Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^7(c+d x)}{7 a d}-\frac {\sin ^8(c+d x)}{8 a d}-\frac {2 \sin ^9(c+d x)}{9 a d}+\frac {\sin ^{10}(c+d x)}{5 a d}+\frac {\sin ^{11}(c+d x)}{11 a d}-\frac {\sin ^{12}(c+d x)}{12 a d} \] Output:
1/7*sin(d*x+c)^7/a/d-1/8*sin(d*x+c)^8/a/d-2/9*sin(d*x+c)^9/a/d+1/5*sin(d*x +c)^10/a/d+1/11*sin(d*x+c)^11/a/d-1/12*sin(d*x+c)^12/a/d
Time = 0.55 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62 \[ \int \frac {\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^7(c+d x) \left (3960-3465 \sin (c+d x)-6160 \sin ^2(c+d x)+5544 \sin ^3(c+d x)+2520 \sin ^4(c+d x)-2310 \sin ^5(c+d x)\right )}{27720 a d} \] Input:
Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^6)/(a + a*Sin[c + d*x]),x]
Output:
(Sin[c + d*x]^7*(3960 - 3465*Sin[c + d*x] - 6160*Sin[c + d*x]^2 + 5544*Sin [c + d*x]^3 + 2520*Sin[c + d*x]^4 - 2310*Sin[c + d*x]^5))/(27720*a*d)
Time = 0.35 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 27, 99, 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.
\(\displaystyle \int \frac {\sin ^6(c+d x) \cos ^7(c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x)^6 \cos (c+d x)^7}{a \sin (c+d x)+a}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \sin ^6(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^7 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int a^6 \sin ^6(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2d(a \sin (c+d x))}{a^{13} d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\int \left (-\sin ^{11}(c+d x) a^{11}+\sin ^{10}(c+d x) a^{11}+2 \sin ^9(c+d x) a^{11}-2 \sin ^8(c+d x) a^{11}-\sin ^7(c+d x) a^{11}+\sin ^6(c+d x) a^{11}\right )d(a \sin (c+d x))}{a^{13} d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{12} a^{12} \sin ^{12}(c+d x)+\frac {1}{11} a^{12} \sin ^{11}(c+d x)+\frac {1}{5} a^{12} \sin ^{10}(c+d x)-\frac {2}{9} a^{12} \sin ^9(c+d x)-\frac {1}{8} a^{12} \sin ^8(c+d x)+\frac {1}{7} a^{12} \sin ^7(c+d x)}{a^{13} d}\) |
Input:
Int[(Cos[c + d*x]^7*Sin[c + d*x]^6)/(a + a*Sin[c + d*x]),x]
Output:
((a^12*Sin[c + d*x]^7)/7 - (a^12*Sin[c + d*x]^8)/8 - (2*a^12*Sin[c + d*x]^ 9)/9 + (a^12*Sin[c + d*x]^10)/5 + (a^12*Sin[c + d*x]^11)/11 - (a^12*Sin[c + d*x]^12)/12)/(a^13*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 1.83 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(-\frac {\frac {\sin \left (d x +c \right )^{12}}{12}-\frac {\sin \left (d x +c \right )^{11}}{11}-\frac {\sin \left (d x +c \right )^{10}}{5}+\frac {2 \sin \left (d x +c \right )^{9}}{9}+\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{7}}{7}}{d a}\) | \(70\) |
default | \(-\frac {\frac {\sin \left (d x +c \right )^{12}}{12}-\frac {\sin \left (d x +c \right )^{11}}{11}-\frac {\sin \left (d x +c \right )^{10}}{5}+\frac {2 \sin \left (d x +c \right )^{9}}{9}+\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{7}}{7}}{d a}\) | \(70\) |
parallelrisch | \(\frac {\left (-\sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+7 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-21 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (-1155 \sin \left (5 d x +5 c \right )+14560 \cos \left (2 d x +2 c \right )-6006 \sin \left (d x +c \right )-5313 \sin \left (3 d x +3 c \right )+2520 \cos \left (4 d x +4 c \right )+14600\right ) \left (\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+7 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+21 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+35 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7096320 a d}\) | \(151\) |
