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

Optimal. Leaf size=73 \[ \frac {2 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \]

[Out]

8/7*(a+a*sin(d*x+c))^(7/2)/a^3/d-8/9*(a+a*sin(d*x+c))^(9/2)/a^4/d+2/11*(a+a*sin(d*x+c))^(11/2)/a^5/d

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Rubi [A]  time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, 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} \[ \frac {2 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \]

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])^(7/2))/(7*a^3*d) - (8*(a + a*Sin[c + d*x])^(9/2))/(9*a^4*d) + (2*(a + a*Sin[c + d*x])^
(11/2))/(11*a^5*d)

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])

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])

Rubi steps

\begin {align*} \int \cos ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^{5/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (a+x)^{5/2}-4 a (a+x)^{7/2}+(a+x)^{9/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {8 (a+a \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {8 (a+a \sin (c+d x))^{9/2}}{9 a^4 d}+\frac {2 (a+a \sin (c+d x))^{11/2}}{11 a^5 d}\\ \end {align*}

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Mathematica [A]  time = 1.02, size = 64, normalized size = 0.88 \[ -\frac {\sqrt {a (\sin (c+d x)+1)} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6 (364 \sin (c+d x)+63 \cos (2 (c+d x))-365)}{693 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/693*((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*Sqrt[a*(1 + Sin[c + d*x])]*(-365 + 63*Cos[2*(c + d*x)] + 364*S
in[c + d*x]))/d

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fricas [A]  time = 0.57, size = 68, normalized size = 0.93 \[ \frac {2 \, {\left (7 \, \cos \left (d x + c\right )^{4} + 16 \, \cos \left (d x + c\right )^{2} + {\left (63 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) + 128\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{693 \, d} \]

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]

2/693*(7*cos(d*x + c)^4 + 16*cos(d*x + c)^2 + (63*cos(d*x + c)^4 + 80*cos(d*x + c)^2 + 128)*sin(d*x + c) + 128
)*sqrt(a*sin(d*x + c) + a)/d

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giac [B]  time = 1.70, size = 189, normalized size = 2.59 \[ \frac {1}{11088} \, \sqrt {2} \sqrt {a} {\left (\frac {77 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} + \frac {693 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} + \frac {6930 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {63 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right )}{d} + \frac {495 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} + \frac {2310 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d}\right )} \]

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]

1/11088*sqrt(2)*sqrt(a)*(77*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(1/4*pi + 9/2*d*x + 9/2*c)/d + 693*sgn(cos(
-1/4*pi + 1/2*d*x + 1/2*c))*sin(1/4*pi + 5/2*d*x + 5/2*c)/d + 6930*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(1/4
*pi + 1/2*d*x + 1/2*c)/d + 63*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 11/2*d*x + 11/2*c)/d + 495*sgn
(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(-1/4*pi + 7/2*d*x + 7/2*c)/d + 2310*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*s
in(-1/4*pi + 3/2*d*x + 3/2*c)/d)

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maple [A]  time = 0.16, size = 41, normalized size = 0.56 \[ -\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}} \left (63 \left (\cos ^{2}\left (d x +c \right )\right )+182 \sin \left (d x +c \right )-214\right )}{693 a^{3} d} \]

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]

-2/693/a^3*(a+a*sin(d*x+c))^(7/2)*(63*cos(d*x+c)^2+182*sin(d*x+c)-214)/d

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maxima [A]  time = 0.30, size = 55, normalized size = 0.75 \[ \frac {2 \, {\left (63 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 308 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 396 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}\right )}}{693 \, a^{5} d} \]

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]

2/693*(63*(a*sin(d*x + c) + a)^(11/2) - 308*(a*sin(d*x + c) + a)^(9/2)*a + 396*(a*sin(d*x + c) + a)^(7/2)*a^2)
/(a^5*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^5\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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