3.281 \(\int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx\)

Optimal. Leaf size=28 \[ \frac {B \sin (c+d x)}{d}-\frac {B \sin ^3(c+d x)}{3 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {21, 2633} \[ \frac {B \sin (c+d x)}{d}-\frac {B \sin ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^3*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin {align*} \int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx &=B \int \cos ^3(c+d x) \, dx\\ &=-\frac {B \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {B \sin (c+d x)}{d}-\frac {B \sin ^3(c+d x)}{3 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 28, normalized size = 1.00 \[ B \left (\frac {\sin (c+d x)}{d}-\frac {\sin ^3(c+d x)}{3 d}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^3*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 25, normalized size = 0.89 \[ \frac {{\left (B \cos \left (d x + c\right )^{2} + 2 \, B\right )} \sin \left (d x + c\right )}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a*B+b*B*cos(d*x+c))/(a+b*cos(d*x+c)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.42, size = 25, normalized size = 0.89 \[ -\frac {B \sin \left (d x + c\right )^{3} - 3 \, B \sin \left (d x + c\right )}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a*B+b*B*cos(d*x+c))/(a+b*cos(d*x+c)),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.08, size = 23, normalized size = 0.82 \[ \frac {B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a*B+b*B*cos(d*x+c))/(a+b*cos(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a*B+b*B*cos(d*x+c))/(a+b*cos(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 0.48, size = 24, normalized size = 0.86 \[ \frac {B\,\left (9\,\sin \left (c+d\,x\right )+\sin \left (3\,c+3\,d\,x\right )\right )}{12\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^3*(B*a + B*b*cos(c + d*x)))/(a + b*cos(c + d*x)),x)

[Out]

(B*(9*sin(c + d*x) + sin(3*c + 3*d*x)))/(12*d)

________________________________________________________________________________________

sympy [A]  time = 1.26, size = 56, normalized size = 2.00 \[ \begin {cases} \frac {2 B \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \cos {\relax (c )}\right ) \cos ^{3}{\relax (c )}}{a + b \cos {\relax (c )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(a*B+b*B*cos(d*x+c))/(a+b*cos(d*x+c)),x)

[Out]

Piecewise((2*B*sin(c + d*x)**3/(3*d) + B*sin(c + d*x)*cos(c + d*x)**2/d, Ne(d, 0)), (x*(B*a + B*b*cos(c))*cos(
c)**3/(a + b*cos(c)), True))

________________________________________________________________________________________