3.2 \(\int \cos ^5(c+d x) (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=72 \[ \frac{(A+3 C) \sin ^5(c+d x)}{5 d}-\frac{(2 A+3 C) \sin ^3(c+d x)}{3 d}+\frac{(A+C) \sin (c+d x)}{d}-\frac{C \sin ^7(c+d x)}{7 d} \]

[Out]

((A + C)*Sin[c + d*x])/d - ((2*A + 3*C)*Sin[c + d*x]^3)/(3*d) + ((A + 3*C)*Sin[c + d*x]^5)/(5*d) - (C*Sin[c +
d*x]^7)/(7*d)

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Rubi [A]  time = 0.0656366, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3013, 373} \[ \frac{(A+3 C) \sin ^5(c+d x)}{5 d}-\frac{(2 A+3 C) \sin ^3(c+d x)}{3 d}+\frac{(A+C) \sin (c+d x)}{d}-\frac{C \sin ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(A + C*Cos[c + d*x]^2),x]

[Out]

((A + C)*Sin[c + d*x])/d - ((2*A + 3*C)*Sin[c + d*x]^3)/(3*d) + ((A + 3*C)*Sin[c + d*x]^5)/(5*d) - (C*Sin[c +
d*x]^7)/(7*d)

Rule 3013

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Dist[f^(-1), Subst[I
nt[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2
, 0]

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \cos ^5(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right )^2 \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (A \left (1+\frac{C}{A}\right )-(2 A+3 C) x^2+(A+3 C) x^4-C x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{(A+C) \sin (c+d x)}{d}-\frac{(2 A+3 C) \sin ^3(c+d x)}{3 d}+\frac{(A+3 C) \sin ^5(c+d x)}{5 d}-\frac{C \sin ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.026855, size = 101, normalized size = 1.4 \[ \frac{A \sin ^5(c+d x)}{5 d}-\frac{2 A \sin ^3(c+d x)}{3 d}+\frac{A \sin (c+d x)}{d}-\frac{C \sin ^7(c+d x)}{7 d}+\frac{3 C \sin ^5(c+d x)}{5 d}-\frac{C \sin ^3(c+d x)}{d}+\frac{C \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(A + C*Cos[c + d*x]^2),x]

[Out]

(A*Sin[c + d*x])/d + (C*Sin[c + d*x])/d - (2*A*Sin[c + d*x]^3)/(3*d) - (C*Sin[c + d*x]^3)/d + (A*Sin[c + d*x]^
5)/(5*d) + (3*C*Sin[c + d*x]^5)/(5*d) - (C*Sin[c + d*x]^7)/(7*d)

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Maple [A]  time = 0.037, size = 74, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{C\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) }+{\frac{A\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+C*cos(d*x+c)^2),x)

[Out]

1/d*(1/7*C*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c)+1/5*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 = 1.04842, size = 81, normalized size = 1.12 \begin{align*} -\frac{15 \, C \sin \left (d x + c\right )^{7} - 21 \,{\left (A + 3 \, C\right )} \sin \left (d x + c\right )^{5} + 35 \,{\left (2 \, A + 3 \, C\right )} \sin \left (d x + c\right )^{3} - 105 \,{\left (A + C\right )} \sin \left (d x + c\right )}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(A+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/105*(15*C*sin(d*x + c)^7 - 21*(A + 3*C)*sin(d*x + c)^5 + 35*(2*A + 3*C)*sin(d*x + c)^3 - 105*(A + C)*sin(d*
x + c))/d

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Fricas [A]  time = 1.61571, size = 162, normalized size = 2.25 \begin{align*} \frac{{\left (15 \, C \cos \left (d x + c\right )^{6} + 3 \,{\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{2} + 56 \, A + 48 \, C\right )} \sin \left (d x + c\right )}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(A+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/105*(15*C*cos(d*x + c)^6 + 3*(7*A + 6*C)*cos(d*x + c)^4 + 4*(7*A + 6*C)*cos(d*x + c)^2 + 56*A + 48*C)*sin(d*
x + c)/d

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Sympy [A]  time = 11.1329, size = 151, normalized size = 2.1 \begin{align*} \begin{cases} \frac{8 A \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 A \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{A \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{16 C \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 C \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 C \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{C \sin{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\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+C*cos(d*x+c)**2),x)

[Out]

Piecewise((8*A*sin(c + d*x)**5/(15*d) + 4*A*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + A*sin(c + d*x)*cos(c + d*x
)**4/d + 16*C*sin(c + d*x)**7/(35*d) + 8*C*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*C*sin(c + d*x)**3*cos(c +
d*x)**4/d + C*sin(c + d*x)*cos(c + d*x)**6/d, Ne(d, 0)), (x*(A + C*cos(c)**2)*cos(c)**5, True))

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Giac [A]  time = 1.17715, size = 103, normalized size = 1.43 \begin{align*} \frac{C \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{{\left (4 \, A + 7 \, C\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (20 \, A + 21 \, C\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{5 \,{\left (8 \, A + 7 \, C\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+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/448*C*sin(7*d*x + 7*c)/d + 1/320*(4*A + 7*C)*sin(5*d*x + 5*c)/d + 1/192*(20*A + 21*C)*sin(3*d*x + 3*c)/d + 5
/64*(8*A + 7*C)*sin(d*x + c)/d