3.266 \(\int \frac{\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=32 \[ \frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d} \]

[Out]

Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d)

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Rubi [A]  time = 0.045625, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2667} \[ \frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d)

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

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

Mathematica [A]  time = 0.0423879, size = 24, normalized size = 0.75 \[ -\frac{(\sin (c+d x)-2) \sin (c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-2 + Sin[c + d*x])*Sin[c + d*x])/(2*a*d)

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Maple [A]  time = 0.016, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{da} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2}}-\sin \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/a*(1/2*sin(d*x+c)^2-sin(d*x+c))

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Maxima [A]  time = 1.01049, size = 34, normalized size = 1.06 \begin{align*} -\frac{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )}{2 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sin(d*x + c)^2 - 2*sin(d*x + c))/(a*d)

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Fricas [A]  time = 1.63596, size = 61, normalized size = 1.91 \begin{align*} \frac{\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right )}{2 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 7.35931, size = 158, normalized size = 4.94 \begin{align*} \begin{cases} \frac{2 \tan ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} - \frac{2 \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} + \frac{2 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d \tan ^{4}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + 2 a d \tan ^{2}{\left (\frac{c}{2} + \frac{d x}{2} \right )} + a d} & \text{for}\: d \neq 0 \\\frac{x \cos ^{3}{\left (c \right )}}{a \sin{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*tan(c/2 + d*x/2)**3/(a*d*tan(c/2 + d*x/2)**4 + 2*a*d*tan(c/2 + d*x/2)**2 + a*d) - 2*tan(c/2 + d*x
/2)**2/(a*d*tan(c/2 + d*x/2)**4 + 2*a*d*tan(c/2 + d*x/2)**2 + a*d) + 2*tan(c/2 + d*x/2)/(a*d*tan(c/2 + d*x/2)*
*4 + 2*a*d*tan(c/2 + d*x/2)**2 + a*d), Ne(d, 0)), (x*cos(c)**3/(a*sin(c) + a), True))

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Giac [A]  time = 1.14697, size = 34, normalized size = 1.06 \begin{align*} -\frac{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right )}{2 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sin(d*x + c)^2 - 2*sin(d*x + c))/(a*d)