\(\int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\) [954]

Optimal result

Integrand size = 29, antiderivative size = 102 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {(A-B) (a+a \sin (c+d x))^4}{a^3 d}-\frac {4 (A-2 B) (a+a \sin (c+d x))^5}{5 a^4 d}+\frac {(A-5 B) (a+a \sin (c+d x))^6}{6 a^5 d}+\frac {B (a+a \sin (c+d x))^7}{7 a^6 d} \]

[Out]

(A-B)*(a+a*sin(d*x+c))^4/a^3/d-4/5*(A-2*B)*(a+a*sin(d*x+c))^5/a^4/d+1/6*(A-5*B)*(a+a*sin(d*x+c))^6/a^5/d+1/7*B
*(a+a*sin(d*x+c))^7/a^6/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2915, 78} \[ \int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {B (a \sin (c+d x)+a)^7}{7 a^6 d}+\frac {(A-5 B) (a \sin (c+d x)+a)^6}{6 a^5 d}-\frac {4 (A-2 B) (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac {(A-B) (a \sin (c+d x)+a)^4}{a^3 d} \]

[In]

Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

((A - B)*(a + a*Sin[c + d*x])^4)/(a^3*d) - (4*(A - 2*B)*(a + a*Sin[c + d*x])^5)/(5*a^4*d) + ((A - 5*B)*(a + a*
Sin[c + d*x])^6)/(6*a^5*d) + (B*(a + a*Sin[c + d*x])^7)/(7*a^6*d)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 (a+x)^3 \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \left (4 a^2 (A-B) (a+x)^3-4 a (A-2 B) (a+x)^4+(A-5 B) (a+x)^5+\frac {B (a+x)^6}{a}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {(A-B) (a+a \sin (c+d x))^4}{a^3 d}-\frac {4 (A-2 B) (a+a \sin (c+d x))^5}{5 a^4 d}+\frac {(A-5 B) (a+a \sin (c+d x))^6}{6 a^5 d}+\frac {B (a+a \sin (c+d x))^7}{7 a^6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a \left ((A-B) (1+\sin (c+d x))^4-\frac {4}{5} (A-2 B) (1+\sin (c+d x))^5+\frac {1}{6} (A-5 B) (1+\sin (c+d x))^6+\frac {1}{7} B (1+\sin (c+d x))^7\right )}{d} \]

[In]

Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

(a*((A - B)*(1 + Sin[c + d*x])^4 - (4*(A - 2*B)*(1 + Sin[c + d*x])^5)/5 + ((A - 5*B)*(1 + Sin[c + d*x])^6)/6 +
(B*(1 + Sin[c + d*x])^7)/7))/d

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(\frac {a \left (\frac {\left (\sin ^{7}\left (d x +c \right )\right ) B}{7}+\frac {\left (A +B \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (A -2 B \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (-2 B -2 A \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (B -2 A \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (A +B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+A \sin \left (d x +c \right )\right )}{d}\)	\(99\)
default	\(\frac {a \left (\frac {\left (\sin ^{7}\left (d x +c \right )\right ) B}{7}+\frac {\left (A +B \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (A -2 B \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (-2 B -2 A \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (B -2 A \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (A +B \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+A \sin \left (d x +c \right )\right )}{d}\)	\(99\)
parallelrisch	\(\frac {5 \left (\frac {\left (-A -B \right ) \cos \left (2 d x +2 c \right )}{8}+\frac {\left (-A -B \right ) \cos \left (4 d x +4 c \right )}{20}+\frac {\left (-A -B \right ) \cos \left (6 d x +6 c \right )}{120}+\frac {\left (A -\frac {B}{20}\right ) \sin \left (3 d x +3 c \right )}{6}+\frac {\left (A -\frac {3 B}{4}\right ) \sin \left (5 d x +5 c \right )}{50}-\frac {B \sin \left (7 d x +7 c \right )}{280}+\left (A +\frac {B}{8}\right ) \sin \left (d x +c \right )+\frac {11 A}{60}+\frac {11 B}{60}\right ) a}{8 d}\)	\(124\)
risch	\(\frac {5 a A \sin \left (d x +c \right )}{8 d}+\frac {5 a B \sin \left (d x +c \right )}{64 d}-\frac {\sin \left (7 d x +7 c \right ) B a}{448 d}-\frac {a \cos \left (6 d x +6 c \right ) A}{192 d}-\frac {a \cos \left (6 d x +6 c \right ) B}{192 d}+\frac {\sin \left (5 d x +5 c \right ) a A}{80 d}-\frac {3 \sin \left (5 d x +5 c \right ) B a}{320 d}-\frac {a \cos \left (4 d x +4 c \right ) A}{32 d}-\frac {a \cos \left (4 d x +4 c \right ) B}{32 d}+\frac {5 a A \sin \left (3 d x +3 c \right )}{48 d}-\frac {\sin \left (3 d x +3 c \right ) B a}{192 d}-\frac {5 a \cos \left (2 d x +2 c \right ) A}{64 d}-\frac {5 a \cos \left (2 d x +2 c \right ) B}{64 d}\)	\(204\)
norman	\(\frac {\frac {\left (2 a A +2 B a \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (2 a A +2 B a \right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (2 a A +2 B a \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (2 a A +2 B a \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {5 \left (4 a A +4 B a \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {5 \left (4 a A +4 B a \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a A \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 a \left (5 A +2 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 a \left (5 A +2 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 a \left (91 A +38 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {2 a \left (113 A -16 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {2 a \left (113 A -16 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\)	\(318\)

