\(\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx\) [567]
Optimal result
Integrand size = 27, antiderivative size = 123 \[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {a \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {2 a \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {2 a \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {a \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {a \sin ^{6+n}(c+d x)}{d (6+n)}
\]
[Out]
a*sin(d*x+c)^(1+n)/d/(1+n)+a*sin(d*x+c)^(2+n)/d/(2+n)-2*a*sin(d*x+c)^(3+n)/d/(3+n)-2*a*sin(d*x+c)^(4+n)/d/(4+n
)+a*sin(d*x+c)^(5+n)/d/(5+n)+a*sin(d*x+c)^(6+n)/d/(6+n)
Rubi [A] (verified)
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2915, 90}
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {2 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {2 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {a \sin ^{n+6}(c+d x)}{d (n+6)}
\]
[In]
Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]
[Out]
(a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (a*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (2*a*Sin[c + d*x]^(3 + n))/(d*(3
+ n)) - (2*a*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (a*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (a*Sin[c + d*x]^(6 +
n))/(d*(6 + n))
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 (a-x)^2 \left (\frac {x}{a}\right )^n (a+x)^3 \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \left (a^5 \left (\frac {x}{a}\right )^n+a^5 \left (\frac {x}{a}\right )^{1+n}-2 a^5 \left (\frac {x}{a}\right )^{2+n}-2 a^5 \left (\frac {x}{a}\right )^{3+n}+a^5 \left (\frac {x}{a}\right )^{4+n}+a^5 \left (\frac {x}{a}\right )^{5+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {a \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {a \sin ^{2+n}(c+d x)}{d (2+n)}-\frac {2 a \sin ^{3+n}(c+d x)}{d (3+n)}-\frac {2 a \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {a \sin ^{5+n}(c+d x)}{d (5+n)}+\frac {a \sin ^{6+n}(c+d x)}{d (6+n)} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.00 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.99
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \sin ^{1+n}(c+d x) (1+\sin (c+d x)) \left (\frac {1}{1+n}+\frac {\sin (c+d x)}{2+n}-\frac {2 \sin ^2(c+d x)}{3+n}-\frac {2 \sin ^3(c+d x)}{4+n}+\frac {\sin ^4(c+d x)}{5+n}+\frac {\sin ^5(c+d x)}{6+n}\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}
\]
[In]
Integrate[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]
[Out]
(a*Sin[c + d*x]^(1 + n)*(1 + Sin[c + d*x])*((1 + n)^(-1) + Sin[c + d*x]/(2 + n) - (2*Sin[c + d*x]^2)/(3 + n) -
(2*Sin[c + d*x]^3)/(4 + n) + Sin[c + d*x]^4/(5 + n) + Sin[c + d*x]^5/(6 + n)))/(d*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^2)
Maple [A] (verified)
Time = 3.97 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.38
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {a \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}+\frac {a \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {a \left (\sin ^{6}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}-\frac {2 a \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}-\frac {2 a \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}\) |
\(170\) |
| default |
\(\frac {a \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}+\frac {a \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {a \left (\sin ^{6}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (6+n \right )}-\frac {2 a \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}-\frac {2 a \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}\) |
\(170\) |
| parallelrisch |
\(\frac {3 \left (\frac {\left (5+n \right ) \left (3+n \right ) \left (1+n \right ) \left (n^{2}+6 n -120\right ) \cos \left (2 d x +2 c \right )}{6}-\frac {\left (n +12\right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (4 d x +4 c \right )}{3}-\frac {\left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \cos \left (6 d x +6 c \right )}{6}+\left (n +\frac {25}{3}\right ) \left (1+n \right ) \left (6+n \right ) \left (2+n \right ) \left (4+n \right ) \sin \left (3 d x +3 c \right )+\frac {\left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) \sin \left (5 d x +5 c \right )}{3}+\left (2400+2584 n +984 n^{2}+\frac {526}{3} n^{3}+16 n^{4}+\frac {2}{3} n^{5}\right ) \sin \left (d x +c \right )+\frac {\left (5+n \right ) \left (3+n \right ) \left (1+n \right ) \left (n^{2}+14 n +88\right )}{3}\right ) \left (\sin ^{n}\left (d x +c \right )\right ) a}{16 \left (n^{3}+12 n^{2}+44 n +48\right ) d \left (n^{3}+9 n^{2}+23 n +15\right )}\) |
\(228\) |
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[In]
int(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
[Out]
a/d/(1+n)*sin(d*x+c)*exp(n*ln(sin(d*x+c)))+a/d/(2+n)*sin(d*x+c)^2*exp(n*ln(sin(d*x+c)))+a/d/(5+n)*sin(d*x+c)^5
*exp(n*ln(sin(d*x+c)))+a/d/(6+n)*sin(d*x+c)^6*exp(n*ln(sin(d*x+c)))-2*a/d/(3+n)*sin(d*x+c)^3*exp(n*ln(sin(d*x+
c)))-2*a/d/(4+n)*sin(d*x+c)^4*exp(n*ln(sin(d*x+c)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (123) = 246\).
