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

Optimal. Leaf size=30 \[ \frac {(A+C) \sin (c+d x)}{d}-\frac {C \sin ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3013} \[ \frac {(A+C) \sin (c+d x)}{d}-\frac {C \sin ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.67 \[ \frac {A \sin (c) \cos (d x)}{d}+\frac {A \cos (c) \sin (d x)}{d}-\frac {C \sin ^3(c+d x)}{3 d}+\frac {C \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 28, normalized size = 0.93 \[ \frac {{\left (C \cos \left (d x + c\right )^{2} + 3 \, A + 2 \, C\right )} \sin \left (d x + c\right )}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(C*cos(d*x + c)^2 + 3*A + 2*C)*sin(d*x + c)/d

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giac [A]  time = 0.19, size = 34, normalized size = 1.13 \[ -\frac {{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C - 3 \, A \sin \left (d x + c\right )}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 33, normalized size = 1.10 \[ \frac {\frac {C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \sin \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/3*C*(2+cos(d*x+c)^2)*sin(d*x+c)+A*sin(d*x+c))

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maxima [A]  time = 0.31, size = 34, normalized size = 1.13 \[ -\frac {{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C - 3 \, A \sin \left (d x + c\right )}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 28, normalized size = 0.93 \[ -\frac {\frac {C\,{\sin \left (c+d\,x\right )}^3}{3}-\sin \left (c+d\,x\right )\,\left (A+C\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.42, size = 56, normalized size = 1.87 \[ \begin {cases} \frac {A \sin {\left (c + d x \right )}}{d} + \frac {2 C \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((A*sin(c + d*x)/d + 2*C*sin(c + d*x)**3/(3*d) + C*sin(c + d*x)*cos(c + d*x)**2/d, Ne(d, 0)), (x*(A +
C*cos(c)**2)*cos(c), True))

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