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]

14/39*sin(d*x+c)/a^3/d/e/(e*cos(d*x+c))^(1/2)-2/13/d/e/(a+a*sin(d*x+c))^3/(e*cos(d*x+c))^(1/2)-14/117/a/d/e/(a
+a*sin(d*x+c))^2/(e*cos(d*x+c))^(1/2)-14/117/d/e/(a^3+a^3*sin(d*x+c))/(e*cos(d*x+c))^(1/2)-14/39*(cos(1/2*d*x+
1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/a^3/d/e^2/cos(d*
x+c)^(1/2)

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Rubi [A]  time = 0.22, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.06, 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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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e \cos \left (d x + c\right )}}{3 \, a^{3} e^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{3} e^{2} \cos \left (d x + c\right )^{2} + {\left (a^{3} e^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{3} e^{2} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]

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)

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maple [B]  time = 4.73, size = 696, normalized size = 3.72 \[ \text {result too large to display} \]

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*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)^12-2688*sin(1/2*d*x+1/2*c)^14*cos(1/2*d*x+1/2*c)-4032*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)^10+8064*cos(1/2*d*x+1/2*c)*sin(
1/2*d*x+1/2*c)^12+5040*(sin(1/2*d*x+1/2*c)^2)^(1/2)*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)^8-10304*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-3360*(sin(1/2*d*x+1/2*c)^2)^(
1/2)*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)^6+7168*cos(1/2*
d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+1260*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(c
os(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*(2*sin(1/2*d*
x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(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*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(si
n(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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