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

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

[Out]

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

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Rubi [A]  time = 0.0745764, 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} \[ \frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0964159, size = 44, normalized size = 0.6 \[ \frac{2 \left (15 \sin ^2(c+d x)-54 \sin (c+d x)+71\right ) (a (\sin (c+d x)+1))^{3/2}}{105 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*(1 + Sin[c + d*x]))^(3/2)*(71 - 54*Sin[c + d*x] + 15*Sin[c + d*x]^2))/(105*a^3*d)

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Maple [A]  time = 0.078, size = 41, normalized size = 0.6 \begin{align*} -{\frac{30\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+108\,\sin \left ( dx+c \right ) -172}{105\,{a}^{3}d} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{3}{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))^(3/2),x)

[Out]

-2/105/a^3*(a+a*sin(d*x+c))^(3/2)*(15*cos(d*x+c)^2+54*sin(d*x+c)-86)/d

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Maxima [A]  time = 0.963007, size = 74, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (15 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} - 84 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a + 140 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{2}\right )}}{105 \, 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))^(3/2),x, algorithm="maxima")

[Out]

2/105*(15*(a*sin(d*x + c) + a)^(7/2) - 84*(a*sin(d*x + c) + a)^(5/2)*a + 140*(a*sin(d*x + c) + a)^(3/2)*a^2)/(
a^5*d)

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Fricas [A]  time = 2.2489, size = 142, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (39 \, \cos \left (d x + c\right )^{2} -{\left (15 \, \cos \left (d x + c\right )^{2} - 32\right )} \sin \left (d x + c\right ) + 32\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{105 \, a^{2} 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))^(3/2),x, algorithm="fricas")

[Out]

2/105*(39*cos(d*x + c)^2 - (15*cos(d*x + c)^2 - 32)*sin(d*x + c) + 32)*sqrt(a*sin(d*x + c) + a)/(a^2*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))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.20317, size = 74, normalized size = 1.01 \begin{align*} \frac{2 \,{\left (15 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} - 84 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a + 140 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{2}\right )}}{105 \, 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))^(3/2),x, algorithm="giac")

[Out]

2/105*(15*(a*sin(d*x + c) + a)^(7/2) - 84*(a*sin(d*x + c) + a)^(5/2)*a + 140*(a*sin(d*x + c) + a)^(3/2)*a^2)/(
a^5*d)