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

Optimal. Leaf size=36 \[ -\frac{2 (e \cos (c+d x))^{5/2}}{5 d e (a \sin (c+d x)+a)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.0705864, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2671} \[ -\frac{2 (e \cos (c+d x))^{5/2}}{5 d e (a \sin (c+d x)+a)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2671

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac{2 (e \cos (c+d x))^{5/2}}{5 d e (a+a \sin (c+d x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.107092, size = 49, normalized size = 1.36 \[ -\frac{2 \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{5/2}}{5 a^3 d e (\sin (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e*Cos[c + d*x])^(5/2)*Sqrt[a*(1 + Sin[c + d*x])])/(5*a^3*d*e*(1 + Sin[c + d*x])^3)

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Maple [A]  time = 0.087, size = 34, normalized size = 0.9 \begin{align*} -{\frac{2\,\cos \left ( dx+c \right ) }{5\,d} \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5/d*(e*cos(d*x+c))^(3/2)*cos(d*x+c)/(a*(1+sin(d*x+c)))^(5/2)

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Maxima [B]  time = 1.58983, size = 177, normalized size = 4.92 \begin{align*} -\frac{2 \,{\left (\sqrt{a} e^{\frac{3}{2}} - \frac{\sqrt{a} e^{\frac{3}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{3}{2}}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{5 \,{\left (a^{3} + \frac{a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

-2/5*(sqrt(a)*e^(3/2) - sqrt(a)*e^(3/2)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(-sin(d*x + c)/(cos(d*x + c) + 1)
+ 1)^(3/2)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/((a^3 + a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*d*(sin(
d*x + c)/(cos(d*x + c) + 1) + 1)^(7/2))

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Fricas [B]  time = 3.42259, size = 174, normalized size = 4.83 \begin{align*} -\frac{2 \, \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}{\left (e \sin \left (d x + c\right ) - e\right )}}{5 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [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((e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out