Integrand size = 14, antiderivative size = 127 \[ \int \frac {\cos (a+b x)}{(c+d x)^4} \, dx=-\frac {\cos (a+b x)}{3 d (c+d x)^3}+\frac {b^2 \cos (a+b x)}{6 d^3 (c+d x)}+\frac {b^3 \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{6 d^4}+\frac {b \sin (a+b x)}{6 d^2 (c+d x)^2}+\frac {b^3 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{6 d^4} \]
-1/3*cos(b*x+a)/d/(d*x+c)^3+1/6*b^2*cos(b*x+a)/d^3/(d*x+c)+1/6*b^3*cos(a-b *c/d)*Si(b*c/d+b*x)/d^4+1/6*b^3*Ci(b*c/d+b*x)*sin(a-b*c/d)/d^4+1/6*b*sin(b *x+a)/d^2/(d*x+c)^2
Time = 0.61 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.13 \[ \int \frac {\cos (a+b x)}{(c+d x)^4} \, dx=\frac {d \cos (b x) \left (\left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a)+b d (c+d x) \sin (a)\right )+d \left (b d (c+d x) \cos (a)-\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a)\right ) \sin (b x)+b^3 (c+d x)^3 \left (\operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )+\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )\right )}{6 d^4 (c+d x)^3} \]
(d*Cos[b*x]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] + b*d*(c + d*x)*Sin[a]) + d *(b*d*(c + d*x)*Cos[a] - (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x] + b^3 *(c + d*x)^3*(CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + Cos[a - (b*c)/d] *SinIntegral[b*(c/d + x)]))/(6*d^4*(c + d*x)^3)
Time = 0.73 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 3778, 25, 3042, 3778, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783}
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.
| \(\displaystyle \int \frac {\cos (a+b x)}{(c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{(c+d x)^4}dx\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {b \int -\frac {\sin (a+b x)}{(c+d x)^3}dx}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \int \frac {\sin (a+b x)}{(c+d x)^3}dx}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {\sin (a+b x)}{(c+d x)^3}dx}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {b \left (\frac {b \int \frac {\cos (a+b x)}{(c+d x)^2}dx}{2 d}-\frac {\sin (a+b x)}{2 d (c+d x)^2}\right )}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (\frac {b \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{(c+d x)^2}dx}{2 d}-\frac {\sin (a+b x)}{2 d (c+d x)^2}\right )}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle -\frac {b \left (\frac {b \left (\frac {b \int -\frac {\sin (a+b x)}{c+d x}dx}{d}-\frac {\cos (a+b x)}{d (c+d x)}\right )}{2 d}-\frac {\sin (a+b x)}{2 d (c+d x)^2}\right )}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \left (\frac {b \left (-\frac {b \int \frac {\sin (a+b x)}{c+d x}dx}{d}-\frac {\cos (a+b x)}{d (c+d x)}\right )}{2 d}-\frac {\sin (a+b x)}{2 d (c+d x)^2}\right )}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (\frac {b \left (-\frac {b \int \frac {\sin (a+b x)}{c+d x}dx}{d}-\frac {\cos (a+b x)}{d (c+d x)}\right )}{2 d}-\frac {\sin (a+b x)}{2 d (c+d x)^2}\right )}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle -\frac {b \left (\frac {b \left (-\frac {b \left (\sin \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x}dx+\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x}dx\right )}{d}-\frac {\cos (a+b x)}{d (c+d x)}\right )}{2 d}-\frac {\sin (a+b x)}{2 d (c+d x)^2}\right )}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (\frac {b \left (-\frac {b \left (\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{c+d x}dx+\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x}dx\right )}{d}-\frac {\cos (a+b x)}{d (c+d x)}\right )}{2 d}-\frac {\sin (a+b x)}{2 d (c+d x)^2}\right )}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle -\frac {b \left (\frac {b \left (-\frac {b \left (\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{c+d x}dx+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d}-\frac {\cos (a+b x)}{d (c+d x)}\right )}{2 d}-\frac {\sin (a+b x)}{2 d (c+d x)^2}\right )}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle -\frac {b \left (\frac {b \left (-\frac {b \left (\frac {\sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d}-\frac {\cos (a+b x)}{d (c+d x)}\right )}{2 d}-\frac {\sin (a+b x)}{2 d (c+d x)^2}\right )}{3 d}-\frac {\cos (a+b x)}{3 d (c+d x)^3}\) |
-1/3*Cos[a + b*x]/(d*(c + d*x)^3) - (b*(-1/2*Sin[a + b*x]/(d*(c + d*x)^2) + (b*(-(Cos[a + b*x]/(d*(c + d*x))) - (b*((CosIntegral[(b*c)/d + b*x]*Sin[ a - (b*c)/d])/d + (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/d))/d))/(2 *d)))/(3*d)
3.1.8.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Time = 1.11 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.45
| method | result | size |
| derivativedivides | \(b^{3} \left (-\frac {\cos \left (b x +a \right )}{3 \left (-a d +b c +d \left (b x +a \right )\right )^{3} d}-\frac {-\frac {\sin \left (b x +a \right )}{2 \left (-a d +b c +d \left (b x +a \right )\right )^{2} d}+\frac {-\frac {\cos \left (b x +a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}}{d}}{2 d}}{3 d}\right )\) | \(184\) |
