\(\int \frac {\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx\) [679]
Optimal result
Integrand size = 29, antiderivative size = 109 \[
\int \frac {\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^5(c+d x)}{5 a d}-\frac {\sin ^6(c+d x)}{6 a d}-\frac {2 \sin ^7(c+d x)}{7 a d}+\frac {\sin ^8(c+d x)}{4 a d}+\frac {\sin ^9(c+d x)}{9 a d}-\frac {\sin ^{10}(c+d x)}{10 a d}
\]
[Out]
1/5*sin(d*x+c)^5/a/d-1/6*sin(d*x+c)^6/a/d-2/7*sin(d*x+c)^7/a/d+1/4*sin(d*x+c)^8/a/d+1/9*sin(d*x+c)^9/a/d-1/10*
sin(d*x+c)^10/a/d
Rubi [A] (verified)
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number
of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 90}
\[
\int \frac {\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sin ^{10}(c+d x)}{10 a d}+\frac {\sin ^9(c+d x)}{9 a d}+\frac {\sin ^8(c+d x)}{4 a d}-\frac {2 \sin ^7(c+d x)}{7 a d}-\frac {\sin ^6(c+d x)}{6 a d}+\frac {\sin ^5(c+d x)}{5 a d}
\]
[In]
Int[(Cos[c + d*x]^7*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
[Out]
Sin[c + d*x]^5/(5*a*d) - Sin[c + d*x]^6/(6*a*d) - (2*Sin[c + d*x]^7)/(7*a*d) + Sin[c + d*x]^8/(4*a*d) + Sin[c
+ d*x]^9/(9*a*d) - Sin[c + d*x]^10/(10*a*d)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 90
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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]))
Rule 2915
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[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] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {(a-x)^3 x^4 (a+x)^2}{a^4} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int (a-x)^3 x^4 (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^{11} d} \\ & = \frac {\text {Subst}\left (\int \left (a^5 x^4-a^4 x^5-2 a^3 x^6+2 a^2 x^7+a x^8-x^9\right ) \, dx,x,a \sin (c+d x)\right )}{a^{11} d} \\ & = \frac {\sin ^5(c+d x)}{5 a d}-\frac {\sin ^6(c+d x)}{6 a d}-\frac {2 \sin ^7(c+d x)}{7 a d}+\frac {\sin ^8(c+d x)}{4 a d}+\frac {\sin ^9(c+d x)}{9 a d}-\frac {\sin ^{10}(c+d x)}{10 a d} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.62
\[
\int \frac {\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^5(c+d x) \left (252-210 \sin (c+d x)-360 \sin ^2(c+d x)+315 \sin ^3(c+d x)+140 \sin ^4(c+d x)-126 \sin ^5(c+d x)\right )}{1260 a d}
\]
[In]
Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^4)/(a + a*Sin[c + d*x]),x]
[Out]
(Sin[c + d*x]^5*(252 - 210*Sin[c + d*x] - 360*Sin[c + d*x]^2 + 315*Sin[c + d*x]^3 + 140*Sin[c + d*x]^4 - 126*S
in[c + d*x]^5))/(1260*a*d)
Maple [A] (verified)
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.64
| | |
| method | result | size |
| | |
| derivativedivides |
\(-\frac {\frac {\left (\sin ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{4}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}}{a d}\) |
\(70\) |
| default |
