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

Optimal. Leaf size=102 \[ \frac{(A-5 B) (a \sin (c+d x)+a)^6}{6 a^5 d}-\frac{4 (A-2 B) (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{a^3 d}+\frac{B (a \sin (c+d x)+a)^7}{7 a^6 d} \]

[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)

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Rubi [A]  time = 0.10771, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2836, 77} \[ \frac{(A-5 B) (a \sin (c+d x)+a)^6}{6 a^5 d}-\frac{4 (A-2 B) (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac{(A-B) (a \sin (c+d x)+a)^4}{a^3 d}+\frac{B (a \sin (c+d x)+a)^7}{7 a^6 d} \]

Antiderivative was successfully verified.

[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 2836

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*x)/b
)^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]

Rule 77

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]))))

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac{\operatorname{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{\operatorname{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]  time = 0.67241, size = 130, normalized size = 1.27 \[ -\frac{a (525 (A+B) \cos (2 (c+d x))+210 (A+B) \cos (4 (c+d x))-4200 A \sin (c+d x)-700 A \sin (3 (c+d x))-84 A \sin (5 (c+d x))+35 A \cos (6 (c+d x))-525 B \sin (c+d x)+35 B \sin (3 (c+d x))+63 B \sin (5 (c+d x))+15 B \sin (7 (c+d x))+35 B \cos (6 (c+d x)))}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(525*(A + B)*Cos[2*(c + d*x)] + 210*(A + B)*Cos[4*(c + d*x)] + 35*A*Cos[6*(c + d*x)] + 35*B*Cos[6*(c + d*x
)] - 4200*A*Sin[c + d*x] - 525*B*Sin[c + d*x] - 700*A*Sin[3*(c + d*x)] + 35*B*Sin[3*(c + d*x)] - 84*A*Sin[5*(c
+ d*x)] + 63*B*Sin[5*(c + d*x)] + 15*B*Sin[7*(c + d*x)]))/(6720*d)

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Maple [A]  time = 0.059, size = 108, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( aB \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{\sin \left ( dx+c \right ) }{35} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) -{\frac{aA \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6}}-{\frac{aB \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6}}+{\frac{aA\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a*B*(-1/7*sin(d*x+c)*cos(d*x+c)^6+1/35*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-1/6*a*A*cos(d*x+c)
^6-1/6*a*B*cos(d*x+c)^6+1/5*a*A*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))

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Maxima [A]  time = 1.05107, size = 140, normalized size = 1.37 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.79138, size = 215, normalized size = 2.11 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 7.57911, size = 178, normalized size = 1.75 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.27205, size = 196, normalized size = 1.92 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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