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

Optimal. Leaf size=24 \[ -\frac {2}{3 a d (a \sin (c+d x)+a)^{3/2}} \]

[Out]

-2/3/a/d/(a+a*sin(d*x+c))^(3/2)

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Rubi [A]  time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2667, 32} \[ -\frac {2}{3 a d (a \sin (c+d x)+a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(3*a*d*(a + a*Sin[c + d*x])^(3/2))

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 \frac {\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac {2}{3 a d (a+a \sin (c+d x))^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 24, normalized size = 1.00 \[ -\frac {2}{3 a d (a \sin (c+d x)+a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(3*a*d*(a + a*Sin[c + d*x])^(3/2))

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fricas [B]  time = 0.77, size = 48, normalized size = 2.00 \[ \frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a}}{3 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

2/3*sqrt(a*sin(d*x + c) + a)/(a^3*d*cos(d*x + c)^2 - 2*a^3*d*sin(d*x + c) - 2*a^3*d)

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giac [A]  time = 1.43, size = 20, normalized size = 0.83 \[ -\frac {2}{3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

-2/3/((a*sin(d*x + c) + a)^(3/2)*a*d)

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maple [A]  time = 0.02, size = 21, normalized size = 0.88 \[ -\frac {2}{3 a d \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/a/d/(a+a*sin(d*x+c))^(3/2)

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maxima [A]  time = 0.61, size = 20, normalized size = 0.83 \[ -\frac {2}{3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-2/3/((a*sin(d*x + c) + a)^(3/2)*a*d)

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mupad [B]  time = 7.52, size = 72, normalized size = 3.00 \[ \frac {8\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}}{3\,a^3\,d\,{\left (-1+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*exp(c*2i + d*x*2i)*(a + a*((exp(- c*1i - d*x*1i)*1i)/2 - (exp(c*1i + d*x*1i)*1i)/2))^(1/2))/(3*a^3*d*(exp(c
*1i + d*x*1i)*1i - 1)^4)

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sympy [A]  time = 25.47, size = 65, normalized size = 2.71 \[ \begin {cases} - \frac {2}{3 a^{2} d \sqrt {a \sin {\left (c + d x \right )} + a} \sin {\left (c + d x \right )} + 3 a^{2} d \sqrt {a \sin {\left (c + d x \right )} + a}} & \text {for}\: d \neq 0 \\\frac {x \cos {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2/(3*a**2*d*sqrt(a*sin(c + d*x) + a)*sin(c + d*x) + 3*a**2*d*sqrt(a*sin(c + d*x) + a)), Ne(d, 0)),
(x*cos(c)/(a*sin(c) + a)**(5/2), True))

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