*sin_integral((b*d*x + a*d)/b) + (b^4*d^3*x^3 - a*b^3*d^3*x ^2 - a^3*b*d^3 + 2*a*b^3*d + (a^2*b^2*d^3 - 6*b^4*d)*x)*cos(d*x + c) - (3* b^4*d^2*x^2 - 2*a*b^3*d^2*x + a^2*b^2*d^2 - 6*b^4)*sin(d*x + c))/(b^5*d^4)
\[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\int \frac {x^{4} \sin {\left (c + d x \right )}}{a + b x}\, dx \]
\[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\int { \frac {x^{4} \sin \left (d x + c\right )}{b x + a} \,d x } \]
-1/2*(((6*a*b^2*(I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_integral _e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + 6*a*b^2*(I*exp_integral_e(2, (I*b* d*x + I*a*d)/b) - I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2 - (a ^3*(-I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(2, -(I*b* d*x + I*a*d)/b))*cos(c)^2 + a^3*(-I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d^2 - 4*(a^2*b*(ex p_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d )/b))*cos(c)^2 + a^2*b*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integ ral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*cos(-(b*c - a*d)/b) - (6*a*b^ 2*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + 6*a*b^2*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + ex p_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2 + (a^3*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*(exp_integral_e(2, (I*b*d*x + I*a*d)/b) + exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d^2 - 4*(a^2*b*(-I*exp_integral_e(2, (I*b*d*x + I* a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^2*b*(-I* exp_integral_e(2, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(2, -(I*b*d*x + I *a*d)/b))*sin(c)^2)*d)*sin(-(b*c - a*d)/b))*cos(d*x + c)^2 + ((6*a*b^2*(I* exp_integral_e(2, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(2, -(I*b*d*x + I *a*d)/b))*cos(c)^2 + 6*a*b^2*(I*exp_integral_e(2, (I*b*d*x + I*a*d)/b) ...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.36 (sec) , antiderivative size = 3337, normalized size of antiderivative = 15.31 \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\text {Too large to display} \]
1/2*(2*b^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b^3*d^3*x^2*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^4 *d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c )^2*tan(1/2*a*d/b)^2 - a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1 /2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^4*d^4*sin_integral(( b*d*x + a*d)/b)*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b ^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 2*a^4*d^4*real_part(cos_i ntegral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^4*d^4*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*ta n(1/2*c)^2*tan(1/2*a*d/b) + 2*b^4*d^3*x^3*tan(1/2*d*x + 1/2*c)^2*tan(1/2*a *d/b)^2 - 2*a^4*d^4*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2 *c)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^4*d^4*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^4*d^3*x ^3*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*b^2*d^3*x*tan(1/2*d*x + 1/2*c)^2* tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b^3*d^3*x^2*tan(1/2*d*x + 1/2*c)^2*tan (1/2*c)^2 - a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2 *c)^2*tan(1/2*c)^2 + a^4*d^4*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2 *d*x + 1/2*c)^2*tan(1/2*c)^2 - 2*a^4*d^4*sin_integral((b*d*x + a*d)/b)*tan (1/2*d*x + 1/2*c)^2*tan(1/2*c)^2 + 4*a^4*d^4*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x + 1/2*c)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^4*d^4*im...
Timed out. \[ \int \frac {x^4 \sin (c+d x)}{a+b x} \, dx=\int \frac {x^4\,\sin \left (c+d\,x\right )}{a+b\,x} \,d x \]