Integrand size = 19, antiderivative size = 22 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {1}{7 b d (a+b \sin (c+d x))^7} \] Output:
-1/7/b/d/(a+b*sin(d*x+c))^7
Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {1}{7 b d (a+b \sin (c+d x))^7} \] Input:
Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x])^8,x]
Output:
-1/7*1/(b*d*(a + b*Sin[c + d*x])^7)
Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3042, 3147, 17}
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.
\(\displaystyle \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8}dx\) |
\(\Big \downarrow \) 3147 |
\(\displaystyle \frac {\int \frac {1}{(a+b \sin (c+d x))^8}d(b \sin (c+d x))}{b d}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -\frac {1}{7 b d (a+b \sin (c+d x))^7}\) |
Input:
Int[Cos[c + d*x]/(a + b*Sin[c + d*x])^8,x]
Output:
-1/7*1/(b*d*(a + b*Sin[c + d*x])^7)
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 4.66 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(-\frac {1}{7 b d \left (a +b \sin \left (d x +c \right )\right )^{7}}\) | \(21\) |
default | \(-\frac {1}{7 b d \left (a +b \sin \left (d x +c \right )\right )^{7}}\) | \(21\) |
risch | \(\frac {128 i {\mathrm e}^{7 i \left (d x +c \right )}}{7 \left ({\mathrm e}^{2 i \left (d x +c \right )} b -b +2 i {\mathrm e}^{i \left (d x +c \right )} a \right )^{7} d b}\) | \(49\) |
parallelrisch | \(\frac {\left (-672 a^{5} b -1120 a^{3} b^{3}-210 a \,b^{5}\right ) \cos \left (2 d x +2 c \right )+\left (-560 a^{4} b^{2}-420 a^{2} b^{4}-21 b^{6}\right ) \sin \left (3 d x +3 c \right )+\left (280 a^{3} b^{3}+84 a \,b^{5}\right ) \cos \left (4 d x +4 c \right )+\left (84 a^{2} b^{4}+7 b^{6}\right ) \sin \left (5 d x +5 c \right )-14 a \,b^{5} \cos \left (6 d x +6 c \right )-b^{6} \sin \left (7 d x +7 c \right )+\left (448 a^{6}+1680 a^{4} b^{2}+840 a^{2} b^{4}+35 b^{6}\right ) \sin \left (d x +c \right )+672 a^{5} b +840 a^{3} b^{3}+140 a \,b^{5}}{448 a^{7} d \left (a +b \sin \left (d x +c \right )\right )^{7}}\) | \(217\) |
norman | \(\frac {\frac {2 \left (245 a^{6}+1400 a^{4} b^{2}+1008 a^{2} b^{4}+64 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7 a^{7} d}+\frac {2 \left (245 a^{6}+1400 a^{4} b^{2}+1008 a^{2} b^{4}+64 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{7 a^{7} d}+\frac {4 \left (45 b \,a^{4}+80 b^{3} a^{2}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{a^{6} d}+\frac {4 \left (45 b \,a^{4}+80 b^{3} a^{2}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{a^{6} d}+\frac {2 \left (120 b \,a^{4}+240 b^{3} a^{2}+64 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{a^{6} d}+\frac {2 \left (21 a^{4}+100 b^{2} a^{2}+48 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{a^{5} d}+\frac {2 \left (21 a^{4}+100 b^{2} a^{2}+48 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{a^{5} d}+\frac {4 \left (18 a^{2} b +20 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{a^{4} d}+\frac {4 \left (18 a^{2} b +20 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{a^{4} d}+\frac {2 \left (7 a^{2}+20 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a^{3} d}+\frac {2 \left (7 a^{2}+20 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{a^{3} d}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{d a}+\frac {12 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2} d}+\frac {12 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{a^{2} d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{7}}\) | \(530\) |
Input:
int(cos(d*x+c)/(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
-1/7/b/d/(a+b*sin(d*x+c))^7
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 218, normalized size of antiderivative = 9.91 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {1}{7 \, {\left (7 \, a b^{7} d \cos \left (d x + c\right )^{6} - 7 \, {\left (5 \, a^{3} b^{5} + 3 \, a b^{7}\right )} d \cos \left (d x + c\right )^{4} + 7 \, {\left (3 \, a^{5} b^{3} + 10 \, a^{3} b^{5} + 3 \, a b^{7}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{7} b + 21 \, a^{5} b^{3} + 35 \, a^{3} b^{5} + 7 \, a b^{7}\right )} d + {\left (b^{8} d \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{4} + 42 \, a^{2} b^{6} + 3 \, b^{8}\right )} d \cos \left (d x + c\right )^{2} - {\left (7 \, a^{6} b^{2} + 35 \, a^{4} b^{4} + 21 \, a^{2} b^{6} + b^{8}\right )} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cos(d*x+c)/(a+b*sin(d*x+c))^8,x, algorithm="fricas")
Output:
1/7/(7*a*b^7*d*cos(d*x + c)^6 - 7*(5*a^3*b^5 + 3*a*b^7)*d*cos(d*x + c)^4 + 7*(3*a^5*b^3 + 10*a^3*b^5 + 3*a*b^7)*d*cos(d*x + c)^2 - (a^7*b + 21*a^5*b ^3 + 35*a^3*b^5 + 7*a*b^7)*d + (b^8*d*cos(d*x + c)^6 - 3*(7*a^2*b^6 + b^8) *d*cos(d*x + c)^4 + (35*a^4*b^4 + 42*a^2*b^6 + 3*b^8)*d*cos(d*x + c)^2 - ( 7*a^6*b^2 + 35*a^4*b^4 + 21*a^2*b^6 + b^8)*d)*sin(d*x + c))
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (19) = 38\).
