3.129 \(\int \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=49 \[ \frac{4 (a \sin (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d} \]

[Out]

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

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Rubi [A]  time = 0.0665718, 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} \[ \frac{4 (a \sin (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.123333, size = 41, normalized size = 0.84 \[ -\frac{2 (\sin (c+d x)+1)^2 (9 \sin (c+d x)-13) (a (\sin (c+d x)+1))^{5/2}}{99 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-2*(1 + Sin[c + d*x])^2*(a*(1 + Sin[c + d*x]))^(5/2)*(-13 + 9*Sin[c + d*x]))/(99*d)

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Maple [A]  time = 0.082, size = 31, normalized size = 0.6 \begin{align*} -{\frac{18\,\sin \left ( dx+c \right ) -26}{99\,{a}^{2}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{9}{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))^(5/2),x)

[Out]

-2/99/a^2*(a+a*sin(d*x+c))^(9/2)*(9*sin(d*x+c)-13)/d

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Maxima [A]  time = 0.953337, size = 51, normalized size = 1.04 \begin{align*} -\frac{2 \,{\left (9 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} - 22 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a\right )}}{99 \, 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))^(5/2),x, algorithm="maxima")

[Out]

-2/99*(9*(a*sin(d*x + c) + a)^(11/2) - 22*(a*sin(d*x + c) + a)^(9/2)*a)/(a^3*d)

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Fricas [B]  time = 1.77678, size = 217, normalized size = 4.43 \begin{align*} -\frac{2 \,{\left (23 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} - 32 \, a^{2} +{\left (9 \, a^{2} \cos \left (d x + c\right )^{4} - 20 \, a^{2} \cos \left (d x + c\right )^{2} - 32 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{99 \, 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))^(5/2),x, algorithm="fricas")

[Out]

-2/99*(23*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 - 32*a^2 + (9*a^2*cos(d*x + c)^4 - 20*a^2*cos(d*x + c)^2 -
32*a^2)*sin(d*x + c))*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))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{3}\,{d x} \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))^(5/2),x, algorithm="giac")

[Out]

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