\(\int \frac {\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [681]
Optimal result
Integrand size = 29, antiderivative size = 91 \[
\int \frac {\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cos ^6(c+d x)}{6 a d}-\frac {\cos ^8(c+d x)}{8 a d}+\frac {\sin ^3(c+d x)}{3 a d}-\frac {2 \sin ^5(c+d x)}{5 a d}+\frac {\sin ^7(c+d x)}{7 a d}
\]
[Out]
1/6*cos(d*x+c)^6/a/d-1/8*cos(d*x+c)^8/a/d+1/3*sin(d*x+c)^3/a/d-2/5*sin(d*x+c)^5/a/d+1/7*sin(d*x+c)^7/a/d
Rubi [A] (verified)
Time = 0.13 (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, 2644, 276, 2645, 14}
\[
\int \frac {\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^7(c+d x)}{7 a d}-\frac {2 \sin ^5(c+d x)}{5 a d}+\frac {\sin ^3(c+d x)}{3 a d}-\frac {\cos ^8(c+d x)}{8 a d}+\frac {\cos ^6(c+d x)}{6 a d}
\]
[In]
Int[(Cos[c + d*x]^7*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
[Out]
Cos[c + d*x]^6/(6*a*d) - Cos[c + d*x]^8/(8*a*d) + Sin[c + d*x]^3/(3*a*d) - (2*Sin[c + d*x]^5)/(5*a*d) + Sin[c
+ d*x]^7/(7*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 ^2(c+d x) \, dx}{a}-\frac {\int \cos ^5(c+d x) \sin ^3(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^2 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, 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 {\cos ^6(c+d x)}{6 a d}-\frac {\cos ^8(c+d x)}{8 a d}+\frac {\sin ^3(c+d x)}{3 a d}-\frac {2 \sin ^5(c+d x)}{5 a d}+\frac {\sin ^7(c+d x)}{7 a d} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.75
\[
\int \frac {\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin ^3(c+d x) \left (280-210 \sin (c+d x)-336 \sin ^2(c+d x)+280 \sin ^3(c+d x)+120 \sin ^4(c+d x)-105 \sin ^5(c+d x)\right )}{840 a d}
\]
[In]
Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^2)/(a + a*Sin[c + d*x]),x]
[Out]
(Sin[c + d*x]^3*(280 - 210*Sin[c + d*x] - 336*Sin[c + d*x]^2 + 280*Sin[c + d*x]^3 + 120*Sin[c + d*x]^4 - 105*S
in[c + d*x]^5))/(840*a*d)
Maple [A] (verified)
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.77
| | |
| method | result | size |
| | |
| derivativedivides |
\(-\frac {\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{3}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}}{a d}\) |
\(70\) |
| default |
\(-\frac {\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{3}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}}{a d}\) |
\(70\) |
| parallelrisch |
\(-\frac {\left (\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (1728 \cos \left (2 d x +2 c \right )-105 \sin \left (5 d x +5 c \right )-1050 \sin \left (d x +c \right )-595 \sin \left (3 d x +3 c \right )+240 \cos \left (4 d x +4 c \right )+2512\right ) \left (\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{26880 a d}\) |
\(105\) |
| risch |
\(\frac {5 \sin \left (d x +c \right )}{64 a d}-\frac {\cos \left (8 d x +8 c \right )}{1024 a d}-\frac {\sin \left (7 d x +7 c \right )}{448 d a}-\frac {\cos \left (6 d x +6 c \right )}{384 a d}-\frac {3 \sin \left (5 d x +5 c \right )}{320 d a}+\frac {\cos \left (4 d x +4 c \right )}{256 a d}-\frac {\sin \left (3 d x +3 c \right )}{192 d a}+\frac {3 \cos \left (2 d x +2 c \right )}{128 a d}\) |
\(135\) |
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[In]
int(cos(d*x+c)^7*sin(d*x+c)^2/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
-1/a/d*(1/8*sin(d*x+c)^8-1/7*sin(d*x+c)^7-1/3*sin(d*x+c)^6+2/5*sin(d*x+c)^5+1/4*sin(d*x+c)^4-1/3*sin(d*x+c)^3)
Fricas [A] (verification not implemented)
none
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.76
\[
\int \frac {\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {105 \, \cos \left (d x + c\right )^{8} - 140 \, \cos \left (d x + c\right )^{6} + 8 \, {\left (15 \, \cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right )}{840 \, a d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
-1/840*(105*cos(d*x + c)^8 - 140*cos(d*x + c)^6 + 8*(15*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. 1719 vs. \(2 (68) = 136\).
