3.1 \(\int \cos ^7(c+d x) (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=92 \[ -\frac {(A+4 C) \sin ^7(c+d x)}{7 d}+\frac {3 (A+2 C) \sin ^5(c+d x)}{5 d}-\frac {(3 A+4 C) \sin ^3(c+d x)}{3 d}+\frac {(A+C) \sin (c+d x)}{d}+\frac {C \sin ^9(c+d x)}{9 d} \]

[Out]

(A+C)*sin(d*x+c)/d-1/3*(3*A+4*C)*sin(d*x+c)^3/d+3/5*(A+2*C)*sin(d*x+c)^5/d-1/7*(A+4*C)*sin(d*x+c)^7/d+1/9*C*si
n(d*x+c)^9/d

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Rubi [A]  time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.00, 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} \[ -\frac {(A+4 C) \sin ^7(c+d x)}{7 d}+\frac {3 (A+2 C) \sin ^5(c+d x)}{5 d}-\frac {(3 A+4 C) \sin ^3(c+d x)}{3 d}+\frac {(A+C) \sin (c+d x)}{d}+\frac {C \sin ^9(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(A + C*Cos[c + d*x]^2),x]

[Out]

((A + C)*Sin[c + d*x])/d - ((3*A + 4*C)*Sin[c + d*x]^3)/(3*d) + (3*(A + 2*C)*Sin[c + d*x]^5)/(5*d) - ((A + 4*C
)*Sin[c + d*x]^7)/(7*d) + (C*Sin[c + d*x]^9)/(9*d)

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]

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]

Rubi steps

\begin {align*} \int \cos ^7(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right )^3 \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 )-(3 A+4 C) x^2+3 (A+2 C) x^4-(A+4 C) x^6+C x^8\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {(A+C) \sin (c+d x)}{d}-\frac {(3 A+4 C) \sin ^3(c+d x)}{3 d}+\frac {3 (A+2 C) \sin ^5(c+d x)}{5 d}-\frac {(A+4 C) \sin ^7(c+d x)}{7 d}+\frac {C \sin ^9(c+d x)}{9 d}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 133, normalized size = 1.45 \[ -\frac {A \sin ^7(c+d x)}{7 d}+\frac {3 A \sin ^5(c+d x)}{5 d}-\frac {A \sin ^3(c+d x)}{d}+\frac {A \sin (c+d x)}{d}+\frac {C \sin ^9(c+d x)}{9 d}-\frac {4 C \sin ^7(c+d x)}{7 d}+\frac {6 C \sin ^5(c+d x)}{5 d}-\frac {4 C \sin ^3(c+d x)}{3 d}+\frac {C \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(A + C*Cos[c + d*x]^2),x]

[Out]

(A*Sin[c + d*x])/d + (C*Sin[c + d*x])/d - (A*Sin[c + d*x]^3)/d - (4*C*Sin[c + d*x]^3)/(3*d) + (3*A*Sin[c + d*x
]^5)/(5*d) + (6*C*Sin[c + d*x]^5)/(5*d) - (A*Sin[c + d*x]^7)/(7*d) - (4*C*Sin[c + d*x]^7)/(7*d) + (C*Sin[c + d
*x]^9)/(9*d)

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fricas [A]  time = 0.48, size = 80, normalized size = 0.87 \[ \frac {{\left (35 \, C \cos \left (d x + c\right )^{8} + 5 \, {\left (9 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (9 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (9 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{2} + 144 \, A + 128 \, C\right )} \sin \left (d x + c\right )}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(A+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/315*(35*C*cos(d*x + c)^8 + 5*(9*A + 8*C)*cos(d*x + c)^6 + 6*(9*A + 8*C)*cos(d*x + c)^4 + 8*(9*A + 8*C)*cos(d
*x + c)^2 + 144*A + 128*C)*sin(d*x + c)/d

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giac [A]  time = 0.19, size = 93, normalized size = 1.01 \[ \frac {C \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {{\left (4 \, A + 9 \, C\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (7 \, A + 9 \, C\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, {\left (A + C\right )} \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {7 \, {\left (10 \, A + 9 \, C\right )} \sin \left (d x + c\right )}{128 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(A+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/2304*C*sin(9*d*x + 9*c)/d + 1/1792*(4*A + 9*C)*sin(7*d*x + 7*c)/d + 1/320*(7*A + 9*C)*sin(5*d*x + 5*c)/d + 7
/64*(A + C)*sin(3*d*x + 3*c)/d + 7/128*(10*A + 9*C)*sin(d*x + c)/d

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maple [A]  time = 0.13, size = 94, normalized size = 1.02 \[ \frac {\frac {C \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}+\frac {A \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(A+C*cos(d*x+c)^2),x)

[Out]

1/d*(1/9*C*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*cos(d*x+c)^2)*sin(d*x+c)+1/7*A*(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))

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maxima [A]  time = 0.33, size = 75, normalized size = 0.82 \[ \frac {35 \, C \sin \left (d x + c\right )^{9} - 45 \, {\left (A + 4 \, C\right )} \sin \left (d x + c\right )^{7} + 189 \, {\left (A + 2 \, C\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (3 \, A + 4 \, C\right )} \sin \left (d x + c\right )^{3} + 315 \, {\left (A + C\right )} \sin \left (d x + c\right )}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(A+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

1/315*(35*C*sin(d*x + c)^9 - 45*(A + 4*C)*sin(d*x + c)^7 + 189*(A + 2*C)*sin(d*x + c)^5 - 105*(3*A + 4*C)*sin(
d*x + c)^3 + 315*(A + C)*sin(d*x + c))/d

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mupad [B]  time = 0.68, size = 74, normalized size = 0.80 \[ \frac {\frac {C\,{\sin \left (c+d\,x\right )}^9}{9}+\left (-\frac {A}{7}-\frac {4\,C}{7}\right )\,{\sin \left (c+d\,x\right )}^7+\left (\frac {3\,A}{5}+\frac {6\,C}{5}\right )\,{\sin \left (c+d\,x\right )}^5+\left (-A-\frac {4\,C}{3}\right )\,{\sin \left (c+d\,x\right )}^3+\left (A+C\right )\,\sin \left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^7*(A + C*cos(c + d*x)^2),x)

[Out]

((C*sin(c + d*x)^9)/9 - sin(c + d*x)^3*(A + (4*C)/3) + sin(c + d*x)*(A + C) + sin(c + d*x)^5*((3*A)/5 + (6*C)/
5) - sin(c + d*x)^7*(A/7 + (4*C)/7))/d

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sympy [A]  time = 14.91, size = 199, normalized size = 2.16 \[ \begin {cases} \frac {16 A \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 A \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 A \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {A \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac {128 C \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {64 C \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {16 C \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {8 C \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {C \sin {\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \cos ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(A+C*cos(d*x+c)**2),x)

[Out]

Piecewise((16*A*sin(c + d*x)**7/(35*d) + 8*A*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*A*sin(c + d*x)**3*cos(c
+ d*x)**4/d + A*sin(c + d*x)*cos(c + d*x)**6/d + 128*C*sin(c + d*x)**9/(315*d) + 64*C*sin(c + d*x)**7*cos(c +
d*x)**2/(35*d) + 16*C*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + 8*C*sin(c + d*x)**3*cos(c + d*x)**6/(3*d) + C*s
in(c + d*x)*cos(c + d*x)**8/d, Ne(d, 0)), (x*(A + C*cos(c)**2)*cos(c)**7, True))

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