\(\int \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [680]
Optimal result
Integrand size = 29, antiderivative size = 91 \[
\int \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\cos ^6(c+d x)}{6 a d}+\frac {\cos ^8(c+d x)}{8 a d}-\frac {\sin ^5(c+d x)}{5 a d}+\frac {2 \sin ^7(c+d x)}{7 a d}-\frac {\sin ^9(c+d x)}{9 a d}
\]
[Out]
-1/6*cos(d*x+c)^6/a/d+1/8*cos(d*x+c)^8/a/d-1/5*sin(d*x+c)^5/a/d+2/7*sin(d*x+c)^7/a/d-1/9*sin(d*x+c)^9/a/d
Rubi [A] (verified)
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2914, 2645, 14, 2644, 276}
\[
\int \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sin ^9(c+d x)}{9 a d}+\frac {2 \sin ^7(c+d x)}{7 a d}-\frac {\sin ^5(c+d x)}{5 a d}+\frac {\cos ^8(c+d x)}{8 a d}-\frac {\cos ^6(c+d x)}{6 a d}
\]
[In]
Int[(Cos[c + d*x]^7*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
[Out]
-1/6*Cos[c + d*x]^6/(a*d) + Cos[c + d*x]^8/(8*a*d) - Sin[c + d*x]^5/(5*a*d) + (2*Sin[c + d*x]^7)/(7*a*d) - Sin
[c + d*x]^9/(9*a*d)
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 276
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rule 2644
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])
Rule 2645
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(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])
Rule 2914
Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))
Rubi steps \begin{align*}
\text {integral}& = \frac {\int \cos ^5(c+d x) \sin ^3(c+d x) \, dx}{a}-\frac {\int \cos ^5(c+d x) \sin ^4(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{a d} \\ & = -\frac {\text {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (c+d x)\right )}{a d} \\ & = -\frac {\cos ^6(c+d x)}{6 a d}+\frac {\cos ^8(c+d x)}{8 a d}-\frac {\sin ^5(c+d x)}{5 a d}+\frac {2 \sin ^7(c+d x)}{7 a d}-\frac {\sin ^9(c+d x)}{9 a d} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75
\[
\int \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^4(c+d x) \left (630-504 \sin (c+d x)-840 \sin ^2(c+d x)+720 \sin ^3(c+d x)+315 \sin ^4(c+d x)-280 \sin ^5(c+d x)\right )}{2520 a d}
\]
[In]
Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
[Out]
(Sin[c + d*x]^4*(630 - 504*Sin[c + d*x] - 840*Sin[c + d*x]^2 + 720*Sin[c + d*x]^3 + 315*Sin[c + d*x]^4 - 280*S
in[c + d*x]^5))/(2520*a*d)
Maple [A] (verified)
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77
| | |
| method | result | size |
| | |
| derivativedivides |
\(-\frac {\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}}{a d}\) |
\(70\) |
| default |
\(-\frac {\frac {\left (\sin ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{3}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}}{a d}\) |
\(70\) |
| parallelrisch |
\(\frac {-7560 \cos \left (2 d x +2 c \right )-140 \sin \left (9 d x +9 c \right )+315 \cos \left (8 d x +8 c \right )-180 \sin \left (7 d x +7 c \right )+1008 \sin \left (5 d x +5 c \right )+840 \cos \left (6 d x +6 c \right )-7560 \sin \left (d x +c \right )+1680 \sin \left (3 d x +3 c \right )-1260 \cos \left (4 d x +4 c \right )+7665}{322560 a d}\) |
\(107\) |
| risch |
\(-\frac {3 \sin \left (d x +c \right )}{128 a d}-\frac {\sin \left (9 d x +9 c \right )}{2304 d a}+\frac {\cos \left (8 d x +8 c \right )}{1024 a d}-\frac {\sin \left (7 d x +7 c \right )}{1792 d a}+\frac {\cos \left (6 d x +6 c \right )}{384 a d}+\frac {\sin \left (5 d x +5 c \right )}{320 d a}-\frac {\cos \left (4 d x +4 c \right )}{256 a d}+\frac {\sin \left (3 d x +3 c \right )}{192 d a}-\frac {3 \cos \left (2 d x +2 c \right )}{128 a d}\) |
\(152\) |
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[In]
int(cos(d*x+c)^7*sin(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
-1/a/d*(1/9*sin(d*x+c)^9-1/8*sin(d*x+c)^8-2/7*sin(d*x+c)^7+1/3*sin(d*x+c)^6+1/5*sin(d*x+c)^5-1/4*sin(d*x+c)^4)
Fricas [A] (verification not implemented)
none
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.87
\[
\int \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {315 \, \cos \left (d x + c\right )^{8} - 420 \, \cos \left (d x + c\right )^{6} - 8 \, {\left (35 \, \cos \left (d x + c\right )^{8} - 50 \, \cos \left (d x + c\right )^{6} + 3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right )}{2520 \, a d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
1/2520*(315*cos(d*x + c)^8 - 420*cos(d*x + c)^6 - 8*(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. 1906 vs. \(2 (68) = 136\).
