\(\int \frac {\cos ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\) [158]

Optimal result

Integrand size = 23, antiderivative size = 73 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {8 (a+a \sin (c+d x))^{5/2}}{5 a^3 d}-\frac {8 (a+a \sin (c+d x))^{7/2}}{7 a^4 d}+\frac {2 (a+a \sin (c+d x))^{9/2}}{9 a^5 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2746, 45} \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {2 (a \sin (c+d x)+a)^{9/2}}{9 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d} \]

[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 45

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])

Rule 2746

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*} \text {integral}& = \frac {\text {Subst}\left (\int (a-x)^2 (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {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] (verified)

Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {2 (1+\sin (c+d x))^3 \left (107-110 \sin (c+d x)+35 \sin ^2(c+d x)\right )}{315 d \sqrt {a (1+\sin (c+d x))}} \]

[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] (verified)

Time = 0.61 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {8 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {8 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5}}{d \,a^{5}}\)	\(56\)
default	\(\frac {\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {8 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {8 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5}}{d \,a^{5}}\)	\(56\)

[In]

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

[Out]

2/d/a^5*(1/9*(a+a*sin(d*x+c))^(9/2)-4/7*a*(a+a*sin(d*x+c))^(7/2)+4/5*a^2*(a+a*sin(d*x+c))^(5/2))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {2 \, {\left (35 \, \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{2} + 8 \, {\left (5 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 64\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{315 \, a d} \]

[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)]

Timed out. \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (61) = 122\).

Time = 0.20 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.19 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {2 \, {\left (315 \, \sqrt {a \sin \left (d x + c\right ) + a} - \frac {42 \, {\left (3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 10 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac {35 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 180 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 378 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{4}}{a^{4}}\right )}}{315 \, a d} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.23 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {32 \, {\left (35 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 90 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}\right )}}{315 \, a d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \]

[In]

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

[Out]

32/315*(35*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2*c)^9 - 90*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2*c
)^7 + 63*sqrt(2)*sqrt(a)*cos(-1/4*pi + 1/2*d*x + 1/2*c)^5)/(a*d*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)))

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^5}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]

[In]

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

[Out]

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