3.3.62 \(\int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3} \, dx\) [262]

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

[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 = 0.15, 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 = {2760, 2762, 2716, 2721, 2719} \begin {gather*} -\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)}} \end {gather*}

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 2716

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 2719

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

Rule 2721

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

Rule 2760

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[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 2762

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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.07, size = 66, normalized size = 0.35 \begin {gather*} \frac {\, _2F_1\left (-\frac {1}{4},\frac {17}{4};\frac {3}{4};\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [4]{1+\sin (c+d x)}}{4 \sqrt [4]{2} a^3 d e \sqrt {e \cos (c+d x)}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(695\) vs. \(2(191)=382\).
time = 12.74, size = 696, normalized size = 3.72

method	result	size
default	\(\text {Expression too large to display}\)	\(696\)

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,method=_RETURNVERBOSE)

[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))*(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)^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))*(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)^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))*(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)^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))*(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)^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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 283, normalized size = 1.51 \begin {gather*} -\frac {21 \, {\left (3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} - 4 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 4 i \, \sqrt {2} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (-3 i \, \sqrt {2} \cos \left (d x + c\right )^{3} + {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} + 4 i \, \sqrt {2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 4 i \, \sqrt {2} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (21 \, \cos \left (d x + c\right )^{4} - 98 \, \cos \left (d x + c\right )^{2} - 63 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 54\right )} \sqrt {\cos \left (d x + c\right )}}{117 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} e^{\frac {3}{2}} - 4 \, a^{3} d \cos \left (d x + c\right ) e^{\frac {3}{2}} + {\left (a^{3} d \cos \left (d x + c\right )^{3} e^{\frac {3}{2}} - 4 \, a^{3} d \cos \left (d x + c\right ) e^{\frac {3}{2}}\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

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]

-1/117*(21*(3*I*sqrt(2)*cos(d*x + c)^3 + (I*sqrt(2)*cos(d*x + c)^3 - 4*I*sqrt(2)*cos(d*x + c))*sin(d*x + c) -
4*I*sqrt(2)*cos(d*x + c))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) +
21*(-3*I*sqrt(2)*cos(d*x + c)^3 + (-I*sqrt(2)*cos(d*x + c)^3 + 4*I*sqrt(2)*cos(d*x + c))*sin(d*x + c) + 4*I*sq
rt(2)*cos(d*x + c))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(21*
cos(d*x + c)^4 - 98*cos(d*x + c)^2 - 63*(cos(d*x + c)^2 - 1)*sin(d*x + c) + 54)*sqrt(cos(d*x + c)))/(3*a^3*d*c
os(d*x + c)^3*e^(3/2) - 4*a^3*d*cos(d*x + c)*e^(3/2) + (a^3*d*cos(d*x + c)^3*e^(3/2) - 4*a^3*d*cos(d*x + c)*e^
(3/2))*sin(d*x + c))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3065 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

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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