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

Optimal. Leaf size=49 \[ \frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \]

[Out]

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

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Rubi [A]  time = 0.0649647, 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} \[ \frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d} \]

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

Mathematica [A]  time = 0.238839, size = 54, normalized size = 1.1 \[ -\frac{2 (5 \sin (c+d x)-9) \sqrt{a (\sin (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}{35 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*Sqrt[a*(1 + Sin[c + d*x])]*(-9 + 5*Sin[c + d*x]))/(35*d)

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Maple [A]  time = 0.086, size = 31, normalized size = 0.6 \begin{align*} -{\frac{10\,\sin \left ( dx+c \right ) -18}{35\,{a}^{2}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)^3*(a+a*sin(d*x+c))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.960687, size = 51, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left (5 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} - 14 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a\right )}}{35 \, 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="maxima")

[Out]

-2/35*(5*(a*sin(d*x + c) + a)^(7/2) - 14*(a*sin(d*x + c) + a)^(5/2)*a)/(a^3*d)

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Fricas [A]  time = 1.59104, size = 124, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (\cos \left (d x + c\right )^{2} +{\left (5 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{35 \, 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]

2/35*(cos(d*x + c)^2 + (5*cos(d*x + c)^2 + 8)*sin(d*x + c) + 8)*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)**3*(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 2.02199, size = 58, normalized size = 1.18 \begin{align*} -\frac{2 \,{\left (\frac{5 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}{a^{2}} - \frac{14 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{a}\right )}}{35 \, 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="giac")

[Out]

-2/35*(5*(a*sin(d*x + c) + a)^(7/2)/a^2 - 14*(a*sin(d*x + c) + a)^(5/2)/a)/(a*d)