Time = 0.29 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.29
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {{\left ({\left (a n^{5} + 15 \, a n^{4} + 85 \, a n^{3} + 225 \, a n^{2} + 274 \, a n + 120 \, a\right )} \cos \left (d x + c\right )^{6} - {\left (a n^{5} + 11 \, a n^{4} + 41 \, a n^{3} + 61 \, a n^{2} + 30 \, a n\right )} \cos \left (d x + c\right )^{4} - 8 \, a n^{3} - 72 \, a n^{2} - 4 \, {\left (a n^{4} + 9 \, a n^{3} + 23 \, a n^{2} + 15 \, a n\right )} \cos \left (d x + c\right )^{2} - 184 \, a n - {\left ({\left (a n^{5} + 16 \, a n^{4} + 95 \, a n^{3} + 260 \, a n^{2} + 324 \, a n + 144 \, a\right )} \cos \left (d x + c\right )^{4} + 8 \, a n^{3} + 96 \, a n^{2} + 4 \, {\left (a n^{4} + 13 \, a n^{3} + 56 \, a n^{2} + 92 \, a n + 48 \, a\right )} \cos \left (d x + c\right )^{2} + 352 \, a n + 384 \, a\right )} \sin \left (d x + c\right ) - 120 \, a\right )} \sin \left (d x + c\right )^{n}}{d n^{6} + 21 \, d n^{5} + 175 \, d n^{4} + 735 \, d n^{3} + 1624 \, d n^{2} + 1764 \, d n + 720 \, d}
\]
[In]
integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="fricas")
[Out]
-((a*n^5 + 15*a*n^4 + 85*a*n^3 + 225*a*n^2 + 274*a*n + 120*a)*cos(d*x + c)^6 - (a*n^5 + 11*a*n^4 + 41*a*n^3 +
61*a*n^2 + 30*a*n)*cos(d*x + c)^4 - 8*a*n^3 - 72*a*n^2 - 4*(a*n^4 + 9*a*n^3 + 23*a*n^2 + 15*a*n)*cos(d*x + c)^
2 - 184*a*n - ((a*n^5 + 16*a*n^4 + 95*a*n^3 + 260*a*n^2 + 324*a*n + 144*a)*cos(d*x + c)^4 + 8*a*n^3 + 96*a*n^2
+ 4*(a*n^4 + 13*a*n^3 + 56*a*n^2 + 92*a*n + 48*a)*cos(d*x + c)^2 + 352*a*n + 384*a)*sin(d*x + c) - 120*a)*sin
(d*x + c)^n/(d*n^6 + 21*d*n^5 + 175*d*n^4 + 735*d*n^3 + 1624*d*n^2 + 1764*d*n + 720*d)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 8675 vs. \(2 (104) = 208\).
