Integrand size = 17, antiderivative size = 46 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {\cos ^8(a+b x)}{8 b}+\frac {\cos ^{10}(a+b x)}{5 b}-\frac {\cos ^{12}(a+b x)}{12 b} \] Output:
-1/8*cos(b*x+a)^8/b+1/5*cos(b*x+a)^10/b-1/12*cos(b*x+a)^12/b
Time = 0.40 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.48 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {600 \cos (2 (a+b x))+75 \cos (4 (a+b x))-100 \cos (6 (a+b x))-30 \cos (8 (a+b x))+12 \cos (10 (a+b x))+5 \cos (12 (a+b x))}{122880 b} \] Input:
Integrate[Cos[a + b*x]^7*Sin[a + b*x]^5,x]
Output:
-1/122880*(600*Cos[2*(a + b*x)] + 75*Cos[4*(a + b*x)] - 100*Cos[6*(a + b*x )] - 30*Cos[8*(a + b*x)] + 12*Cos[10*(a + b*x)] + 5*Cos[12*(a + b*x)])/b
Time = 0.38 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3042, 3045, 243, 49, 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.
\(\displaystyle \int \sin ^5(a+b x) \cos ^7(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^5 \cos (a+b x)^7dx\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle -\frac {\int \cos ^7(a+b x) \left (1-\cos ^2(a+b x)\right )^2d\cos (a+b x)}{b}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {\int \cos ^6(a+b x) \left (1-\cos ^2(a+b x)\right )^2d\cos ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {\int \left (\cos ^{10}(a+b x)-2 \cos ^8(a+b x)+\cos ^6(a+b x)\right )d\cos ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {1}{6} \cos ^{12}(a+b x)-\frac {2}{5} \cos ^{10}(a+b x)+\frac {1}{4} \cos ^8(a+b x)}{2 b}\) |
Input:
Int[Cos[a + b*x]^7*Sin[a + b*x]^5,x]
Output:
-1/2*(Cos[a + b*x]^8/4 - (2*Cos[a + b*x]^10)/5 + Cos[a + b*x]^12/6)/b
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Time = 43.96 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(-\frac {\frac {\sin \left (b x +a \right )^{12}}{12}-\frac {3 \sin \left (b x +a \right )^{10}}{10}+\frac {3 \sin \left (b x +a \right )^{8}}{8}-\frac {\sin \left (b x +a \right )^{6}}{6}}{b}\) | \(47\) |
default | \(-\frac {\frac {\sin \left (b x +a \right )^{12}}{12}-\frac {3 \sin \left (b x +a \right )^{10}}{10}+\frac {3 \sin \left (b x +a \right )^{8}}{8}-\frac {\sin \left (b x +a \right )^{6}}{6}}{b}\) | \(47\) |
parallelrisch | \(\frac {-600 \cos \left (2 b x +2 a \right )+100 \cos \left (6 b x +6 a \right )+30 \cos \left (8 b x +8 a \right )-75 \cos \left (4 b x +4 a \right )+562-5 \cos \left (12 b x +12 a \right )-12 \cos \left (10 b x +10 a \right )}{122880 b}\) | \(74\) |
risch | \(-\frac {\cos \left (12 b x +12 a \right )}{24576 b}-\frac {\cos \left (10 b x +10 a \right )}{10240 b}+\frac {\cos \left (8 b x +8 a \right )}{4096 b}+\frac {5 \cos \left (6 b x +6 a \right )}{6144 b}-\frac {5 \cos \left (4 b x +4 a \right )}{8192 b}-\frac {5 \cos \left (2 b x +2 a \right )}{1024 b}\) | \(86\) |
orering | \(-\frac {5369 \left (-7 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{6} b +5 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{4} b \right )}{14400 b^{2}}-\frac {37037 \left (-210 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{8} b^{3}+868 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{6} b^{3}-590 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{4} b^{3}+60 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right )^{2} b^{3}\right )}{1036800 b^{4}}-\frac {44473 \left (-2520 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{10} b^{5}+42000 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{8} b^{5}-117712 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{6} b^{5}+76280 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{4} b^{5}-10200 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right )^{2} b^{5}+120 \cos \left (b x +a \right )^{12} b^{5}\right )}{33177600 b^{6}}-\frac {1001 \left (-5040 \sin \left (b x +a \right )^{12} b^{7}+635040 \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{10} b^{7}-6950160 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{8} b^{7}+16511488 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{6} b^{7}-10246640 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{4} b^{7}+1461600 