risch | \(\frac {5 \sin \left (d x +c \right )}{512 a d}-\frac {\sin \left (11 d x +11 c \right )}{11264 d a}-\frac {\cos \left (12 d x +12 c \right )}{24576 a d}+\frac {\cos \left (10 d x +10 c \right )}{10240 a d}+\frac {\sin \left (9 d x +9 c \right )}{9216 d a}+\frac {\cos \left (8 d x +8 c \right )}{4096 a d}+\frac {5 \sin \left (7 d x +7 c \right )}{7168 d a}-\frac {5 \cos \left (6 d x +6 c \right )}{6144 a d}-\frac {\sin \left (5 d x +5 c \right )}{1024 d a}-\frac {5 \cos \left (4 d x +4 c \right )}{8192 a d}-\frac {5 \sin \left (3 d x +3 c \right )}{1536 d a}+\frac {5 \cos \left (2 d x +2 c \right )}{1024 a d}\) | \(203\) |
Input:
int(cos(d*x+c)^7*sin(d*x+c)^6/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/d/a*(1/12*sin(d*x+c)^12-1/11*sin(d*x+c)^11-1/5*sin(d*x+c)^10+2/9*sin(d* x+c)^9+1/8*sin(d*x+c)^8-1/7*sin(d*x+c)^7)
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2310 \, \cos \left (d x + c\right )^{12} - 8316 \, \cos \left (d x + c\right )^{10} + 10395 \, \cos \left (d x + c\right )^{8} - 4620 \, \cos \left (d x + c\right )^{6} + 40 \, {\left (63 \, \cos \left (d x + c\right )^{10} - 161 \, \cos \left (d x + c\right )^{8} + 113 \, \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right )}{27720 \, a d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
-1/27720*(2310*cos(d*x + c)^12 - 8316*cos(d*x + c)^10 + 10395*cos(d*x + c) ^8 - 4620*cos(d*x + c)^6 + 40*(63*cos(d*x + c)^10 - 161*cos(d*x + c)^8 + 1 13*cos(d*x + c)^6 - 3*cos(d*x + c)^4 - 4*cos(d*x + c)^2 - 8)*sin(d*x + c)) /(a*d)
Timed out. \[ \int \frac {\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**7*sin(d*x+c)**6/(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2310 \, \sin \left (d x + c\right )^{12} - 2520 \, \sin \left (d x + c\right )^{11} - 5544 \, \sin \left (d x + c\right )^{10} + 6160 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} - 3960 \, \sin \left (d x + c\right )^{7}}{27720 \, a d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/27720*(2310*sin(d*x + c)^12 - 2520*sin(d*x + c)^11 - 5544*sin(d*x + c)^ 10 + 6160*sin(d*x + c)^9 + 3465*sin(d*x + c)^8 - 3960*sin(d*x + c)^7)/(a*d )
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {2310 \, \sin \left (d x + c\right )^{12} - 2520 \, \sin \left (d x + c\right )^{11} - 5544 \, \sin \left (d x + c\right )^{10} + 6160 \, \sin \left (d x + c\right )^{9} + 3465 \, \sin \left (d x + c\right )^{8} - 3960 \, \sin \left (d x + c\right )^{7}}{27720 \, a d} \] Input:
integrate(cos(d*x+c)^7*sin(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/27720*(2310*sin(d*x + c)^12 - 2520*sin(d*x + c)^11 - 5544*sin(d*x + c)^ 10 + 6160*sin(d*x + c)^9 + 3465*sin(d*x + c)^8 - 3960*sin(d*x + c)^7)/(a*d )
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^7}{7\,a}-\frac {{\sin \left (c+d\,x\right )}^8}{8\,a}-\frac {2\,{\sin \left (c+d\,x\right )}^9}{9\,a}+\frac {{\sin \left (c+d\,x\right )}^{10}}{5\,a}+\frac {{\sin \left (c+d\,x\right )}^{11}}{11\,a}-\frac {{\sin \left (c+d\,x\right )}^{12}}{12\,a}}{d} \] Input:
int((cos(c + d*x)^7*sin(c + d*x)^6)/(a + a*sin(c + d*x)),x)
Output:
(sin(c + d*x)^7/(7*a) - sin(c + d*x)^8/(8*a) - (2*sin(c + d*x)^9)/(9*a) + sin(c + d*x)^10/(5*a) + sin(c + d*x)^11/(11*a) - sin(c + d*x)^12/(12*a))/d
Time = 0.83 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {\cos ^7(c+d x) \sin ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin \left (d x +c \right )^{7} \left (-2310 \sin \left (d x +c \right )^{5}+2520 \sin \left (d x +c \right )^{4}+5544 \sin \left (d x +c \right )^{3}-6160 \sin \left (d x +c \right )^{2}-3465 \sin \left (d x +c \right )+3960\right )}{27720 a d} \] Input:
int(cos(d*x+c)^7*sin(d*x+c)^6/(a+a*sin(d*x+c)),x)
Output:
(sin(c + d*x)**7*( - 2310*sin(c + d*x)**5 + 2520*sin(c + d*x)**4 + 5544*si n(c + d*x)**3 - 6160*sin(c + d*x)**2 - 3465*sin(c + d*x) + 3960))/(27720*a *d)