Time = 8.35 (sec) , antiderivative size = 8675, normalized size of antiderivative = 70.53
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)**5*sin(d*x+c)**n*(a+a*sin(d*x+c)),x)
[Out]
Piecewise((x*(a*sin(c) + a)*sin(c)**n*cos(c)**5, Eq(d, 0)), (a*log(sin(c + d*x))/d - 8*a/(15*d*sin(c + d*x)) +
a*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) + 4*a*cos(c + d*x)**2/(15*d*sin(c + d*x)**3) - a*cos(c + d*x)**4/(4*d
*sin(c + d*x)**4) - a*cos(c + d*x)**4/(5*d*sin(c + d*x)**5), Eq(n, -6)), (a*log(sin(c + d*x))/d + 8*a*sin(c +
d*x)/(3*d) + 4*a*cos(c + d*x)**2/(3*d*sin(c + d*x)) + a*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) - a*cos(c + d*x)
**4/(3*d*sin(c + d*x)**3) - a*cos(c + d*x)**4/(4*d*sin(c + d*x)**4), Eq(n, -5)), (48*a*log(tan(c/2 + d*x/2)**2
+ 1)*tan(c/2 + d*x/2)**7/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 9
6*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**5/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24
*d*tan(c/2 + d*x/2)**3) + 48*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**3/(24*d*tan(c/2 + d*x/2)**7 + 48
*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 48*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7/(24*d*tan(
c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 96*a*log(tan(c/2 + d*x/2))*tan(c/2 +
d*x/2)**5/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 48*a*log(tan(c/2
+ d*x/2))*tan(c/2 + d*x/2)**3/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3)
- a*tan(c/2 + d*x/2)**10/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 3
*a*tan(c/2 + d*x/2)**9/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 19*a
*tan(c/2 + d*x/2)**8/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 110*a*
tan(c/2 + d*x/2)**6/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 54*a*ta
n(c/2 + d*x/2)**5/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 110*a*tan
(c/2 + d*x/2)**4/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 19*a*tan(c
/2 + d*x/2)**2/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 3*a*tan(c/2
+ d*x/2)/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - a/(24*d*tan(c/2 +
d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3), Eq(n, -4)), (48*a*log(tan(c/2 + d*x/2)**2 +
1)*tan(c/2 + d*x/2)**8/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*
tan(c/2 + d*x/2)**2) + 144*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(24*d*tan(c/2 + d*x/2)**8 + 72*d
*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) + 144*a*log(tan(c/2 + d*x/2)**2 +
1)*tan(c/2 + d*x/2)**4/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*
tan(c/2 + d*x/2)**2) + 48*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**2/(24*d*tan(c/2 + d*x/2)**8 + 72*d*
tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 48*a*log(tan(c/2 + d*x/2))*tan(c/
2 + d*x/2)**8/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 +
d*x/2)**2) - 144*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**6/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2
)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 144*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/
(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) -
48*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan
(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 3*a*tan(c/2 + d*x/2)**10/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c
/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 12*a*tan(c/2 + d*x/2)**9/(24*d*tan(c/2
+ d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 144*a*tan(c/2
+ d*x/2)**7/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 +
d*x/2)**2) + 63*a*tan(c/2 + d*x/2)**6/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*
x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 200*a*tan(c/2 + d*x/2)**5/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x
/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) + 63*a*tan(c/2 + d*x/2)**4/(24*d*tan(c/2 + d*x/2
)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 144*a*tan(c/2 + d*x/2
)**3/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**
2) - 12*a*tan(c/2 + d*x/2)/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 2
4*d*tan(c/2 + d*x/2)**2) - 3*a/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4
+ 24*d*tan(c/2 + d*x/2)**2), Eq(n, -3)), (-6*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**9/(6*d*tan(c/2
+ d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d
*x/2)) - 24*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**7/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2
)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 36*a*log(tan(c/2 + d*x/2)
**2 + 1)*tan(c/2 + d*x/2)**5/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 +
24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 24*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**3/(6*d*
tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan
(c/2 + d*x/2)) - 6*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d
*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) + 6*a*log(tan(c/2 + d*x
/2))*tan(c/2 + d*x/2)**9/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d
*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) + 24*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7/(6*d*tan(c/2 + d
*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/
2)) + 36*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**5/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*
d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) + 24*a*log(tan(c/2 + d*x/2))*tan(c/2
+ d*x/2)**3/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*
x/2)**3 + 6*d*tan(c/2 + d*x/2)) + 6*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)/(6*d*tan(c/2 + d*x/2)**9 + 24*d*t
an(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 3*a*tan(c/2
+ d*x/2)**10/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 +
d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 39*a*tan(c/2 + d*x/2)**8/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)*
*7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 24*a*tan(c/2 + d*x/2)**7/(6
*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*
tan(c/2 + d*x/2)) - 86*a*tan(c/2 + d*x/2)**6/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/
2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 24*a*tan(c/2 + d*x/2)**5/(6*d*tan(c/2 + d*x
/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)
) - 86*a*tan(c/2 + d*x/2)**4/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 +
24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 24*a*tan(c/2 + d*x/2)**3/(6*d*tan(c/2 + d*x/2)**9 + 24*d*ta
n(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 39*a*tan(c/2
+ d*x/2)**2/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d
*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 3*a/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*
x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)), Eq(n, -2)), (-15*a*log(tan(c/2 + d*x/2)**2 + 1)*ta
n(c/2 + d*x/2)**10/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*t
