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*(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)

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Rubi [A]  time = 0.0726169, 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} \[ \frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \]

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 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 \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*}

Mathematica [A]  time = 0.998606, 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]

-((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*Sin[c +
d*x]))/(693*d)

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Maple [A]  time = 0.086, size = 41, normalized size = 0.6 \begin{align*} -{\frac{126\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+364\,\sin \left ( dx+c \right ) -428}{693\,{a}^{3}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{7}{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]

-2/693/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 = 0.966128, size = 74, normalized size = 1.01 \begin{align*} \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} \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]

2/693*(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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Fricas [A]  time = 1.71955, size = 189, normalized size = 2.59 \begin{align*} \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} \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]

2/693*(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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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

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Giac [A]  time = 1.7412, size = 81, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (\frac{63 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}}}{a^{4}} - \frac{308 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}}}{a^{3}} + \frac{396 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}{a^{2}}\right )}}{693 \, 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="giac")

[Out]

2/693*(63*(a*sin(d*x + c) + a)^(11/2)/a^4 - 308*(a*sin(d*x + c) + a)^(9/2)/a^3 + 396*(a*sin(d*x + c) + a)^(7/2
)/a^2)/(a*d)