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

Optimal. Leaf size=49 \[ \frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0625084, antiderivative size = 49, 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} \[ \frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0637287, size = 34, normalized size = 0.69 \[ -\frac{2 (3 \sin (c+d x)-7) (a (\sin (c+d x)+1))^{3/2}}{15 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a*(1 + Sin[c + d*x]))^(3/2)*(-7 + 3*Sin[c + d*x]))/(15*a^2*d)

________________________________________________________________________________________

Maple [A]  time = 0.088, size = 31, normalized size = 0.6 \begin{align*} -{\frac{6\,\sin \left ( dx+c \right ) -14}{15\,{a}^{2}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}} \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))^(1/2),x)

[Out]

-2/15/a^2*(a+a*sin(d*x+c))^(3/2)*(3*sin(d*x+c)-7)/d

________________________________________________________________________________________

Maxima [A]  time = 0.953746, size = 101, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{a \sin \left (d x + c\right ) + a} - \frac{3 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{2}}{a^{2}}\right )}}{15 \, 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))^(1/2),x, algorithm="maxima")

[Out]

2/15*(15*sqrt(a*sin(d*x + c) + a) - (3*(a*sin(d*x + c) + a)^(5/2) - 10*(a*sin(d*x + c) + a)^(3/2)*a + 15*sqrt(
a*sin(d*x + c) + a)*a^2)/a^2)/(a*d)

________________________________________________________________________________________

Fricas [A]  time = 1.86179, size = 104, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) + 4\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{15 \, 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))^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*cos(d*x + c)^2 + 4*sin(d*x + c) + 4)*sqrt(a*sin(d*x + c) + a)/(a*d)

________________________________________________________________________________________

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))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.15475, size = 51, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left (3 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a\right )}}{15 \, 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))^(1/2),x, algorithm="giac")

[Out]

-2/15*(3*(a*sin(d*x + c) + a)^(5/2) - 10*(a*sin(d*x + c) + a)^(3/2)*a)/(a^3*d)