\(-\frac {\frac {\left (\sin ^{10}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{4}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}}{a d}\) |
\(70\) |
| parallelrisch |
\(\frac {\left (\sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-5 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (880 \cos \left (2 d x +2 c \right )-63 \sin \left (5 d x +5 c \right )-420 \sin \left (d x +c \right )-315 \sin \left (3 d x +3 c \right )+140 \cos \left (4 d x +4 c \right )+996\right ) \left (\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+5 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+10 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80640 a d}\) |
\(127\) |
| risch |
\(\frac {3 \sin \left (d x +c \right )}{128 a d}+\frac {\cos \left (10 d x +10 c \right )}{5120 a d}+\frac {\sin \left (9 d x +9 c \right )}{2304 d a}+\frac {\sin \left (7 d x +7 c \right )}{1792 d a}-\frac {5 \cos \left (6 d x +6 c \right )}{3072 a d}-\frac {\sin \left (5 d x +5 c \right )}{320 d a}-\frac {\sin \left (3 d x +3 c \right )}{192 d a}+\frac {5 \cos \left (2 d x +2 c \right )}{512 a d}\) |
\(135\) |
| | |
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[In]
int(cos(d*x+c)^7*sin(d*x+c)^4/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
-1/a/d*(1/10*sin(d*x+c)^10-1/9*sin(d*x+c)^9-1/4*sin(d*x+c)^8+2/7*sin(d*x+c)^7+1/6*sin(d*x+c)^6-1/5*sin(d*x+c)^
5)
Fricas [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82
\[
\int \frac {\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {126 \, \cos \left (d x + c\right )^{10} - 315 \, \cos \left (d x + c\right )^{8} + 210 \, \cos \left (d x + c\right )^{6} + 4 \, {\left (35 \, \cos \left (d x + c\right )^{8} - 50 \, \cos \left (d x + c\right )^{6} + 3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right )}{1260 \, a d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
1/1260*(126*cos(d*x + c)^10 - 315*cos(d*x + c)^8 + 210*cos(d*x + c)^6 + 4*(35*cos(d*x + c)^8 - 50*cos(d*x + c)
^6 + 3*cos(d*x + c)^4 + 4*cos(d*x + c)^2 + 8)*sin(d*x + c))/(a*d)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 2093 vs. \(2 (82) = 164\).
Time = 119.04 (sec) , antiderivative size = 2093, normalized size of antiderivative = 19.20
\[
\int \frac {\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)**7*sin(d*x+c)**4/(a+a*sin(d*x+c)),x)
[Out]
Piecewise((2016*tan(c/2 + d*x/2)**15/(315*a*d*tan(c/2 + d*x/2)**20 + 3150*a*d*tan(c/2 + d*x/2)**18 + 14175*a*d
*tan(c/2 + d*x/2)**16 + 37800*a*d*tan(c/2 + d*x/2)**14 + 66150*a*d*tan(c/2 + d*x/2)**12 + 79380*a*d*tan(c/2 +
d*x/2)**10 + 66150*a*d*tan(c/2 + d*x/2)**8 + 37800*a*d*tan(c/2 + d*x/2)**6 + 14175*a*d*tan(c/2 + d*x/2)**4 + 3
150*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) - 3360*tan(c/2 + d*x/2)**14/(315*a*d*tan(c/2 + d*x/2)**20 + 3150*a*d*ta
