3.266 \(\int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=120 \[ \frac{28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{42 e^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{7/2}}{5 a d (a \sin (c+d x)+a)^3} \]

[Out]

(42*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^4*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(
7/2))/(5*a*d*(a + a*Sin[c + d*x])^3) + (28*e^3*(e*Cos[c + d*x])^(3/2))/(5*d*(a^4 + a^4*Sin[c + d*x]))

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Rubi [A]  time = 0.133817, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2680, 2640, 2639} \[ \frac{28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left (a^4 \sin (c+d x)+a^4\right )}+\frac{42 e^4 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{7/2}}{5 a d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^4,x]

[Out]

(42*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^4*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(
7/2))/(5*a*d*(a + a*Sin[c + d*x])^3) + (28*e^3*(e*Cos[c + d*x])^(3/2))/(5*d*(a^4 + a^4*Sin[c + d*x]))

Rule 2680

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

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))^4} \, dx &=-\frac{4 e (e \cos (c+d x))^{7/2}}{5 a d (a+a \sin (c+d x))^3}-\frac{\left (7 e^2\right ) \int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{4 e (e \cos (c+d x))^{7/2}}{5 a d (a+a \sin (c+d x))^3}+\frac{28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac{\left (21 e^4\right ) \int \sqrt{e \cos (c+d x)} \, dx}{5 a^4}\\ &=-\frac{4 e (e \cos (c+d x))^{7/2}}{5 a d (a+a \sin (c+d x))^3}+\frac{28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac{\left (21 e^4 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 a^4 \sqrt{\cos (c+d x)}}\\ &=\frac{42 e^4 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{7/2}}{5 a d (a+a \sin (c+d x))^3}+\frac{28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 0.0905803, size = 66, normalized size = 0.55 \[ -\frac{(e \cos (c+d x))^{11/2} \, _2F_1\left (\frac{9}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{11 \sqrt [4]{2} a^4 d e (\sin (c+d x)+1)^{11/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^4,x]

[Out]

-((e*Cos[c + d*x])^(11/2)*Hypergeometric2F1[9/4, 11/4, 15/4, (1 - Sin[c + d*x])/2])/(11*2^(1/4)*a^4*d*e*(1 + S
in[c + d*x])^(11/4))

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Maple [B]  time = 2.181, size = 332, normalized size = 2.8 \begin{align*}{\frac{2\,{e}^{5}}{5\,{a}^{4}d} \left ( 84\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-128\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -84\,{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+128\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +80\, \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 ) -16\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -80\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+12\,\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( 4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{-1} \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))^4,x)

[Out]

2/5/(4*sin(1/2*d*x+1/2*c)^4-4*sin(1/2*d*x+1/2*c)^2+1)/a^4/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/
2)*(84*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin
(1/2*d*x+1/2*c)^4-128*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-84*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(
1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+128*sin(1/2*d*x+1/2*c)^4*cos(1/2*d
*x+1/2*c)+80*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(c
os(1/2*d*x+1/2*c),2^(1/2))-16*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-80*sin(1/2*d*x+1/2*c)^3+12*sin(1/2*d*x+1
/2*c))*e^5/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}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}\,{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))^4,x, algorithm="maxima")

[Out]

integrate((e*cos(d*x + c))^(9/2)/(a*sin(d*x + c) + a)^4, 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^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \,{\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )}, 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))^4,x, algorithm="fricas")

[Out]

integral(sqrt(e*cos(d*x + c))*e^4*cos(d*x + c)^4/(a^4*cos(d*x + c)^4 - 8*a^4*cos(d*x + c)^2 + 8*a^4 - 4*(a^4*c
os(d*x + c)^2 - 2*a^4)*sin(d*x + c)), 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))**4,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))^(9/2)/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

Timed out