an(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 75*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**8/
(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4
+ 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 150*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(15*d*tan(c/2 + d*
x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d
*x/2)**2 + 15*d) - 150*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**4/(15*d*tan(c/2 + d*x/2)**10 + 75*d*ta
n(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) -
75*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**2/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 +
150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 15*a*log(tan(c/2 +
d*x/2)**2 + 1)/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(
c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 15*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**10/(15*d*tan
(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*ta
n(c/2 + d*x/2)**2 + 15*d) + 75*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**8/(15*d*tan(c/2 + d*x/2)**10 + 75*d*t
an(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d)
+ 150*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**6/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*
d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 150*a*log(tan(c/2 + d*x
/2))*tan(c/2 + d*x/2)**4/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 1
50*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 75*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(
15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 +
75*d*tan(c/2 + d*x/2)**2 + 15*d) + 15*a*log(tan(c/2 + d*x/2))/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x
/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 30*a*tan(c
/2 + d*x/2)**9/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c
/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 60*a*tan(c/2 + d*x/2)**8/(15*d*tan(c/2 + d*x/2)**10 + 75*d
*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d
) + 40*a*tan(c/2 + d*x/2)**7/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6
+ 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 120*a*tan(c/2 + d*x/2)**6/(15*d*tan(c/2 + d*
x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d
*x/2)**2 + 15*d) + 116*a*tan(c/2 + d*x/2)**5/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan
(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 120*a*tan(c/2 + d*x/2)**4/(1
5*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 +
75*d*tan(c/2 + d*x/2)**2 + 15*d) + 40*a*tan(c/2 + d*x/2)**3/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)
**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 60*a*tan(c/2
+ d*x/2)**2/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2
+ d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 30*a*tan(c/2 + d*x/2)/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c
/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d), Eq(
n, -1)), (a*n**5*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3
+ 1624*d*n**2 + 1764*d*n + 720*d) + a*n**5*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 +
175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 4*a*n**4*sin(c + d*x)**4*sin(c + d*x)**n*cos(c +
d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 4*a*n**4*sin(c + d*x
)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n +
720*d) + 19*a*n**4*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n*
*3 + 1624*d*n**2 + 1764*d*n + 720*d) + 20*a*n**4*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n
**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 8*a*n**3*sin(c + d*x)**6*sin(c + d*x)**n/(d*
n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 8*a*n**3*sin(c + d*x)**5*sin(c
+ d*x)**n/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 60*a*n**3*sin(c +
d*x)**4*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n
+ 720*d) + 68*a*n**3*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d
*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 137*a*n**3*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 +
21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 155*a*n**3*sin(c + d*x)*sin(c + d*x)*
*n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 72*a*n**2
*sin(c + d*x)**6*sin(c + d*x)**n/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*
d) + 96*a*n**2*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1
764*d*n + 720*d) + 308*a*n**2*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4
+ 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 416*a*n**2*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(
d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 461*a*n**2*sin(c + d*x)**2*si
n(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d)
+ 580*a*n**2*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624
*d*n**2 + 1764*d*n + 720*d) + 184*a*n*sin(c + d*x)**6*sin(c + d*x)**n/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d
*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 352*a*n*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**6 + 21*d*n**5 + 175*d*
n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 612*a*n*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**2/
(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 1072*a*n*sin(c + d*x)**3*sin
(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) +
702*a*n*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*
d*n**2 + 1764*d*n + 720*d) + 1044*a*n*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d
*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 120*a*sin(c + d*x)**6*sin(c + d*x)**n/(d*n**6 + 21*d*n*
*5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 384*a*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**6
+ 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 360*a*sin(c + d*x)**4*sin(c + d*x)*
*n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 960*a*sin
(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 176
4*d*n + 720*d) + 360*a*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*
d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 720*a*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n
**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d), True))
Maxima [A] (verification not implemented)
none
Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.89
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {\frac {a \sin \left (d x + c\right )^{n + 6}}{n + 6} + \frac {a \sin \left (d x + c\right )^{n + 5}}{n + 5} - \frac {2 \, a \sin \left (d x + c\right )^{n + 4}}{n + 4} - \frac {2 \, a \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {a \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d}
\]
[In]
integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="maxima")
[Out]
(a*sin(d*x + c)^(n + 6)/(n + 6) + a*sin(d*x + c)^(n + 5)/(n + 5) - 2*a*sin(d*x + c)^(n + 4)/(n + 4) - 2*a*sin(
d*x + c)^(n + 3)/(n + 3) + a*sin(d*x + c)^(n + 2)/(n + 2) + a*sin(d*x + c)^(n + 1)/(n + 1))/d
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (123) = 246\).
