Integrand size = 17, antiderivative size = 233 \[ \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx=\frac {2 \cos (c+d x)}{b^2 d^3}-\frac {3 a^2 \cos (c+d x)}{b^4 d}+\frac {2 a x \cos (c+d x)}{b^3 d}-\frac {x^2 \cos (c+d x)}{b^2 d}+\frac {a^4 d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^6}-\frac {4 a^3 \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 x \sin (c+d x)}{b^2 d^2}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {a^4 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^6} \]
a^4*d*Ci(a*d/b+d*x)*cos(-c+a*d/b)/b^6+2*cos(d*x+c)/b^2/d^3-3*a^2*cos(d*x+c )/b^4/d+2*a*x*cos(d*x+c)/b^3/d-x^2*cos(d*x+c)/b^2/d-4*a^3*cos(-c+a*d/b)*Si (a*d/b+d*x)/b^5+4*a^3*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^5+a^4*d*Si(a*d/b+d*x)* sin(-c+a*d/b)/b^6-2*a*sin(d*x+c)/b^3/d^2+2*x*sin(d*x+c)/b^2/d^2-a^4*sin(d* x+c)/b^5/(b*x+a)
Time = 0.67 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.76 \[ \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx=\frac {a^3 \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cos \left (c-\frac {a d}{b}\right )-4 b \sin \left (c-\frac {a d}{b}\right )\right )-\frac {b \left (b (a+b x) \left (3 a^2 d^2-2 a b d^2 x+b^2 \left (-2+d^2 x^2\right )\right ) \cos (c+d x)+d \left (2 a^2 b^2+a^4 d^2-2 b^4 x^2\right ) \sin (c+d x)\right )}{d^3 (a+b x)}-a^3 \left (4 b \cos \left (c-\frac {a d}{b}\right )+a d \sin \left (c-\frac {a d}{b}\right )\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^6} \]
(a^3*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 4*b*Sin[c - (a*d)/b] ) - (b*(b*(a + b*x)*(3*a^2*d^2 - 2*a*b*d^2*x + b^2*(-2 + d^2*x^2))*Cos[c + d*x] + d*(2*a^2*b^2 + a^4*d^2 - 2*b^4*x^2)*Sin[c + d*x]))/(d^3*(a + b*x)) - a^3*(4*b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^6
Time = 0.73 (sec) , antiderivative size = 233, 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.
| \(\displaystyle \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {a^4 \sin (c+d x)}{b^4 (a+b x)^2}-\frac {4 a^3 \sin (c+d x)}{b^4 (a+b x)}+\frac {3 a^2 \sin (c+d x)}{b^4}-\frac {2 a x \sin (c+d x)}{b^3}+\frac {x^2 \sin (c+d x)}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^4 d \cos \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^6}-\frac {a^4 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^6}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {3 a^2 \cos (c+d x)}{b^4 d}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 a x \cos (c+d x)}{b^3 d}+\frac {2 \cos (c+d x)}{b^2 d^3}+\frac {2 x \sin (c+d x)}{b^2 d^2}-\frac {x^2 \cos (c+d x)}{b^2 d}\) |
(2*Cos[c + d*x])/(b^2*d^3) - (3*a^2*Cos[c + d*x])/(b^4*d) + (2*a*x*Cos[c + d*x])/(b^3*d) - (x^2*Cos[c + d*x])/(b^2*d) + (a^4*d*Cos[c - (a*d)/b]*CosI ntegral[(a*d)/b + d*x])/b^6 - (4*a^3*CosIntegral[(a*d)/b + d*x]*Sin[c - (a *d)/b])/b^5 - (2*a*Sin[c + d*x])/(b^3*d^2) + (2*x*Sin[c + d*x])/(b^2*d^2) - (a^4*Sin[c + d*x])/(b^5*(a + b*x)) - (4*a^3*Cos[c - (a*d)/b]*SinIntegral [(a*d)/b + d*x])/b^5 - (a^4*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x]) /b^6
3.1.26.3.1 Defintions of rubi rules used
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 734, normalized size of antiderivative = 3.15
| method | result | size |
