3.158 \(\int \frac{\cos ^5(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=73 \[ \frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]

[Out]

(8*(a + a*Sin[c + d*x])^(5/2))/(5*a^3*d) - (8*(a + a*Sin[c + d*x])^(7/2))/(7*a^4*d) + (2*(a + a*Sin[c + d*x])^
(9/2))/(9*a^5*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0672223, antiderivative size = 73, 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} \[ \frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(8*(a + a*Sin[c + d*x])^(5/2))/(5*a^3*d) - (8*(a + a*Sin[c + d*x])^(7/2))/(7*a^4*d) + (2*(a + a*Sin[c + d*x])^
(9/2))/(9*a^5*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 ^5(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{3/2}-4 a (a+x)^{5/2}+(a+x)^{7/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{8 (a+a \sin (c+d x))^{5/2}}{5 a^3 d}-\frac{8 (a+a \sin (c+d x))^{7/2}}{7 a^4 d}+\frac{2 (a+a \sin (c+d x))^{9/2}}{9 a^5 d}\\ \end{align*}

Mathematica [A]  time = 0.13584, size = 51, normalized size = 0.7 \[ \frac{2 (\sin (c+d x)+1)^3 \left (35 \sin ^2(c+d x)-110 \sin (c+d x)+107\right )}{315 d \sqrt{a (\sin (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(2*(1 + Sin[c + d*x])^3*(107 - 110*Sin[c + d*x] + 35*Sin[c + d*x]^2))/(315*d*Sqrt[a*(1 + Sin[c + d*x])])

________________________________________________________________________________________

Maple [A]  time = 0.079, size = 41, normalized size = 0.6 \begin{align*} -{\frac{70\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+220\,\sin \left ( dx+c \right ) -284}{315\,{a}^{3}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5/(a+a*sin(d*x+c))^(1/2),x)

[Out]

-2/315/a^3*(a+a*sin(d*x+c))^(5/2)*(35*cos(d*x+c)^2+110*sin(d*x+c)-142)/d

________________________________________________________________________________________

Maxima [B]  time = 0.953644, size = 216, normalized size = 2.96 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{a \sin \left (d x + c\right ) + a} - \frac{42 \,{\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 + 15 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac{35 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 378 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{a \sin \left (d x + c\right ) + a} a^{4}}{a^{4}}\right )}}{315 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

2/315*(315*sqrt(a*sin(d*x + c) + a) - 42*(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 + (35*(a*sin(d*x + c) + a)^(9/2) - 180*(a*sin(d*x + c) + a)^(7/2)*a + 378*(a
*sin(d*x + c) + a)^(5/2)*a^2 - 420*(a*sin(d*x + c) + a)^(3/2)*a^3 + 315*sqrt(a*sin(d*x + c) + a)*a^4)/a^4)/(a*
d)

________________________________________________________________________________________

Fricas [A]  time = 1.84219, size = 165, normalized size = 2.26 \begin{align*} \frac{2 \,{\left (35 \, \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{2} + 8 \,{\left (5 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 64\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{315 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*cos(d*x + c)^4 + 8*cos(d*x + c)^2 + 8*(5*cos(d*x + c)^2 + 8)*sin(d*x + c) + 64)*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)**5/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.23034, size = 74, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (35 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 252 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2}\right )}}{315 \, a^{5} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

2/315*(35*(a*sin(d*x + c) + a)^(9/2) - 180*(a*sin(d*x + c) + a)^(7/2)*a + 252*(a*sin(d*x + c) + a)^(5/2)*a^2)/
(a^5*d)