3.1.4 \(\int \cos (c+d x) (A+C \cos ^2(c+d x)) \, dx\) [4]

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 = {3092} \begin {gather*} \frac {(A+C) \sin (c+d x)}{d}-\frac {C \sin ^3(c+d x)}{3 d} \end {gather*}

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

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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Maple [A]
time = 0.10, size = 33, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {\frac {C \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}+A \sin \left (d x +c \right )}{d}\)	\(33\)
default	\(\frac {\frac {C \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}+A \sin \left (d x +c \right )}{d}\)	\(33\)
risch	\(\frac {\sin \left (d x +c \right ) A}{d}+\frac {3 C \sin \left (d x +c \right )}{4 d}+\frac {\sin \left (3 d x +3 c \right ) C}{12 d}\)	\(40\)
norman	\(\frac {\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 \left (A +C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (3 A +C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 34, normalized size = 1.13 \begin {gather*} -\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} \end {gather*}

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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Fricas [A]
time = 0.37, size = 28, normalized size = 0.93 \begin {gather*} \frac {{\left (C \cos \left (d x + c\right )^{2} + 3 \, A + 2 \, C\right )} \sin \left (d x + c\right )}{3 \, d} \end {gather*}

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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
time = 0.12, size = 56, normalized size = 1.87 \begin {gather*} \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}{\left (c \right )}\right ) \cos {\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)*(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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Giac [A]
time = 0.39, size = 34, normalized size = 1.13 \begin {gather*} -\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} \end {gather*}

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

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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