| risch | \(-\frac {i \left (-2 i b^{6} d^{5} x^{5}+4 i a \,b^{5} d^{5} x^{4}-6 i b^{6} c \,d^{4} x^{4}-8 i a^{3} b^{3} d^{5} x^{2}+10 i a^{4} b^{2} d^{5} x -24 i a^{3} b^{3} c \,d^{4} x +4 i b^{6} d^{3} x^{3}+12 i a^{5} b \,d^{5}-18 i a^{4} b^{2} c \,d^{4}+12 i b^{6} c \,d^{2} x^{2}-12 i a^{2} b^{4} d^{3} x +24 i a \,b^{5} c \,d^{2} x -8 i a^{3} b^{3} d^{3}+12 i a^{2} b^{4} c \,d^{2}\right ) \cos \left (d x +c \right )}{2 b^{5} d^{4} \left (b x +a \right ) \left (-d x b -d a \right ) \left (-d x b +2 d a -3 c b \right )}+\frac {\left (-2 a^{4} b^{2} d^{6} x^{2}+4 b^{6} d^{4} x^{4}+2 a^{5} b \,d^{6} x -6 a^{4} b^{2} c \,d^{5} x -4 a \,b^{5} d^{4} x^{3}+12 b^{6} c \,d^{3} x^{3}+4 a^{6} d^{6}-6 a^{5} b c \,d^{5}-12 a^{2} b^{4} d^{4} x^{2}+12 a \,b^{5} c \,d^{3} x^{2}+4 a^{3} b^{3} d^{4} x -12 a^{2} b^{4} c \,d^{3} x +8 a^{4} b^{2} d^{4}-12 a^{3} b^{3} c \,d^{3}\right ) \sin \left (d x +c \right )}{2 b^{5} d^{4} \left (b x +a \right ) \left (-d x b -d a \right ) \left (-d x b +2 d a -3 c b \right )}-\frac {2 i \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{b^{5}}-\frac {d \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right ) a^{4}}{2 b^{6}}+\frac {2 i \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{b^{5}}-\frac {d \cos \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right ) a^{4}}{2 b^{6}}-\frac {2 \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{b^{5}}+\frac {i d \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (-\frac {i d \left (b x +a \right )}{b}\right ) a^{4}}{2 b^{6}}-\frac {2 \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{b^{5}}-\frac {i d \sin \left (\frac {d a -c b}{b}\right ) \operatorname {Ei}_{1}\left (\frac {i d \left (b x +a \right )}{b}\right ) a^{4}}{2 b^{6}}\) | \(734\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1216\) |
| default | \(\text {Expression too large to display}\) | \(1216\) |
-1/2*I/b^5/d^4*(-8*I*a^3*b^3*d^5*x^2+4*I*a*b^5*d^5*x^4-6*I*b^6*c*d^4*x^4+1 0*I*a^4*b^2*d^5*x-18*I*a^4*b^2*c*d^4+12*I*b^6*c*d^2*x^2+24*I*a*b^5*c*d^2*x -24*I*a^3*b^3*c*d^4*x+4*I*b^6*d^3*x^3+12*I*a^5*b*d^5-8*I*a^3*b^3*d^3-2*I*b ^6*d^5*x^5-12*I*a^2*b^4*d^3*x+12*I*a^2*b^4*c*d^2)/(b*x+a)/(-b*d*x-a*d)/(-b *d*x+2*a*d-3*b*c)*cos(d*x+c)+1/2/b^5/d^4*(-2*a^4*b^2*d^6*x^2+4*b^6*d^4*x^4 +2*a^5*b*d^6*x-6*a^4*b^2*c*d^5*x-4*a*b^5*d^4*x^3+12*b^6*c*d^3*x^3+4*a^6*d^ 6-6*a^5*b*c*d^5-12*a^2*b^4*d^4*x^2+12*a*b^5*c*d^3*x^2+4*a^3*b^3*d^4*x-12*a ^2*b^4*c*d^3*x+8*a^4*b^2*d^4-12*a^3*b^3*c*d^3)/(b*x+a)/(-b*d*x-a*d)/(-b*d* x+2*a*d-3*b*c)*sin(d*x+c)-2*I/b^5*cos((a*d-b*c)/b)*Ei(1,-I*d*(b*x+a)/b)*a^ 3-1/2*d/b^6*cos((a*d-b*c)/b)*Ei(1,-I*d*(b*x+a)/b)*a^4+2*I/b^5*cos((a*d-b*c )/b)*Ei(1,I*d*(b*x+a)/b)*a^3-1/2*d/b^6*cos((a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/b )*a^4-2/b^5*sin((a*d-b*c)/b)*Ei(1,-I*d*(b*x+a)/b)*a^3+1/2*I*d/b^6*sin((a*d -b*c)/b)*Ei(1,-I*d*(b*x+a)/b)*a^4-2/b^5*sin((a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/ b)*a^3-1/2*I*d/b^6*sin((a*d-b*c)/b)*Ei(1,I*d*(b*x+a)/b)*a^4
Time = 0.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.23 \[ \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx=-\frac {{\left (b^{5} d^{2} x^{3} - a b^{4} d^{2} x^{2} + 3 \, a^{3} b^{2} d^{2} - 2 \, a b^{4} + {\left (a^{2} b^{3} d^{2} - 2 \, b^{5}\right )} x\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) - 4 \, {\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + {\left (a^{4} b d^{3} - 2 \, b^{5} d x^{2} + 2 \, a^{2} b^{3} d\right )} \sin \left (d x + c\right ) - {\left (4 \, {\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{b^{7} d^{3} x + a b^{6} d^{3}} \]
-((b^5*d^2*x^3 - a*b^4*d^2*x^2 + 3*a^3*b^2*d^2 - 2*a*b^4 + (a^2*b^3*d^2 - 2*b^5)*x)*cos(d*x + c) - ((a^4*b*d^4*x + a^5*d^4)*cos_integral((b*d*x + a* d)/b) - 4*(a^3*b^2*d^3*x + a^4*b*d^3)*sin_integral((b*d*x + a*d)/b))*cos(- (b*c - a*d)/b) + (a^4*b*d^3 - 2*b^5*d*x^2 + 2*a^2*b^3*d)*sin(d*x + c) - (4 *(a^3*b^2*d^3*x + a^4*b*d^3)*cos_integral((b*d*x + a*d)/b) + (a^4*b*d^4*x + a^5*d^4)*sin_integral((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(b^7*d^3*x + a*b^6*d^3)
