3.272 \(\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=225 \[ -\frac{42 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{221 a^4 d e^2 \sqrt{\cos (c+d x)}}+\frac{42 \sin (c+d x)}{221 a^4 d e \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left (a^4 \sin (c+d x)+a^4\right ) \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt{e \cos (c+d x)}}-\frac{18}{221 a d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}-\frac{2}{17 d e (a \sin (c+d x)+a)^4 \sqrt{e \cos (c+d x)}} \]

[Out]

(-42*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(221*a^4*d*e^2*Sqrt[Cos[c + d*x]]) + (42*Sin[c + d*x])/(2
21*a^4*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(17*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4) - 18/(221*a*d*e*Sqrt
[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) - 14/(221*d*e*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x])^2) - 14/(
221*d*e*Sqrt[e*Cos[c + d*x]]*(a^4 + a^4*Sin[c + d*x]))

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Rubi [A]  time = 0.299081, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{42 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{221 a^4 d e^2 \sqrt{\cos (c+d x)}}+\frac{42 \sin (c+d x)}{221 a^4 d e \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left (a^4 \sin (c+d x)+a^4\right ) \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt{e \cos (c+d x)}}-\frac{18}{221 a d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}-\frac{2}{17 d e (a \sin (c+d x)+a)^4 \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-42*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(221*a^4*d*e^2*Sqrt[Cos[c + d*x]]) + (42*Sin[c + d*x])/(2
21*a^4*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(17*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4) - 18/(221*a*d*e*Sqrt
[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) - 14/(221*d*e*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x])^2) - 14/(
221*d*e*Sqrt[e*Cos[c + d*x]]*(a^4 + a^4*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))^4} \, dx &=-\frac{2}{17 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4}+\frac{9 \int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3} \, dx}{17 a}\\ &=-\frac{2}{17 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac{18}{221 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}+\frac{63 \int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx}{221 a^2}\\ &=-\frac{2}{17 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac{18}{221 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{221 d e \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac{35 \int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))} \, dx}{221 a^3}\\ &=-\frac{2}{17 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac{18}{221 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{221 d e \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{14}{221 d e \sqrt{e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}+\frac{21 \int \frac{1}{(e \cos (c+d x))^{3/2}} \, dx}{221 a^4}\\ &=\frac{42 \sin (c+d x)}{221 a^4 d e \sqrt{e \cos (c+d x)}}-\frac{2}{17 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac{18}{221 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{221 d e \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{14}{221 d e \sqrt{e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}-\frac{21 \int \sqrt{e \cos (c+d x)} \, dx}{221 a^4 e^2}\\ &=\frac{42 \sin (c+d x)}{221 a^4 d e \sqrt{e \cos (c+d x)}}-\frac{2}{17 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac{18}{221 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{221 d e \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{14}{221 d e \sqrt{e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\left (21 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{221 a^4 e^2 \sqrt{\cos (c+d x)}}\\ &=-\frac{42 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{221 a^4 d e^2 \sqrt{\cos (c+d x)}}+\frac{42 \sin (c+d x)}{221 a^4 d e \sqrt{e \cos (c+d x)}}-\frac{2}{17 d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4}-\frac{18}{221 a d e \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3}-\frac{14}{221 d e \sqrt{e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac{14}{221 d e \sqrt{e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 0.0876748, size = 66, normalized size = 0.29 \[ \frac{\sqrt [4]{\sin (c+d x)+1} \, _2F_1\left (-\frac{1}{4},\frac{21}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{8 \sqrt [4]{2} a^4 d e \sqrt{e \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Hypergeometric2F1[-1/4, 21/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(8*2^(1/4)*a^4*d*e*Sqrt[e*
Cos[c + d*x]])

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Maple [B]  time = 5.447, size = 878, normalized size = 3.9 \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))^4,x)

[Out]

-2/221/(256*sin(1/2*d*x+1/2*c)^16-1024*sin(1/2*d*x+1/2*c)^14+1792*sin(1/2*d*x+1/2*c)^12-1792*sin(1/2*d*x+1/2*c
)^10+1120*sin(1/2*d*x+1/2*c)^8-448*sin(1/2*d*x+1/2*c)^6+112*sin(1/2*d*x+1/2*c)^4-16*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)/e*(5376*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)^16-10752*sin(1/2*d*x+1/2*c)^18*cos(1
/2*d*x+1/2*c)-21504*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)^14+43008*sin(1/2*d*x+1/2*c)^16*cos(1/2*d*x+1/2*c)+37632*(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)^12-76160*sin(1/2
*d*x+1/2*c)^14*cos(1/2*d*x+1/2*c)-37632*(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)^10+77952*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+23520*Ellip
ticE(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)^8-50560*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-9408*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+21376*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*
c)^8+2352*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)^4-5656*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-336*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+792*sin(1/2*d*x+1/2*c)^4*cos(
1/2*d*x+1/2*c)-272*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)*Ellip
ticE(cos(1/2*d*x+1/2*c),2^(1/2))-242*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+272*sin(1/2*d*x+1/2*c)^3+36*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))^4,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 )}}{a^{4} e^{2} \cos \left (d x + c\right )^{6} - 8 \, a^{4} e^{2} \cos \left (d x + c\right )^{4} + 8 \, a^{4} e^{2} \cos \left (d x + c\right )^{2} - 4 \,{\left (a^{4} e^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{4} 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))^4,x, algorithm="fricas")

[Out]

integral(sqrt(e*cos(d*x + c))/(a^4*e^2*cos(d*x + c)^6 - 8*a^4*e^2*cos(d*x + c)^4 + 8*a^4*e^2*cos(d*x + c)^2 -
4*(a^4*e^2*cos(d*x + c)^4 - 2*a^4*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))**4,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 )}^{4}}\,{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))^4,x, algorithm="giac")

[Out]

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