[In]

int(cos(d*x+c)^5*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

a/d*(1/7*sin(d*x+c)^7*B+1/6*(A+B)*sin(d*x+c)^6+1/5*(A-2*B)*sin(d*x+c)^5+1/4*(-2*B-2*A)*sin(d*x+c)^4+1/3*(B-2*A
)*sin(d*x+c)^3+1/2*(A+B)*sin(d*x+c)^2+A*sin(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {35 \, {\left (A + B\right )} a \cos \left (d x + c\right )^{6} + 2 \, {\left (15 \, B a \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, A + B\right )} a \cos \left (d x + c\right )^{4} - 4 \, {\left (7 \, A + B\right )} a \cos \left (d x + c\right )^{2} - 8 \, {\left (7 \, A + B\right )} a\right )} \sin \left (d x + c\right )}{210 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/210*(35*(A + B)*a*cos(d*x + c)^6 + 2*(15*B*a*cos(d*x + c)^6 - 3*(7*A + B)*a*cos(d*x + c)^4 - 4*(7*A + B)*a*
cos(d*x + c)^2 - 8*(7*A + B)*a)*sin(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.75 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {8 A a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {A a \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {8 B a \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 B a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {B a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {B a \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a \sin {\left (c \right )} + a\right ) \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((8*A*a*sin(c + d*x)**5/(15*d) + 4*A*a*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + A*a*sin(c + d*x)*cos(c
+ d*x)**4/d - A*a*cos(c + d*x)**6/(6*d) + 8*B*a*sin(c + d*x)**7/(105*d) + 4*B*a*sin(c + d*x)**5*cos(c + d*x)*
*2/(15*d) + B*a*sin(c + d*x)**3*cos(c + d*x)**4/(3*d) - B*a*cos(c + d*x)**6/(6*d), Ne(d, 0)), (x*(A + B*sin(c)
)*(a*sin(c) + a)*cos(c)**5, True))

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {30 \, B a \sin \left (d x + c\right )^{7} + 35 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{6} + 42 \, {\left (A - 2 \, B\right )} a \sin \left (d x + c\right )^{5} - 105 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{4} - 70 \, {\left (2 \, A - B\right )} a \sin \left (d x + c\right )^{3} + 105 \, {\left (A + B\right )} a \sin \left (d x + c\right )^{2} + 210 \, A a \sin \left (d x + c\right )}{210 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/210*(30*B*a*sin(d*x + c)^7 + 35*(A + B)*a*sin(d*x + c)^6 + 42*(A - 2*B)*a*sin(d*x + c)^5 - 105*(A + B)*a*sin
(d*x + c)^4 - 70*(2*A - B)*a*sin(d*x + c)^3 + 105*(A + B)*a*sin(d*x + c)^2 + 210*A*a*sin(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.42 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {B a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (A a + B a\right )} \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {{\left (A a + B a\right )} \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {5 \, {\left (A a + B a\right )} \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (4 \, A a - 3 \, B a\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (20 \, A a - B a\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, {\left (8 \, A a + B a\right )} \sin \left (d x + c\right )}{64 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/448*B*a*sin(7*d*x + 7*c)/d - 1/192*(A*a + B*a)*cos(6*d*x + 6*c)/d - 1/32*(A*a + B*a)*cos(4*d*x + 4*c)/d - 5
/64*(A*a + B*a)*cos(2*d*x + 2*c)/d + 1/320*(4*A*a - 3*B*a)*sin(5*d*x + 5*c)/d + 1/192*(20*A*a - B*a)*sin(3*d*x
+ 3*c)/d + 5/64*(8*A*a + B*a)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {\frac {B\,a\,{\sin \left (c+d\,x\right )}^7}{7}+\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,\left (A-2\,B\right )\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^4}{2}-\frac {a\,\left (2\,A-B\right )\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {a\,\left (A+B\right )\,{\sin \left (c+d\,x\right )}^2}{2}+A\,a\,\sin \left (c+d\,x\right )}{d} \]

[In]

int(cos(c + d*x)^5*(A + B*sin(c + d*x))*(a + a*sin(c + d*x)),x)

[Out]

(A*a*sin(c + d*x) - (a*sin(c + d*x)^3*(2*A - B))/3 + (a*sin(c + d*x)^2*(A + B))/2 - (a*sin(c + d*x)^4*(A + B))
/2 + (a*sin(c + d*x)^6*(A + B))/6 + (a*sin(c + d*x)^5*(A - 2*B))/5 + (B*a*sin(c + d*x)^7)/7)/d