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

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 72, 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 = {3092, 380} \begin {gather*} \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} \end {gather*}

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 380

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 3092

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 ^5(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx &=-\frac {\text {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 {\text {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*}

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Mathematica [A]
time = 0.04, size = 109, normalized size = 1.51 \begin {gather*} \frac {5 A \sin (c+d x)}{8 d}+\frac {35 C \sin (c+d x)}{64 d}+\frac {5 A \sin (3 (c+d x))}{48 d}+\frac {7 C \sin (3 (c+d x))}{64 d}+\frac {A \sin (5 (c+d x))}{80 d}+\frac {7 C \sin (5 (c+d x))}{320 d}+\frac {C \sin (7 (c+d x))}{448 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*A*Sin[c + d*x])/(8*d) + (35*C*Sin[c + d*x])/(64*d) + (5*A*Sin[3*(c + d*x)])/(48*d) + (7*C*Sin[3*(c + d*x)])
/(64*d) + (A*Sin[5*(c + d*x)])/(80*d) + (7*C*Sin[5*(c + d*x)])/(320*d) + (C*Sin[7*(c + d*x)])/(448*d)

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Maple [A]
time = 0.18, size = 74, normalized size = 1.03

method	result	size
derivativedivides	\(\frac {\frac {C \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\)	\(74\)
default	\(\frac {\frac {C \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\)	\(74\)
risch	\(\frac {5 \sin \left (d x +c \right ) A}{8 d}+\frac {35 C \sin \left (d x +c \right )}{64 d}+\frac {\sin \left (7 d x +7 c \right ) C}{448 d}+\frac {\sin \left (5 d x +5 c \right ) A}{80 d}+\frac {7 \sin \left (5 d x +5 c \right ) C}{320 d}+\frac {5 \sin \left (3 d x +3 c \right ) A}{48 d}+\frac {7 \sin \left (3 d x +3 c \right ) C}{64 d}\)	\(101\)
norman	\(\frac {\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 \left (A +C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (5 A +3 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 \left (5 A +3 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 \left (91 A +53 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {2 \left (113 A +129 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {2 \left (113 A +129 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\)	\(169\)

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,method=_RETURNVERBOSE)

[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 = 0.28, size = 60, normalized size = 0.83 \begin {gather*} -\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 {gather*}

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 = 0.36, size = 63, normalized size = 0.88 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (60) = 120\).
time = 0.58, size = 151, normalized size = 2.10 \begin {gather*} \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 {gather*}

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 = 0.42, size = 76, normalized size = 1.06 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.66, size = 59, normalized size = 0.82 \begin {gather*} -\frac {\frac {C\,{\sin \left (c+d\,x\right )}^7}{7}+\left (-\frac {A}{5}-\frac {3\,C}{5}\right )\,{\sin \left (c+d\,x\right )}^5+\left (\frac {2\,A}{3}+C\right )\,{\sin \left (c+d\,x\right )}^3+\left (-A-C\right )\,\sin \left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5*(A + C*cos(c + d*x)^2),x)

[Out]

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

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