3.189 \(\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{2 \sqrt{a \sin (c+d x)+a}}{a^3 d}-\frac{4}{a^2 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

-4/(a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Sqrt[a + a*Sin[c + d*x]])/(a^3*d)

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Rubi [A]  time = 0.0676261, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2667, 43} \[ -\frac{2 \sqrt{a \sin (c+d x)+a}}{a^3 d}-\frac{4}{a^2 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

-4/(a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Sqrt[a + a*Sin[c + d*x]])/(a^3*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])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-x}{(a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{2 a}{(a+x)^{3/2}}-\frac{1}{\sqrt{a+x}}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac{4}{a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \sqrt{a+a \sin (c+d x)}}{a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.0544934, size = 30, normalized size = 0.67 \[ -\frac{2 (\sin (c+d x)+3)}{a^2 d \sqrt{a (\sin (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-2*(3 + Sin[c + d*x]))/(a^2*d*Sqrt[a*(1 + Sin[c + d*x])])

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Maple [A]  time = 0.087, size = 29, normalized size = 0.6 \begin{align*} -2\,{\frac{3+\sin \left ( dx+c \right ) }{{a}^{2}\sqrt{a+a\sin \left ( dx+c \right ) }d}} \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))^(5/2),x)

[Out]

-2/a^2/(a+a*sin(d*x+c))^(1/2)*(3+sin(d*x+c))/d

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Maxima [A]  time = 0.942827, size = 57, normalized size = 1.27 \begin{align*} -\frac{2 \,{\left (\frac{\sqrt{a \sin \left (d x + c\right ) + a}}{a^{2}} + \frac{2}{\sqrt{a \sin \left (d x + c\right ) + a} a}\right )}}{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))^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.13332, size = 104, normalized size = 2.31 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a}{\left (\sin \left (d x + c\right ) + 3\right )}}{a^{3} d \sin \left (d x + c\right ) + a^{3} 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))^(5/2),x, algorithm="fricas")

[Out]

-2*sqrt(a*sin(d*x + c) + a)*(sin(d*x + c) + 3)/(a^3*d*sin(d*x + c) + a^3*d)

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Sympy [A]  time = 30.4713, size = 267, normalized size = 5.93 \begin{align*} \begin{cases} \text{NaN} & \text{for}\: \left (c = \frac{3 \pi }{2} \vee c = - d x + \frac{3 \pi }{2}\right ) \wedge \left (c = - d x + \frac{3 \pi }{2} \vee d = 0\right ) \\\frac{x \cos ^{3}{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{\frac{5}{2}}} & \text{for}\: d = 0 \\- \frac{8 \sqrt{a \sin{\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin{\left (c + d x \right )} + 3 a^{3} d} - \frac{24 \sqrt{a \sin{\left (c + d x \right )} + a} \sin{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin{\left (c + d x \right )} + 3 a^{3} d} - \frac{2 \sqrt{a \sin{\left (c + d x \right )} + a} \cos ^{2}{\left (c + d x \right )}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin{\left (c + d x \right )} + 3 a^{3} d} - \frac{16 \sqrt{a \sin{\left (c + d x \right )} + a}}{3 a^{3} d \sin ^{2}{\left (c + d x \right )} + 6 a^{3} d \sin{\left (c + d x \right )} + 3 a^{3} d} & \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))**(5/2),x)

[Out]

Piecewise((nan, (Eq(d, 0) | Eq(c, -d*x + 3*pi/2)) & (Eq(c, 3*pi/2) | Eq(c, -d*x + 3*pi/2))), (x*cos(c)**3/(a*s
in(c) + a)**(5/2), Eq(d, 0)), (-8*sqrt(a*sin(c + d*x) + a)*sin(c + d*x)**2/(3*a**3*d*sin(c + d*x)**2 + 6*a**3*
d*sin(c + d*x) + 3*a**3*d) - 24*sqrt(a*sin(c + d*x) + a)*sin(c + d*x)/(3*a**3*d*sin(c + d*x)**2 + 6*a**3*d*sin
(c + d*x) + 3*a**3*d) - 2*sqrt(a*sin(c + d*x) + a)*cos(c + d*x)**2/(3*a**3*d*sin(c + d*x)**2 + 6*a**3*d*sin(c
+ d*x) + 3*a**3*d) - 16*sqrt(a*sin(c + d*x) + a)/(3*a**3*d*sin(c + d*x)**2 + 6*a**3*d*sin(c + d*x) + 3*a**3*d)
, True))

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Giac [A]  time = 1.1691, size = 49, normalized size = 1.09 \begin{align*} -\frac{2 \,{\left (\sqrt{a \sin \left (d x + c\right ) + a} + \frac{2 \, a}{\sqrt{a \sin \left (d x + c\right ) + a}}\right )}}{a^{3} 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))^(5/2),x, algorithm="giac")

[Out]

-2*(sqrt(a*sin(d*x + c) + a) + 2*a/sqrt(a*sin(d*x + c) + a))/(a^3*d)