Time = 0.33 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.08
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {\frac {{\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 6 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 8 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - 16 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} - 24 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 10 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 24 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}\right )} a}{n^{3} + 12 \, n^{2} + 44 \, n + 48} + \frac {{\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 4 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - 12 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) - 10 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 8 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 15 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )\right )} a}{n^{3} + 9 \, n^{2} + 23 \, n + 15}}{d}
\]
[In]
integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="giac")
[Out]
((n^2*sin(d*x + c)^n*sin(d*x + c)^6 + 6*n*sin(d*x + c)^n*sin(d*x + c)^6 - 2*n^2*sin(d*x + c)^n*sin(d*x + c)^4
+ 8*sin(d*x + c)^n*sin(d*x + c)^6 - 16*n*sin(d*x + c)^n*sin(d*x + c)^4 + n^2*sin(d*x + c)^n*sin(d*x + c)^2 - 2
4*sin(d*x + c)^n*sin(d*x + c)^4 + 10*n*sin(d*x + c)^n*sin(d*x + c)^2 + 24*sin(d*x + c)^n*sin(d*x + c)^2)*a/(n^
3 + 12*n^2 + 44*n + 48) + (n^2*sin(d*x + c)^n*sin(d*x + c)^5 + 4*n*sin(d*x + c)^n*sin(d*x + c)^5 - 2*n^2*sin(d
*x + c)^n*sin(d*x + c)^3 + 3*sin(d*x + c)^n*sin(d*x + c)^5 - 12*n*sin(d*x + c)^n*sin(d*x + c)^3 + n^2*sin(d*x
+ c)^n*sin(d*x + c) - 10*sin(d*x + c)^n*sin(d*x + c)^3 + 8*n*sin(d*x + c)^n*sin(d*x + c) + 15*sin(d*x + c)^n*s
in(d*x + c))*a/(n^3 + 9*n^2 + 23*n + 15))/d
Mupad [B] (verification not implemented)
Time = 16.24 (sec) , antiderivative size = 550, normalized size of antiderivative = 4.47
\[
\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a\,{\sin \left (c+d\,x\right )}^n\,\left (n^5+23\,n^4+237\,n^3+1129\,n^2+2234\,n+1320\right )}{16\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (6\,c+6\,d\,x\right )\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{32\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (4\,c+4\,d\,x\right )\,\left (n^5+23\,n^4+173\,n^3+553\,n^2+762\,n+360\right )}{16\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,\sin \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n\,\left (n^5\,1{}\mathrm {i}+n^4\,24{}\mathrm {i}+n^3\,263{}\mathrm {i}+n^2\,1476{}\mathrm {i}+n\,3876{}\mathrm {i}+3600{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (2\,c+2\,d\,x\right )\,\left (-n^5-15\,n^4+43\,n^3+927\,n^2+2670\,n+1800\right )}{32\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\sin \left (5\,c+5\,d\,x\right )\,\left (n^5\,1{}\mathrm {i}+n^4\,16{}\mathrm {i}+n^3\,95{}\mathrm {i}+n^2\,260{}\mathrm {i}+n\,324{}\mathrm {i}+144{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\sin \left (3\,c+3\,d\,x\right )\,\left (n^5\,3{}\mathrm {i}+n^4\,64{}\mathrm {i}+n^3\,493{}\mathrm {i}+n^2\,1676{}\mathrm {i}+n\,2444{}\mathrm {i}+1200{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}
\]
[In]
int(cos(c + d*x)^5*sin(c + d*x)^n*(a + a*sin(c + d*x)),x)
[Out]
(a*sin(c + d*x)^n*(2234*n + 1129*n^2 + 237*n^3 + 23*n^4 + n^5 + 1320))/(16*d*(1764*n + 1624*n^2 + 735*n^3 + 17
5*n^4 + 21*n^5 + n^6 + 720)) - (a*sin(c + d*x)^n*cos(6*c + 6*d*x)*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 1
20))/(32*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (a*sin(c + d*x)^n*cos(4*c + 4*d*x)*
(762*n + 553*n^2 + 173*n^3 + 23*n^4 + n^5 + 360))/(16*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6
+ 720)) - (a*sin(c + d*x)*sin(c + d*x)^n*(n*3876i + n^2*1476i + n^3*263i + n^4*24i + n^5*1i + 3600i)*1i)/(8*d*
(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (a*sin(c + d*x)^n*cos(2*c + 2*d*x)*(2670*n + 9
27*n^2 + 43*n^3 - 15*n^4 - n^5 + 1800))/(32*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) -
(a*sin(c + d*x)^n*sin(5*c + 5*d*x)*(n*324i + n^2*260i + n^3*95i + n^4*16i + n^5*1i + 144i)*1i)/(16*d*(1764*n +
1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (a*sin(c + d*x)^n*sin(3*c + 3*d*x)*(n*2444i + n^2*1676i
+ n^3*493i + n^4*64i + n^5*3i + 1200i)*1i)/(16*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)
)