Time = 78.79 (sec) , antiderivative size = 1906, normalized size of antiderivative = 20.95
\[
\int \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)**7*sin(d*x+c)**3/(a+a*sin(d*x+c)),x)
[Out]
Piecewise((1260*tan(c/2 + d*x/2)**14/(315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d
*tan(c/2 + d*x/2)**14 + 26460*a*d*tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 +
d*x/2)**8 + 26460*a*d*tan(c/2 + d*x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315
*a*d) - 2016*tan(c/2 + d*x/2)**13/(315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*ta
n(c/2 + d*x/2)**14 + 26460*a*d*tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x
/2)**8 + 26460*a*d*tan(c/2 + d*x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315*a*
d) - 420*tan(c/2 + d*x/2)**12/(315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*tan(c/
2 + d*x/2)**14 + 26460*a*d*tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x/2)*
*8 + 26460*a*d*tan(c/2 + d*x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) +
3456*tan(c/2 + d*x/2)**11/(315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*tan(c/2 +
d*x/2)**14 + 26460*a*d*tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x/2)**8
+ 26460*a*d*tan(c/2 + d*x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) + 25
20*tan(c/2 + d*x/2)**10/(315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*tan(c/2 + d*
x/2)**14 + 26460*a*d*tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x/2)**8 + 2
6460*a*d*tan(c/2 + d*x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) - 6976*
tan(c/2 + d*x/2)**9/(315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*tan(c/2 + d*x/2)
**14 + 26460*a*d*tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x/2)**8 + 26460
*a*d*tan(c/2 + d*x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) + 2520*tan(
c/2 + d*x/2)**8/(315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*tan(c/2 + d*x/2)**14
+ 26460*a*d*tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x/2)**8 + 26460*a*d
*tan(c/2 + d*x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) + 3456*tan(c/2
+ d*x/2)**7/(315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*tan(c/2 + d*x/2)**14 + 2
6460*a*d*tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x/2)**8 + 26460*a*d*tan
(c/2 + d*x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) - 420*tan(c/2 + d*x
/2)**6/(315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*tan(c/2 + d*x/2)**14 + 26460*
a*d*tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x/2)**8 + 26460*a*d*tan(c/2
+ d*x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*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)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*tan(c/2 + d*x/2)**14 + 26460*a*d*
tan(c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x/2)**8 + 26460*a*d*tan(c/2 + d*
x/2)**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315*a*d) + 1260*tan(c/2 + d*x/2)**4/(
315*a*d*tan(c/2 + d*x/2)**18 + 2835*a*d*tan(c/2 + d*x/2)**16 + 11340*a*d*tan(c/2 + d*x/2)**14 + 26460*a*d*tan(
c/2 + d*x/2)**12 + 39690*a*d*tan(c/2 + d*x/2)**10 + 39690*a*d*tan(c/2 + d*x/2)**8 + 26460*a*d*tan(c/2 + d*x/2)
**6 + 11340*a*d*tan(c/2 + d*x/2)**4 + 2835*a*d*tan(c/2 + d*x/2)**2 + 315*a*d), Ne(d, 0)), (x*sin(c)**3*cos(c)*
*7/(a*sin(c) + a), True))
Maxima [A] (verification not implemented)
none
Time = 0.22 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.76
\[
\int \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {280 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} - 720 \, \sin \left (d x + c\right )^{7} + 840 \, \sin \left (d x + c\right )^{6} + 504 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4}}{2520 \, a d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
-1/2520*(280*sin(d*x + c)^9 - 315*sin(d*x + c)^8 - 720*sin(d*x + c)^7 + 840*sin(d*x + c)^6 + 504*sin(d*x + c)^
5 - 630*sin(d*x + c)^4)/(a*d)
Giac [A] (verification not implemented)
none
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.76
\[
\int \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {280 \, \sin \left (d x + c\right )^{9} - 315 \, \sin \left (d x + c\right )^{8} - 720 \, \sin \left (d x + c\right )^{7} + 840 \, \sin \left (d x + c\right )^{6} + 504 \, \sin \left (d x + c\right )^{5} - 630 \, \sin \left (d x + c\right )^{4}}{2520 \, a d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
-1/2520*(280*sin(d*x + c)^9 - 315*sin(d*x + c)^8 - 720*sin(d*x + c)^7 + 840*sin(d*x + c)^6 + 504*sin(d*x + c)^
5 - 630*sin(d*x + c)^4)/(a*d)
Mupad [B] (verification not implemented)
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91
\[
\int \frac {\cos ^7(c+d x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^4}{4\,a}-\frac {{\sin \left (c+d\,x\right )}^5}{5\,a}-\frac {{\sin \left (c+d\,x\right )}^6}{3\,a}+\frac {2\,{\sin \left (c+d\,x\right )}^7}{7\,a}+\frac {{\sin \left (c+d\,x\right )}^8}{8\,a}-\frac {{\sin \left (c+d\,x\right )}^9}{9\,a}}{d}
\]
[In]
int((cos(c + d*x)^7*sin(c + d*x)^3)/(a + a*sin(c + d*x)),x)
[Out]
(sin(c + d*x)^4/(4*a) - sin(c + d*x)^5/(5*a) - sin(c + d*x)^6/(3*a) + (2*sin(c + d*x)^7)/(7*a) + sin(c + d*x)^
8/(8*a) - sin(c + d*x)^9/(9*a))/d