Integrand size = 15, antiderivative size = 69 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=-\frac {\cos (c+d x)}{b d}-\frac {a \operatorname {CosIntegral}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^2}-\frac {a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^2} \] Output:
-cos(d*x+c)/b/d+a*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^2-a*cos(-c+a*d/b)*Si(a*d/b +d*x)/b^2
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=-\frac {b \cos (c+d x)+a d \operatorname {CosIntegral}\left (d \left (\frac {a}{b}+x\right )\right ) \sin \left (c-\frac {a d}{b}\right )+a d \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^2 d} \] Input:
Integrate[(x*Sin[c + d*x])/(a + b*x),x]
Output:
-((b*Cos[c + d*x] + a*d*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + a*d*Co s[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/(b^2*d))
Time = 0.37 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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.
\(\displaystyle \int \frac {x \sin (c+d x)}{a+b x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sin (c+d x)}{b}-\frac {a \sin (c+d x)}{b (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \sin \left (c-\frac {a d}{b}\right ) \operatorname {CosIntegral}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {a \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^2}-\frac {\cos (c+d x)}{b d}\) |
Input:
Int[(x*Sin[c + d*x])/(a + b*x),x]
Output:
-(Cos[c + d*x]/(b*d)) - (a*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/b^ 2 - (a*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^2
Result contains complex when optimal does not.
Time = 0.89 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.17
method | result | size |
risch | \(-\frac {\cos \left (d x +c \right )}{b d}+\frac {i a \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right )}{2 b^{2}}-\frac {i a \cos \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right )}{2 b^{2}}-\frac {a \sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (\frac {i \left (b x +a \right ) d}{b}\right )}{2 b^{2}}-\frac {a \sin \left (\frac {a d -b c}{b}\right ) \operatorname {expIntegral}_{1}\left (-\frac {i \left (b x +a \right ) d}{b}\right )}{2 b^{2}}\) | \(150\) |
derivativedivides | \(\frac {-\frac {\left (a d -b c \right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{b}-\frac {d \cos \left (d x +c \right )}{b}-d c \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{d^{2}}\) | \(180\) |
default | \(\frac {-\frac {\left (a d -b c \right ) d \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{b}-\frac {d \cos \left (d x +c \right )}{b}-d c \left (\frac {\operatorname {Si}\left (d x +c +\frac {a d -b c}{b}\right ) \cos \left (\frac {a d -b c}{b}\right )}{b}-\frac {\operatorname {Ci}\left (d x +c +\frac {a d -b c}{b}\right ) \sin \left (\frac {a d -b c}{b}\right )}{b}\right )}{d^{2}}\) | \(180\) |
Input:
int(x*sin(d*x+c)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
-cos(d*x+c)/b/d+1/2*I*a/b^2*cos((a*d-b*c)/b)*Ei(1,I*(b*x+a)*d/b)-1/2*I*a/b ^2*cos((a*d-b*c)/b)*Ei(1,-I*(b*x+a)*d/b)-1/2*a/b^2*sin((a*d-b*c)/b)*Ei(1,I *(b*x+a)*d/b)-1/2*a/b^2*sin((a*d-b*c)/b)*Ei(1,-I*(b*x+a)*d/b)
Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=\frac {a d \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) \sin \left (-\frac {b c - a d}{b}\right ) - a d \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) - b \cos \left (d x + c\right )}{b^{2} d} \] Input:
integrate(x*sin(d*x+c)/(b*x+a),x, algorithm="fricas")
Output:
(a*d*cos_integral((b*d*x + a*d)/b)*sin(-(b*c - a*d)/b) - a*d*cos(-(b*c - a *d)/b)*sin_integral((b*d*x + a*d)/b) - b*cos(d*x + c))/(b^2*d)
\[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=\int \frac {x \sin {\left (c + d x \right )}}{a + b x}\, dx \] Input:
integrate(x*sin(d*x+c)/(b*x+a),x)
Output:
Integral(x*sin(c + d*x)/(a + b*x), x)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 776, normalized size of antiderivative = 11.25 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx =\text {Too large to display} \] Input:
integrate(x*sin(d*x+c)/(b*x+a),x, algorithm="maxima")
Output:
-1/2*((d*(-I*exp_integral_e(1, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + I*exp_ integral_e(1, -(I*(d*x + c)*b - I*b*c + I*a*d)/b))*cos(-(b*c - a*d)/b) + d *(exp_integral_e(1, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp_integral_e(1, -(I*(d*x + c)*b - I*b*c + I*a*d)/b))*sin(-(b*c - a*d)/b))*c/b + ((d*x + c )*b*d*cos(d*x + c)^3 + (d*x + c)*b*d*cos(d*x + c) - ((b*c*d*(exp_integral_ e(2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp_integral_e(2, -(I*(d*x + c)* b - I*b*c + I*a*d)/b)) - a*d^2*(exp_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp_integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)))*cos(- (b*c - a*d)/b) - (a*d^2*(I*exp_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a* d)/b) - I*exp_integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)) + b*c*d*( -I*exp_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + I*exp_integral_e (2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)))*sin(-(b*c - a*d)/b))*cos(d*x + c )^2 + ((d*x + c)*b*d*cos(d*x + c) - (b*c*d*(exp_integral_e(2, (I*(d*x + c) *b - I*b*c + I*a*d)/b) + exp_integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d )/b)) - a*d^2*(exp_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp_ integral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)))*cos(-(b*c - a*d)/b) + (a*d^2*(I*exp_integral_e(2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) - I*exp_int egral_e(2, -(I*(d*x + c)*b - I*b*c + I*a*d)/b)) + b*c*d*(-I*exp_integral_e (2, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + I*exp_integral_e(2, -(I*(d*x + c) *b - I*b*c + I*a*d)/b)))*sin(-(b*c - a*d)/b))*sin(d*x + c)^2)/(((d*x + ...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 1647, normalized size of antiderivative = 23.87 \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=\text {Too large to display} \] Input:
integrate(x*sin(d*x+c)/(b*x+a),x, algorithm="giac")
Output:
-1/2*(a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 *tan(1/2*a*d/b)^2 - a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x) ^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*d*sin_integral((b*d*x + a*d)/b)*tan (1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*d*real_part(cos_integral(d *x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a*d*real_part( cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2 *a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/ 2*a*d/b)^2 - 2*a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*ta n(1/2*c)*tan(1/2*a*d/b)^2 - a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1 /2*d*x)^2*tan(1/2*c)^2 + a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2 *d*x)^2*tan(1/2*c)^2 - 2*a*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2* tan(1/2*c)^2 + 4*a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*t an(1/2*c)*tan(1/2*a*d/b) - 4*a*d*imag_part(cos_integral(-d*x - a*d/b))*tan (1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 8*a*d*sin_integral((b*d*x + a*d)/b )*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - a*d*imag_part(cos_integral(d* x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a*d*imag_part(cos_integral(- d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*d*sin_integral((b*d*x + ...
Timed out. \[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=\int \frac {x\,\sin \left (c+d\,x\right )}{a+b\,x} \,d x \] Input:
int((x*sin(c + d*x))/(a + b*x),x)
Output:
int((x*sin(c + d*x))/(a + b*x), x)
\[ \int \frac {x \sin (c+d x)}{a+b x} \, dx=\frac {-\cos \left (d x +c \right )-\left (\int \frac {\sin \left (d x +c \right )}{b x +a}d x \right ) a d}{b d} \] Input:
int(x*sin(d*x+c)/(b*x+a),x)
Output:
( - (cos(c + d*x) + int(sin(c + d*x)/(a + b*x),x)*a*d))/(b*d)