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right )^{2} b^{7}-21840 \cos \left (b x +a \right )^{12} b^{7}\right )}{44236800 b^{8}}-\frac {91 \left (-117089280 \sin \left (b x +a \right )^{10} b^{9} \cos \left (b x +a \right )^{2}+1330560 \sin \left (b x +a \right )^{12} b^{9}+1080710400 \cos \left (b x +a \right )^{4} \sin \left (b x +a \right )^{8} b^{9}-2349985792 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{6} b^{9}+1405633280 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{4} b^{9}-201845760 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right )^{2} b^{9}+3185280 \cos \left (b x +a \right )^{12} b^{9}\right )}{530841600 b^{10}}-\frac {-163171599360 \sin \left (b x +a \right )^{8} b^{11} \cos \left (b x +a \right )^{4}+19232801280 \sin \left (b x +a \right )^{10} b^{11} \cos \left (b x +a \right )^{2}-250145280 \sin \left (b x +a \right )^{12} b^{11}+336634052608 \cos \left (b x +a \right )^{6} \sin \left (b x +a \right )^{6} b^{11}-195493821440 \cos \left (b x +a \right )^{8} \sin \left (b x +a \right )^{4} b^{11}+27784035840 \cos \left (b x +a \right )^{10} \sin \left (b x +a \right )^{2} b^{11}-441914880 \cos \left (b x +a \right )^{12} b^{11}}{2123366400 b^{12}}\) | \(671\) |
Input:
int(cos(b*x+a)^7*sin(b*x+a)^5,x,method=_RETURNVERBOSE)
Output:
-1/b*(1/12*sin(b*x+a)^12-3/10*sin(b*x+a)^10+3/8*sin(b*x+a)^8-1/6*sin(b*x+a )^6)
Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {10 \, \cos \left (b x + a\right )^{12} - 24 \, \cos \left (b x + a\right )^{10} + 15 \, \cos \left (b x + a\right )^{8}}{120 \, b} \] Input:
integrate(cos(b*x+a)^7*sin(b*x+a)^5,x, algorithm="fricas")
Output:
-1/120*(10*cos(b*x + a)^12 - 24*cos(b*x + a)^10 + 15*cos(b*x + a)^8)/b
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (34) = 68\).
Time = 2.46 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.80 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=\begin {cases} \frac {\sin ^{12}{\left (a + b x \right )}}{120 b} + \frac {\sin ^{10}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{20 b} + \frac {\sin ^{8}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{8 b} + \frac {\sin ^{6}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{6 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (a \right )} \cos ^{7}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(b*x+a)**7*sin(b*x+a)**5,x)
Output:
Piecewise((sin(a + b*x)**12/(120*b) + sin(a + b*x)**10*cos(a + b*x)**2/(20 *b) + sin(a + b*x)**8*cos(a + b*x)**4/(8*b) + sin(a + b*x)**6*cos(a + b*x) **6/(6*b), Ne(b, 0)), (x*sin(a)**5*cos(a)**7, True))
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {10 \, \sin \left (b x + a\right )^{12} - 36 \, \sin \left (b x + a\right )^{10} + 45 \, \sin \left (b x + a\right )^{8} - 20 \, \sin \left (b x + a\right )^{6}}{120 \, b} \] Input:
integrate(cos(b*x+a)^7*sin(b*x+a)^5,x, algorithm="maxima")
Output:
-1/120*(10*sin(b*x + a)^12 - 36*sin(b*x + a)^10 + 45*sin(b*x + a)^8 - 20*s in(b*x + a)^6)/b
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {10 \, \cos \left (b x + a\right )^{12} - 24 \, \cos \left (b x + a\right )^{10} + 15 \, \cos \left (b x + a\right )^{8}}{120 \, b} \] Input:
integrate(cos(b*x+a)^7*sin(b*x+a)^5,x, algorithm="giac")
Output:
-1/120*(10*cos(b*x + a)^12 - 24*cos(b*x + a)^10 + 15*cos(b*x + a)^8)/b
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=-\frac {{\cos \left (a+b\,x\right )}^8\,\left (10\,{\cos \left (a+b\,x\right )}^4-24\,{\cos \left (a+b\,x\right )}^2+15\right )}{120\,b} \] Input:
int(cos(a + b*x)^7*sin(a + b*x)^5,x)
Output:
-(cos(a + b*x)^8*(10*cos(a + b*x)^4 - 24*cos(a + b*x)^2 + 15))/(120*b)
Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \cos ^7(a+b x) \sin ^5(a+b x) \, dx=\frac {\sin \left (b x +a \right )^{6} \left (-10 \sin \left (b x +a \right )^{6}+36 \sin \left (b x +a \right )^{4}-45 \sin \left (b x +a \right )^{2}+20\right )}{120 b} \] Input:
int(cos(b*x+a)^7*sin(b*x+a)^5,x)
Output:
(sin(a + b*x)**6*( - 10*sin(a + b*x)**6 + 36*sin(a + b*x)**4 - 45*sin(a + b*x)**2 + 20))/(120*b)