n(c/2 + d*x/2)**18 + 14175*a*d*tan(c/2 + d*x/2)**16 + 37800*a*d*tan(c/2 + d*x/2)**14 + 66150*a*d*tan(c/2 + d*x
/2)**12 + 79380*a*d*tan(c/2 + d*x/2)**10 + 66150*a*d*tan(c/2 + d*x/2)**8 + 37800*a*d*tan(c/2 + d*x/2)**6 + 141
75*a*d*tan(c/2 + d*x/2)**4 + 3150*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) - 1440*tan(c/2 + d*x/2)**13/(315*a*d*tan(
c/2 + d*x/2)**20 + 3150*a*d*tan(c/2 + d*x/2)**18 + 14175*a*d*tan(c/2 + d*x/2)**16 + 37800*a*d*tan(c/2 + d*x/2)
**14 + 66150*a*d*tan(c/2 + d*x/2)**12 + 79380*a*d*tan(c/2 + d*x/2)**10 + 66150*a*d*tan(c/2 + d*x/2)**8 + 37800
*a*d*tan(c/2 + d*x/2)**6 + 14175*a*d*tan(c/2 + d*x/2)**4 + 3150*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) + 6720*tan(
c/2 + d*x/2)**12/(315*a*d*tan(c/2 + d*x/2)**20 + 3150*a*d*tan(c/2 + d*x/2)**18 + 14175*a*d*tan(c/2 + d*x/2)**1
6 + 37800*a*d*tan(c/2 + d*x/2)**14 + 66150*a*d*tan(c/2 + d*x/2)**12 + 79380*a*d*tan(c/2 + d*x/2)**10 + 66150*a
*d*tan(c/2 + d*x/2)**8 + 37800*a*d*tan(c/2 + d*x/2)**6 + 14175*a*d*tan(c/2 + d*x/2)**4 + 3150*a*d*tan(c/2 + d*
x/2)**2 + 315*a*d) + 3520*tan(c/2 + d*x/2)**11/(315*a*d*tan(c/2 + d*x/2)**20 + 3150*a*d*tan(c/2 + d*x/2)**18 +
14175*a*d*tan(c/2 + d*x/2)**16 + 37800*a*d*tan(c/2 + d*x/2)**14 + 66150*a*d*tan(c/2 + d*x/2)**12 + 79380*a*d*
tan(c/2 + d*x/2)**10 + 66150*a*d*tan(c/2 + d*x/2)**8 + 37800*a*d*tan(c/2 + d*x/2)**6 + 14175*a*d*tan(c/2 + d*x
/2)**4 + 3150*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) - 12096*tan(c/2 + d*x/2)**10/(315*a*d*tan(c/2 + d*x/2)**20 +
3150*a*d*tan(c/2 + d*x/2)**18 + 14175*a*d*tan(c/2 + d*x/2)**16 + 37800*a*d*tan(c/2 + d*x/2)**14 + 66150*a*d*ta
n(c/2 + d*x/2)**12 + 79380*a*d*tan(c/2 + d*x/2)**10 + 66150*a*d*tan(c/2 + d*x/2)**8 + 37800*a*d*tan(c/2 + d*x/
2)**6 + 14175*a*d*tan(c/2 + d*x/2)**4 + 3150*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) + 3520*tan(c/2 + d*x/2)**9/(31
5*a*d*tan(c/2 + d*x/2)**20 + 3150*a*d*tan(c/2 + d*x/2)**18 + 14175*a*d*tan(c/2 + d*x/2)**16 + 37800*a*d*tan(c/
2 + d*x/2)**14 + 66150*a*d*tan(c/2 + d*x/2)**12 + 79380*a*d*tan(c/2 + d*x/2)**10 + 66150*a*d*tan(c/2 + d*x/2)*
*8 + 37800*a*d*tan(c/2 + d*x/2)**6 + 14175*a*d*tan(c/2 + d*x/2)**4 + 3150*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) +
6720*tan(c/2 + d*x/2)**8/(315*a*d*tan(c/2 + d*x/2)**20 + 3150*a*d*tan(c/2 + d*x/2)**18 + 14175*a*d*tan(c/2 +
d*x/2)**16 + 37800*a*d*tan(c/2 + d*x/2)**14 + 66150*a*d*tan(c/2 + d*x/2)**12 + 79380*a*d*tan(c/2 + d*x/2)**10
+ 66150*a*d*tan(c/2 + d*x/2)**8 + 37800*a*d*tan(c/2 + d*x/2)**6 + 14175*a*d*tan(c/2 + d*x/2)**4 + 3150*a*d*tan
(c/2 + d*x/2)**2 + 315*a*d) - 1440*tan(c/2 + d*x/2)**7/(315*a*d*tan(c/2 + d*x/2)**20 + 3150*a*d*tan(c/2 + d*x/
2)**18 + 14175*a*d*tan(c/2 + d*x/2)**16 + 37800*a*d*tan(c/2 + d*x/2)**14 + 66150*a*d*tan(c/2 + d*x/2)**12 + 79
380*a*d*tan(c/2 + d*x/2)**10 + 66150*a*d*tan(c/2 + d*x/2)**8 + 37800*a*d*tan(c/2 + d*x/2)**6 + 14175*a*d*tan(c