Time = 50.94 (sec) , antiderivative size = 1719, normalized size of antiderivative = 18.89
\[
\int \frac {\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)**7*sin(d*x+c)**2/(a+a*sin(d*x+c)),x)
[Out]
Piecewise((280*tan(c/2 + d*x/2)**13/(105*a*d*tan(c/2 + d*x/2)**16 + 840*a*d*tan(c/2 + d*x/2)**14 + 2940*a*d*ta
n(c/2 + d*x/2)**12 + 5880*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 5880*a*d*tan(c/2 + d*x/2)*
*6 + 2940*a*d*tan(c/2 + d*x/2)**4 + 840*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) - 420*tan(c/2 + d*x/2)**12/(105*a*d
*tan(c/2 + d*x/2)**16 + 840*a*d*tan(c/2 + d*x/2)**14 + 2940*a*d*tan(c/2 + d*x/2)**12 + 5880*a*d*tan(c/2 + d*x/
2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 5880*a*d*tan(c/2 + d*x/2)**6 + 2940*a*d*tan(c/2 + d*x/2)**4 + 840*a*d*
tan(c/2 + d*x/2)**2 + 105*a*d) + 56*tan(c/2 + d*x/2)**11/(105*a*d*tan(c/2 + d*x/2)**16 + 840*a*d*tan(c/2 + d*x
/2)**14 + 2940*a*d*tan(c/2 + d*x/2)**12 + 5880*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 5880*
a*d*tan(c/2 + d*x/2)**6 + 2940*a*d*tan(c/2 + d*x/2)**4 + 840*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) + 560*tan(c/2
+ d*x/2)**10/(105*a*d*tan(c/2 + d*x/2)**16 + 840*a*d*tan(c/2 + d*x/2)**14 + 2940*a*d*tan(c/2 + d*x/2)**12 + 58
80*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 5880*a*d*tan(c/2 + d*x/2)**6 + 2940*a*d*tan(c/2 +
d*x/2)**4 + 840*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) + 688*tan(c/2 + d*x/2)**9/(105*a*d*tan(c/2 + d*x/2)**16 +
840*a*d*tan(c/2 + d*x/2)**14 + 2940*a*d*tan(c/2 + d*x/2)**12 + 5880*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/
2 + d*x/2)**8 + 5880*a*d*tan(c/2 + d*x/2)**6 + 2940*a*d*tan(c/2 + d*x/2)**4 + 840*a*d*tan(c/2 + d*x/2)**2 + 10
5*a*d) - 1400*tan(c/2 + d*x/2)**8/(105*a*d*tan(c/2 + d*x/2)**16 + 840*a*d*tan(c/2 + d*x/2)**14 + 2940*a*d*tan(
c/2 + d*x/2)**12 + 5880*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 5880*a*d*tan(c/2 + d*x/2)**6
+ 2940*a*d*tan(c/2 + d*x/2)**4 + 840*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) + 688*tan(c/2 + d*x/2)**7/(105*a*d*ta
n(c/2 + d*x/2)**16 + 840*a*d*tan(c/2 + d*x/2)**14 + 2940*a*d*tan(c/2 + d*x/2)**12 + 5880*a*d*tan(c/2 + d*x/2)*
*10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 5880*a*d*tan(c/2 + d*x/2)**6 + 2940*a*d*tan(c/2 + d*x/2)**4 + 840*a*d*tan
(c/2 + d*x/2)**2 + 105*a*d) + 560*tan(c/2 + d*x/2)**6/(105*a*d*tan(c/2 + d*x/2)**16 + 840*a*d*tan(c/2 + d*x/2)
**14 + 2940*a*d*tan(c/2 + d*x/2)**12 + 5880*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 5880*a*d
*tan(c/2 + d*x/2)**6 + 2940*a*d*tan(c/2 + d*x/2)**4 + 840*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) + 56*tan(c/2 + d*
