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

Optimal. Leaf size=47 \[ \frac{4 \sqrt{a \sin (c+d x)+a}}{a^2 d}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \]

[Out]

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

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Rubi [A]  time = 0.0665267, antiderivative size = 47, 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} \[ \frac{4 \sqrt{a \sin (c+d x)+a}}{a^2 d}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[a + a*Sin[c + d*x]])/(a^2*d) - (2*(a + a*Sin[c + d*x])^(3/2))/(3*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))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a-x}{\sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{2 a}{\sqrt{a+x}}-\sqrt{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{4 \sqrt{a+a \sin (c+d x)}}{a^2 d}-\frac{2 (a+a \sin (c+d x))^{3/2}}{3 a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.0500463, size = 32, normalized size = 0.68 \[ -\frac{2 (\sin (c+d x)-5) \sqrt{a (\sin (c+d x)+1)}}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-5 + Sin[c + d*x])*Sqrt[a*(1 + Sin[c + d*x])])/(3*a^2*d)

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Maple [A]  time = 0.08, size = 29, normalized size = 0.6 \begin{align*} -{\frac{2\,\sin \left ( dx+c \right ) -10}{3\,{a}^{2}d}\sqrt{a+a\sin \left ( dx+c \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))^(3/2),x)

[Out]

-2/3/a^2*(a+a*sin(d*x+c))^(1/2)*(sin(d*x+c)-5)/d

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Maxima [A]  time = 0.964622, size = 49, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{a \sin \left (d x + c\right ) + a} a\right )}}{3 \, 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))^(3/2),x, algorithm="maxima")

[Out]

-2/3*((a*sin(d*x + c) + a)^(3/2) - 6*sqrt(a*sin(d*x + c) + a)*a)/(a^3*d)

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Fricas [A]  time = 2.20186, size = 78, normalized size = 1.66 \begin{align*} -\frac{2 \, \sqrt{a \sin \left (d x + c\right ) + a}{\left (\sin \left (d x + c\right ) - 5\right )}}{3 \, a^{2} 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))^(3/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(a*sin(d*x + c) + a)*(sin(d*x + c) - 5)/(a^2*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.15639, size = 49, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{a \sin \left (d x + c\right ) + a} a\right )}}{3 \, 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))^(3/2),x, algorithm="giac")

[Out]

-2/3*((a*sin(d*x + c) + a)^(3/2) - 6*sqrt(a*sin(d*x + c) + a)*a)/(a^3*d)