/2 + d*x/2)**4 + 3150*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) - 3360*tan(c/2 + d*x/2)**6/(315*a*d*tan(c/2 + d*x/2)*
*20 + 3150*a*d*tan(c/2 + d*x/2)**18 + 14175*a*d*tan(c/2 + d*x/2)**16 + 37800*a*d*tan(c/2 + d*x/2)**14 + 66150*
a*d*tan(c/2 + d*x/2)**12 + 79380*a*d*tan(c/2 + d*x/2)**10 + 66150*a*d*tan(c/2 + d*x/2)**8 + 37800*a*d*tan(c/2
+ d*x/2)**6 + 14175*a*d*tan(c/2 + d*x/2)**4 + 3150*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) + 2016*tan(c/2 + d*x/2)*
*5/(315*a*d*tan(c/2 + d*x/2)**20 + 3150*a*d*tan(c/2 + d*x/2)**18 + 14175*a*d*tan(c/2 + d*x/2)**16 + 37800*a*d*
tan(c/2 + d*x/2)**14 + 66150*a*d*tan(c/2 + d*x/2)**12 + 79380*a*d*tan(c/2 + d*x/2)**10 + 66150*a*d*tan(c/2 + d
*x/2)**8 + 37800*a*d*tan(c/2 + d*x/2)**6 + 14175*a*d*tan(c/2 + d*x/2)**4 + 3150*a*d*tan(c/2 + d*x/2)**2 + 315*
a*d), Ne(d, 0)), (x*sin(c)**4*cos(c)**7/(a*sin(c) + a), True))
Maxima [A] (verification not implemented)
none
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
\[
\int \frac {\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {126 \, \sin \left (d x + c\right )^{10} - 140 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} + 360 \, \sin \left (d x + c\right )^{7} + 210 \, \sin \left (d x + c\right )^{6} - 252 \, \sin \left (d x + c\right )^{5}}{1260 \, a d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
-1/1260*(126*sin(d*x + c)^10 - 140*sin(d*x + c)^9 - 315*sin(d*x + c)^8 + 360*sin(d*x + c)^7 + 210*sin(d*x + c)
^6 - 252*sin(d*x + c)^5)/(a*d)
Giac [A] (verification not implemented)
none
Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
\[
\int \frac {\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {126 \, \sin \left (d x + c\right )^{10} - 140 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} + 360 \, \sin \left (d x + c\right )^{7} + 210 \, \sin \left (d x + c\right )^{6} - 252 \, \sin \left (d x + c\right )^{5}}{1260 \, a d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
-1/1260*(126*sin(d*x + c)^10 - 140*sin(d*x + c)^9 - 315*sin(d*x + c)^8 + 360*sin(d*x + c)^7 + 210*sin(d*x + c)
^6 - 252*sin(d*x + c)^5)/(a*d)
Mupad [B] (verification not implemented)
Time = 10.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.76
\[
\int \frac {\cos ^7(c+d x) \sin ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^5}{5\,a}-\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}-\frac {2\,{\sin \left (c+d\,x\right )}^7}{7\,a}+\frac {{\sin \left (c+d\,x\right )}^8}{4\,a}+\frac {{\sin \left (c+d\,x\right )}^9}{9\,a}-\frac {{\sin \left (c+d\,x\right )}^{10}}{10\,a}}{d}
\]
[In]
int((cos(c + d*x)^7*sin(c + d*x)^4)/(a + a*sin(c + d*x)),x)
[Out]
(sin(c + d*x)^5/(5*a) - sin(c + d*x)^6/(6*a) - (2*sin(c + d*x)^7)/(7*a) + sin(c + d*x)^8/(4*a) + sin(c + d*x)^
9/(9*a) - sin(c + d*x)^10/(10*a))/d