Time = 9.22 (sec) , antiderivative size = 167, normalized size of antiderivative = 7.59 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\begin {cases} \frac {x \cos {\left (c \right )}}{a^{8}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin {\left (c + d x \right )}}{a^{8} d} & \text {for}\: b = 0 \\\frac {x \cos {\left (c \right )}}{\left (a + b \sin {\left (c \right )}\right )^{8}} & \text {for}\: d = 0 \\- \frac {1}{7 a^{7} b d + 49 a^{6} b^{2} d \sin {\left (c + d x \right )} + 147 a^{5} b^{3} d \sin ^{2}{\left (c + d x \right )} + 245 a^{4} b^{4} d \sin ^{3}{\left (c + d x \right )} + 245 a^{3} b^{5} d \sin ^{4}{\left (c + d x \right )} + 147 a^{2} b^{6} d \sin ^{5}{\left (c + d x \right )} + 49 a b^{7} d \sin ^{6}{\left (c + d x \right )} + 7 b^{8} d \sin ^{7}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)/(a+b*sin(d*x+c))**8,x)
Output:
Piecewise((x*cos(c)/a**8, Eq(b, 0) & Eq(d, 0)), (sin(c + d*x)/(a**8*d), Eq (b, 0)), (x*cos(c)/(a + b*sin(c))**8, Eq(d, 0)), (-1/(7*a**7*b*d + 49*a**6 *b**2*d*sin(c + d*x) + 147*a**5*b**3*d*sin(c + d*x)**2 + 245*a**4*b**4*d*s in(c + d*x)**3 + 245*a**3*b**5*d*sin(c + d*x)**4 + 147*a**2*b**6*d*sin(c + d*x)**5 + 49*a*b**7*d*sin(c + d*x)**6 + 7*b**8*d*sin(c + d*x)**7), True))
Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {1}{7 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b d} \] Input:
integrate(cos(d*x+c)/(a+b*sin(d*x+c))^8,x, algorithm="maxima")
Output:
-1/7/((b*sin(d*x + c) + a)^7*b*d)
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {1}{7 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b d} \] Input:
integrate(cos(d*x+c)/(a+b*sin(d*x+c))^8,x, algorithm="giac")
Output:
-1/7/((b*sin(d*x + c) + a)^7*b*d)
Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 5.41 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {1}{d\,\left (7\,a^7\,b+49\,a^6\,b^2\,\sin \left (c+d\,x\right )+147\,a^5\,b^3\,{\sin \left (c+d\,x\right )}^2+245\,a^4\,b^4\,{\sin \left (c+d\,x\right )}^3+245\,a^3\,b^5\,{\sin \left (c+d\,x\right )}^4+147\,a^2\,b^6\,{\sin \left (c+d\,x\right )}^5+49\,a\,b^7\,{\sin \left (c+d\,x\right )}^6+7\,b^8\,{\sin \left (c+d\,x\right )}^7\right )} \] Input:
int(cos(c + d*x)/(a + b*sin(c + d*x))^8,x)
Output:
-1/(d*(7*a^7*b + 7*b^8*sin(c + d*x)^7 + 49*a^6*b^2*sin(c + d*x) + 49*a*b^7 *sin(c + d*x)^6 + 147*a^5*b^3*sin(c + d*x)^2 + 245*a^4*b^4*sin(c + d*x)^3 + 245*a^3*b^5*sin(c + d*x)^4 + 147*a^2*b^6*sin(c + d*x)^5))
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 5.27 \[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {1}{7 b d \left (\sin \left (d x +c \right )^{7} b^{7}+7 \sin \left (d x +c \right )^{6} a \,b^{6}+21 \sin \left (d x +c \right )^{5} a^{2} b^{5}+35 \sin \left (d x +c \right )^{4} a^{3} b^{4}+35 \sin \left (d x +c \right )^{3} a^{4} b^{3}+21 \sin \left (d x +c \right )^{2} a^{5} b^{2}+7 \sin \left (d x +c \right ) a^{6} b +a^{7}\right )} \] Input:
int(cos(d*x+c)/(a+b*sin(d*x+c))^8,x)
Output:
( - 1)/(7*b*d*(sin(c + d*x)**7*b**7 + 7*sin(c + d*x)**6*a*b**6 + 21*sin(c + d*x)**5*a**2*b**5 + 35*sin(c + d*x)**4*a**3*b**4 + 35*sin(c + d*x)**3*a* *4*b**3 + 21*sin(c + d*x)**2*a**5*b**2 + 7*sin(c + d*x)*a**6*b + a**7))