| default | \(b^{3} \left (-\frac {\cos \left (b x +a \right )}{3 \left (-a d +b c +d \left (b x +a \right )\right )^{3} d}-\frac {-\frac {\sin \left (b x +a \right )}{2 \left (-a d +b c +d \left (b x +a \right )\right )^{2} d}+\frac {-\frac {\cos \left (b x +a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}}{d}}{2 d}}{3 d}\right )\) | \(184\) |
| risch | \(-\frac {i b^{3} {\mathrm e}^{-\frac {i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (i x b +i a -\frac {i \left (a d -b c \right )}{d}\right )}{12 d^{4}}+\frac {i b^{3} {\mathrm e}^{\frac {i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (-i x b -i a -\frac {-i a d +i b c}{d}\right )}{12 d^{4}}-\frac {\left (-2 b^{5} d^{5} x^{5}-10 b^{5} c \,d^{4} x^{4}-20 b^{5} c^{2} d^{3} x^{3}-20 b^{5} c^{3} d^{2} x^{2}-10 b^{5} c^{4} d x +4 b^{3} d^{5} x^{3}-2 b^{5} c^{5}+12 b^{3} c \,d^{4} x^{2}+12 b^{3} c^{2} d^{3} x +4 b^{3} c^{3} d^{2}\right ) \cos \left (b x +a \right )}{12 d^{3} \left (d x +c \right )^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {i \left (2 i b^{4} d^{5} x^{4}+8 i b^{4} c \,d^{4} x^{3}+12 i b^{4} c^{2} d^{3} x^{2}+8 i b^{4} c^{3} d^{2} x +2 i b^{4} c^{4} d \right ) \sin \left (b x +a \right )}{12 d^{3} \left (d x +c \right )^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}\) | \(405\) |
b^3*(-1/3*cos(b*x+a)/(-a*d+b*c+d*(b*x+a))^3/d-1/3*(-1/2*sin(b*x+a)/(-a*d+b *c+d*(b*x+a))^2/d+1/2*(-cos(b*x+a)/(-a*d+b*c+d*(b*x+a))/d-(-Si(-b*x-a-(-a* d+b*c)/d)*cos((-a*d+b*c)/d)/d-Ci(b*x+a+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d)/ d)/d)/d)
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (117) = 234\).
Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.85 \[ \int \frac {\cos (a+b x)}{(c+d x)^4} \, dx=\frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right ) + {\left (b d^{3} x + b c d^{2}\right )} \sin \left (b x + a\right )}{6 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]
1/6*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos_integra l((b*d*x + b*c)/d)*sin(-(b*c - a*d)/d) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos(-(b*c - a*d)/d)*sin_integral((b*d*x + b*c)/d) + (b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d - 2*d^3)*cos(b*x + a) + (b*d^3 *x + b*c*d^2)*sin(b*x + a))/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4 )
\[ \int \frac {\cos (a+b x)}{(c+d x)^4} \, dx=\int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \]
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.96 \[ \int \frac {\cos (a+b x)}{(c+d x)^4} \, dx=-\frac {b^{4} {\left (E_{4}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{4}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b^{4} {\left (-i \, E_{4}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + i \, E_{4}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + {\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left (b x + a\right )}\right )} b} \]
-1/2*(b^4*(exp_integral_e(4, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_inte gral_e(4, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*cos(-(b*c - a*d)/d) + b^4*( -I*exp_integral_e(4, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + I*exp_integral_e (4, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d))/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)) *b)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.56 (sec) , antiderivative size = 8378, normalized size of antiderivative = 65.97 \[ \int \frac {\cos (a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \]
1/12*(b^3*d^3*x^3*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan( 1/2*a)^2*tan(1/2*b*c/d)^2 - b^3*d^3*x^3*imag_part(cos_integral(-b*x - b*c/ d))*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 2*b^3*d^3*x^3*sin_integ ral((b*d*x + b*c)/d)*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 2*b^3* d^3*x^3*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2*t an(1/2*b*c/d) + 2*b^3*d^3*x^3*real_part(cos_integral(-b*x - b*c/d))*tan(1/ 2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d) - 2*b^3*d^3*x^3*real_part(cos_integra l(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d)^2 - 2*b^3*d^3*x^3 *real_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b *c/d)^2 + 3*b^3*c*d^2*x^2*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x )^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 - 3*b^3*c*d^2*x^2*imag_part(cos_integral (-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + 6*b^3*c*d^2 *x^2*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c /d)^2 - b^3*d^3*x^3*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*ta n(1/2*a)^2 + b^3*d^3*x^3*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x )^2*tan(1/2*a)^2 - 2*b^3*d^3*x^3*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*x )^2*tan(1/2*a)^2 + 4*b^3*d^3*x^3*imag_part(cos_integral(b*x + b*c/d))*tan( 1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d) - 4*b^3*d^3*x^3*imag_part(cos_integra l(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d) + 8*b^3*d^3*x^3* sin_integral((b*d*x + b*c)/d)*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d) ...
Timed out. \[ \int \frac {\cos (a+b x)}{(c+d x)^4} \, dx=\int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^4} \,d x \]