3.9 \(\int \cos ^6(c+d x) (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=117 \[ \frac{(8 A+7 C) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 (8 A+7 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} x (8 A+7 C)+\frac{C \sin (c+d x) \cos ^7(c+d x)}{8 d} \]

[Out]

(5*(8*A + 7*C)*x)/128 + (5*(8*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*(8*A + 7*C)*Cos[c + d*x]^3*Sin[
c + d*x])/(192*d) + ((8*A + 7*C)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (C*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)

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Rubi [A]  time = 0.0670029, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3014, 2635, 8} \[ \frac{(8 A+7 C) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 (8 A+7 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} x (8 A+7 C)+\frac{C \sin (c+d x) \cos ^7(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(A + C*Cos[c + d*x]^2),x]

[Out]

(5*(8*A + 7*C)*x)/128 + (5*(8*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*(8*A + 7*C)*Cos[c + d*x]^3*Sin[
c + d*x])/(192*d) + ((8*A + 7*C)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (C*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)

Rule 3014

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[
e + f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[(A*(m + 2) + C*(m + 1))/(m + 2), Int[(b*Sin[e + f*
x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !LtQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^6(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{8} (8 A+7 C) \int \cos ^6(c+d x) \, dx\\ &=\frac{(8 A+7 C) \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{C \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{48} (5 (8 A+7 C)) \int \cos ^4(c+d x) \, dx\\ &=\frac{5 (8 A+7 C) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{(8 A+7 C) \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{C \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{64} (5 (8 A+7 C)) \int \cos ^2(c+d x) \, dx\\ &=\frac{5 (8 A+7 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 (8 A+7 C) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{(8 A+7 C) \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{C \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{128} (5 (8 A+7 C)) \int 1 \, dx\\ &=\frac{5}{128} (8 A+7 C) x+\frac{5 (8 A+7 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 (8 A+7 C) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{(8 A+7 C) \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{C \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.156183, size = 93, normalized size = 0.79 \[ \frac{48 (15 A+14 C) \sin (2 (c+d x))+24 (6 A+7 C) \sin (4 (c+d x))+16 A \sin (6 (c+d x))+960 A c+960 A d x+32 C \sin (6 (c+d x))+3 C \sin (8 (c+d x))+840 c C+840 C d x}{3072 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(A + C*Cos[c + d*x]^2),x]

[Out]

(960*A*c + 840*c*C + 960*A*d*x + 840*C*d*x + 48*(15*A + 14*C)*Sin[2*(c + d*x)] + 24*(6*A + 7*C)*Sin[4*(c + d*x
)] + 16*A*Sin[6*(c + d*x)] + 32*C*Sin[6*(c + d*x)] + 3*C*Sin[8*(c + d*x)])/(3072*d)

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Maple [A]  time = 0.037, size = 106, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( C \left ({\frac{\sin \left ( dx+c \right ) }{8} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( dx+c \right ) }{16}} \right ) }+{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) +A \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(A+C*cos(d*x+c)^2),x)

[Out]

1/d*(C*(1/8*(cos(d*x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+35/128*d*x+35/128*c
)+A*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c))

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Maxima [A]  time = 1.56528, size = 176, normalized size = 1.5 \begin{align*} \frac{15 \,{\left (d x + c\right )}{\left (8 \, A + 7 \, C\right )} + \frac{15 \,{\left (8 \, A + 7 \, C\right )} \tan \left (d x + c\right )^{7} + 55 \,{\left (8 \, A + 7 \, C\right )} \tan \left (d x + c\right )^{5} + 73 \,{\left (8 \, A + 7 \, C\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (88 \, A + 93 \, C\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{8} + 4 \, \tan \left (d x + c\right )^{6} + 6 \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{2} + 1}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(A+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

1/384*(15*(d*x + c)*(8*A + 7*C) + (15*(8*A + 7*C)*tan(d*x + c)^7 + 55*(8*A + 7*C)*tan(d*x + c)^5 + 73*(8*A + 7
*C)*tan(d*x + c)^3 + 3*(88*A + 93*C)*tan(d*x + c))/(tan(d*x + c)^8 + 4*tan(d*x + c)^6 + 6*tan(d*x + c)^4 + 4*t
an(d*x + c)^2 + 1))/d

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Fricas [A]  time = 1.69715, size = 216, normalized size = 1.85 \begin{align*} \frac{15 \,{\left (8 \, A + 7 \, C\right )} d x +{\left (48 \, C \cos \left (d x + c\right )^{7} + 8 \,{\left (8 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{5} + 10 \,{\left (8 \, A + 7 \, C\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (8 \, A + 7 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(A+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/384*(15*(8*A + 7*C)*d*x + (48*C*cos(d*x + c)^7 + 8*(8*A + 7*C)*cos(d*x + c)^5 + 10*(8*A + 7*C)*cos(d*x + c)^
3 + 15*(8*A + 7*C)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 19.1781, size = 354, normalized size = 3.03 \begin{align*} \begin{cases} \frac{5 A x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{15 A x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{15 A x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{5 A x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{5 A \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{5 A \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac{11 A \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac{35 C x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{35 C x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{105 C x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{35 C x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{35 C x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{35 C \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{385 C \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac{511 C \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac{93 C \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text{for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \cos ^{6}{\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)**6*(A+C*cos(d*x+c)**2),x)

[Out]

Piecewise((5*A*x*sin(c + d*x)**6/16 + 15*A*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*A*x*sin(c + d*x)**2*cos(c
+ d*x)**4/16 + 5*A*x*cos(c + d*x)**6/16 + 5*A*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*A*sin(c + d*x)**3*cos(c
+ d*x)**3/(6*d) + 11*A*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 35*C*x*sin(c + d*x)**8/128 + 35*C*x*sin(c + d*x)
**6*cos(c + d*x)**2/32 + 105*C*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 35*C*x*sin(c + d*x)**2*cos(c + d*x)**6/3
2 + 35*C*x*cos(c + d*x)**8/128 + 35*C*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 385*C*sin(c + d*x)**5*cos(c + d*x
)**3/(384*d) + 511*C*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) + 93*C*sin(c + d*x)*cos(c + d*x)**7/(128*d), Ne(d
, 0)), (x*(A + C*cos(c)**2)*cos(c)**6, True))

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Giac [A]  time = 1.15112, size = 117, normalized size = 1. \begin{align*} \frac{5}{128} \,{\left (8 \, A + 7 \, C\right )} x + \frac{C \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{{\left (A + 2 \, C\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (6 \, A + 7 \, C\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{{\left (15 \, A + 14 \, C\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(A+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

5/128*(8*A + 7*C)*x + 1/1024*C*sin(8*d*x + 8*c)/d + 1/192*(A + 2*C)*sin(6*d*x + 6*c)/d + 1/128*(6*A + 7*C)*sin
(4*d*x + 4*c)/d + 1/64*(15*A + 14*C)*sin(2*d*x + 2*c)/d