\[ \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^{4} \sin {\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \]
\[ \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx=\int { \frac {x^{4} \sin \left (d x + c\right )}{{\left (b x + a\right )}^{2}} \,d x } \]
1/2*(2*((2*a^2*b*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e( 3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + 2*a^2*b*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2 + (a^3*(-I* exp_integral_e(3, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(3, -(I*b*d*x + I *a*d)/b))*cos(c)^2 + a^3*(-I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) + I*ex p_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*cos(-(b*c - a*d)/b) + (2*a^2*b*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, - (I*b*d*x + I*a*d)/b))*cos(c)^2 + 2*a^2*b*(I*exp_integral_e(3, (I*b*d*x + I *a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2 + (a^3*(exp _integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d) /b))*cos(c)^2 + a^3*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral _e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*sin(-(b*c - a*d)/b))*cos(d*x + c )^2 + 2*((2*a^2*b*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e (3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + 2*a^2*b*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2 + (a^3*(-I *exp_integral_e(3, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*(-I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) + I*e xp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*cos(-(b*c - a*d)/b) + (2*a^2*b*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + 2*a^2*b*(I*exp_integral_e(3, (I*b*d*x...
Leaf count of result is larger than twice the leaf count of optimal. 1973 vs. \(2 (236) = 472\).
Time = 0.37 (sec) , antiderivative size = 1973, normalized size of antiderivative = 8.47 \[ \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx=\text {Too large to display} \]
((b*x + a)*a^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^4*cos(-(b*c - a*d)/b) *cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/ b) - a^4*b*c*d^4*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x + a ) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^5*d^5*cos(-(b*c - a*d)/b)*cos_i ntegral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + ( b*x + a)*a^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^4*sin(-(b*c - a*d)/b)*s in_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - a^4*b*c*d^4*sin(-(b*c - a*d)/b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^5*d^5*sin(-(b*c - a*d)/b)*sin_int egral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + 4*( b*x + a)*a^3*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c - a*d)/ b) - 4*a^3*b^2*c*d^3*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a ) + d) - b*c + a*d)/b)*sin(-(b*c - a*d)/b) + 4*a^4*b*d^4*cos_integral(((b* x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c - a*d )/b) - 4*(b*x + a)*a^3*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*cos(-(b*c - a*d)/b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b *c + a*d)/b) + 4*a^3*b^2*c*d^3*cos(-(b*c - a*d)/b)*sin_integral(((b*x + a) *(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - 4*a^4*b*d^4*cos(-(b *c - a*d)/b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d...
Timed out. \[ \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx=\int \frac {x^4\,\sin \left (c+d\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]