3.234 \(\int \frac{(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=101 \[ \frac{2 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 a d}+\frac{6 e^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 a d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{7/2}}{7 a d} \]

[Out]

(2*e*(e*Cos[c + d*x])^(7/2))/(7*a*d) + (6*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a*d*Sqrt[Cos[
c + d*x]]) + (2*e^3*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)

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Rubi [A]  time = 0.0954738, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2682, 2635, 2640, 2639} \[ \frac{2 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 a d}+\frac{6 e^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 a d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{7/2}}{7 a d} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x]),x]

[Out]

(2*e*(e*Cos[c + d*x])^(7/2))/(7*a*d) + (6*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a*d*Sqrt[Cos[
c + d*x]]) + (2*e^3*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)

Rule 2682

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx &=\frac{2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac{e^2 \int (e \cos (c+d x))^{5/2} \, dx}{a}\\ &=\frac{2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac{2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac{\left (3 e^4\right ) \int \sqrt{e \cos (c+d x)} \, dx}{5 a}\\ &=\frac{2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac{2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac{\left (3 e^4 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 a \sqrt{\cos (c+d x)}}\\ &=\frac{2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac{6 e^4 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a d \sqrt{\cos (c+d x)}}+\frac{2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}\\ \end{align*}

Mathematica [C]  time = 0.0919839, size = 66, normalized size = 0.65 \[ -\frac{4\ 2^{3/4} (e \cos (c+d x))^{11/2} \, _2F_1\left (-\frac{3}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{11 a d e (\sin (c+d x)+1)^{11/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x]),x]

[Out]

(-4*2^(3/4)*(e*Cos[c + d*x])^(11/2)*Hypergeometric2F1[-3/4, 11/4, 15/4, (1 - Sin[c + d*x])/2])/(11*a*d*e*(1 +
Sin[c + d*x])^(11/4))

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Maple [A]  time = 0.598, size = 216, normalized size = 2.1 \begin{align*}{\frac{2\,{e}^{5}}{35\,da} \left ( 80\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}+56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -160\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}-56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +120\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+21\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +14\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -40\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+5\,\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c)),x)

[Out]

2/35/a/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^5*(80*sin(1/2*d*x+1/2*c)^9+56*sin(1/2*d*x+1/2*
c)^6*cos(1/2*d*x+1/2*c)-160*sin(1/2*d*x+1/2*c)^7-56*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+120*sin(1/2*d*x+1/
2*c)^5+21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+
14*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-40*sin(1/2*d*x+1/2*c)^3+5*sin(1/2*d*x+1/2*c))/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{9}{2}}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(9/2)/(a*sin(d*x + c) + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e \cos \left (d x + c\right )} e^{4} \cos \left (d x + c\right )^{4}}{a \sin \left (d x + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

integral(sqrt(e*cos(d*x + c))*e^4*cos(d*x + c)^4/(a*sin(d*x + c) + a), x)

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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))**(9/2)/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{9}{2}}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate((e*cos(d*x + c))^(9/2)/(a*sin(d*x + c) + a), x)