3.262 \(\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=187 \[ -\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{39 a^3 d e^2 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{39 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{14}{117 d e \left (a^3 \sin (c+d x)+a^3\right ) \sqrt{e \cos (c+d x)}}-\frac{14}{117 a d e (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}} \]

[Out]

(-14*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(39*a^3*d*e^2*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(39
*a^3*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(13*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) - 14/(117*a*d*e*Sqrt[e
*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2) - 14/(117*d*e*Sqrt[e*Cos[c + d*x]]*(a^3 + a^3*Sin[c + d*x]))

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Rubi [A]  time = 0.224771, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2681, 2683, 2636, 2640, 2639} \[ -\frac{14 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{39 a^3 d e^2 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{39 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{14}{117 d e \left (a^3 \sin (c+d x)+a^3\right ) \sqrt{e \cos (c+d x)}}-\frac{14}{117 a d e (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3),x]

[Out]

(-14*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(39*a^3*d*e^2*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(39
*a^3*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(13*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) - 14/(117*a*d*e*Sqrt[e
*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2) - 14/(117*d*e*Sqrt[e*Cos[c + d*x]]*(a^3 + a^3*Sin[c + d*x]))

Rule 2681

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*(2*m + p + 1)), x] + Dist[(m + p + 1)/(a*(2*m + p + 1)),
Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && IntegersQ[2*m, 2*p]

Rule 2683

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

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[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{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3} \, dx &=-\frac{2}{13 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}+\frac{7 \int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx}{13 a}\\ &=-\frac{2}{13 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{117 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2}+\frac{35 \int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))} \, dx}{117 a^2}\\ &=-\frac{2}{13 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{117 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac{14}{117 d e \sqrt{e \cos (c+d x)} \left (a^3+a^3 \sin (c+d x)\right )}+\frac{7 \int \frac{1}{(e \cos (c+d x))^{3/2}} \, dx}{39 a^3}\\ &=\frac{14 \sin (c+d x)}{39 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{117 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac{14}{117 d e \sqrt{e \cos (c+d x)} \left (a^3+a^3 \sin (c+d x)\right )}-\frac{7 \int \sqrt{e \cos (c+d x)} \, dx}{39 a^3 e^2}\\ &=\frac{14 \sin (c+d x)}{39 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{117 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac{14}{117 d e \sqrt{e \cos (c+d x)} \left (a^3+a^3 \sin (c+d x)\right )}-\frac{\left (7 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{39 a^3 e^2 \sqrt{\cos (c+d x)}}\\ &=-\frac{14 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{39 a^3 d e^2 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{39 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{117 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2}-\frac{14}{117 d e \sqrt{e \cos (c+d x)} \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 0.061958, size = 66, normalized size = 0.35 \[ \frac{\sqrt [4]{\sin (c+d x)+1} \, _2F_1\left (-\frac{1}{4},\frac{17}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{4 \sqrt [4]{2} a^3 d e \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3),x]

[Out]

(Hypergeometric2F1[-1/4, 17/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(4*2^(1/4)*a^3*d*e*Sqrt[e*
Cos[c + d*x]])

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Maple [B]  time = 4.014, size = 696, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/117/(64*sin(1/2*d*x+1/2*c)^12-192*sin(1/2*d*x+1/2*c)^10+240*sin(1/2*d*x+1/2*c)^8-160*sin(1/2*d*x+1/2*c)^6+6
0*sin(1/2*d*x+1/2*c)^4-12*sin(1/2*d*x+1/2*c)^2+1)/a^3/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e
*(1344*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin
(1/2*d*x+1/2*c)^12-2688*sin(1/2*d*x+1/2*c)^14*cos(1/2*d*x+1/2*c)-4032*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipti
cE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^10+8064*cos(1/2*d*x+1/2*c)*sin(
1/2*d*x+1/2*c)^12+5040*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(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)^8-10304*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-3360*EllipticE(cos(1/2*d*x+1/
2*c),2^(1/2))*(sin(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)^6+7168*cos(1/2*
d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+1260*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)*(s
in(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-2896*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-252*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+65
6*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-52*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))-138*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+52*sin(
1/2*d*x+1/2*c)^3+23*sin(1/2*d*x+1/2*c))/d

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Maxima [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(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

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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 )}}{3 \, a^{3} e^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{3} e^{2} \cos \left (d x + c\right )^{2} +{\left (a^{3} e^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{3} e^{2} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(-sqrt(e*cos(d*x + c))/(3*a^3*e^2*cos(d*x + c)^4 - 4*a^3*e^2*cos(d*x + c)^2 + (a^3*e^2*cos(d*x + c)^4
- 4*a^3*e^2*cos(d*x + c)^2)*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(1/(e*cos(d*x+c))**(3/2)/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

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