x/2)**5/(105*a*d*tan(c/2 + d*x/2)**16 + 840*a*d*tan(c/2 + d*x/2)**14 + 2940*a*d*tan(c/2 + d*x/2)**12 + 5880*a*
d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 5880*a*d*tan(c/2 + d*x/2)**6 + 2940*a*d*tan(c/2 + d*x/
2)**4 + 840*a*d*tan(c/2 + d*x/2)**2 + 105*a*d) - 420*tan(c/2 + d*x/2)**4/(105*a*d*tan(c/2 + d*x/2)**16 + 840*a
*d*tan(c/2 + d*x/2)**14 + 2940*a*d*tan(c/2 + d*x/2)**12 + 5880*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d
*x/2)**8 + 5880*a*d*tan(c/2 + d*x/2)**6 + 2940*a*d*tan(c/2 + d*x/2)**4 + 840*a*d*tan(c/2 + d*x/2)**2 + 105*a*d
) + 280*tan(c/2 + d*x/2)**3/(105*a*d*tan(c/2 + d*x/2)**16 + 840*a*d*tan(c/2 + d*x/2)**14 + 2940*a*d*tan(c/2 +
d*x/2)**12 + 5880*a*d*tan(c/2 + d*x/2)**10 + 7350*a*d*tan(c/2 + d*x/2)**8 + 5880*a*d*tan(c/2 + d*x/2)**6 + 294
0*a*d*tan(c/2 + d*x/2)**4 + 840*a*d*tan(c/2 + d*x/2)**2 + 105*a*d), Ne(d, 0)), (x*sin(c)**2*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
\[
\int \frac {\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {105 \, \sin \left (d x + c\right )^{8} - 120 \, \sin \left (d x + c\right )^{7} - 280 \, \sin \left (d x + c\right )^{6} + 336 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} - 280 \, \sin \left (d x + c\right )^{3}}{840 \, a d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
-1/840*(105*sin(d*x + c)^8 - 120*sin(d*x + c)^7 - 280*sin(d*x + c)^6 + 336*sin(d*x + c)^5 + 210*sin(d*x + c)^4
- 280*sin(d*x + c)^3)/(a*d)
Giac [A] (verification not implemented)
none
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.76
\[
\int \frac {\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {105 \, \sin \left (d x + c\right )^{8} - 120 \, \sin \left (d x + c\right )^{7} - 280 \, \sin \left (d x + c\right )^{6} + 336 \, \sin \left (d x + c\right )^{5} + 210 \, \sin \left (d x + c\right )^{4} - 280 \, \sin \left (d x + c\right )^{3}}{840 \, a d}
\]
[In]
integrate(cos(d*x+c)^7*sin(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
-1/840*(105*sin(d*x + c)^8 - 120*sin(d*x + c)^7 - 280*sin(d*x + c)^6 + 336*sin(d*x + c)^5 + 210*sin(d*x + c)^4
- 280*sin(d*x + c)^3)/(a*d)
Mupad [B] (verification not implemented)
Time = 10.65 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.91
\[
\int \frac {\cos ^7(c+d x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {{\sin \left (c+d\,x\right )}^3}{3\,a}-\frac {{\sin \left (c+d\,x\right )}^4}{4\,a}-\frac {2\,{\sin \left (c+d\,x\right )}^5}{5\,a}+\frac {{\sin \left (c+d\,x\right )}^6}{3\,a}+\frac {{\sin \left (c+d\,x\right )}^7}{7\,a}-\frac {{\sin \left (c+d\,x\right )}^8}{8\,a}}{d}
\]
[In]
int((cos(c + d*x)^7*sin(c + d*x)^2)/(a + a*sin(c + d*x)),x)
[Out]
(sin(c + d*x)^3/(3*a) - sin(c + d*x)^4/(4*a) - (2*sin(c + d*x)^5)/(5*a) + sin(c + d*x)^6/(3*a) + sin(c + d*x)^
7/(7*a) - sin(c + d*x)^8/(8*a))/d