3.3.34 \(\int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx\) [234]

Optimal. Leaf size=101 \[ \frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {6 e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)}}+\frac {2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d} \]

[Out]

2/7*e*(e*cos(d*x+c))^(7/2)/a/d+2/5*e^3*(e*cos(d*x+c))^(3/2)*sin(d*x+c)/a/d+6/5*e^4*(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/d/cos(d*x+c)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2761, 2715, 2721, 2719} \begin {gather*} \frac {6 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 a d \sqrt {\cos (c+d x)}}+\frac {2 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 a d}+\frac {2 e (e \cos (c+d x))^{7/2}}{7 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(e*Cos[c + d*x])^(7/2))/(7*a*d) + (6*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a*d*Sqrt[Cos[
c + d*x]]) + (2*e^3*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[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 2761

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

Rubi steps

\begin {align*} \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx &=\frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {e^2 \int (e \cos (c+d x))^{5/2} \, dx}{a}\\ &=\frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac {\left (3 e^4\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 a}\\ &=\frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}+\frac {\left (3 e^4 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a \sqrt {\cos (c+d x)}}\\ &=\frac {2 e (e \cos (c+d x))^{7/2}}{7 a d}+\frac {6 e^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)}}+\frac {2 e^3 (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 a d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.11, size = 66, normalized size = 0.65 \begin {gather*} -\frac {4\ 2^{3/4} (e \cos (c+d x))^{11/2} \, _2F_1\left (-\frac {3}{4},\frac {11}{4};\frac {15}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{11 a d e (1+\sin (c+d x))^{11/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*2^(3/4)*(e*Cos[c + d*x])^(11/2)*Hypergeometric2F1[-3/4, 11/4, 15/4, (1 - Sin[c + d*x])/2])/(11*a*d*e*(1 +
Sin[c + d*x])^(11/4))

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Maple [A]
time = 2.48, size = 216, normalized size = 2.14

method	result	size
default	\(\frac {2 e^{5} \left (80 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-160 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+120 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+14 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-40 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(216\)

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)),x,method=_RETURNVERBOSE)

[Out]

2/35/a/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^5*(80*sin(1/2*d*x+1/2*c)^9+56*sin(1/2*d*x+1/2*
c)^6*cos(1/2*d*x+1/2*c)-160*sin(1/2*d*x+1/2*c)^7-56*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+120*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))+
14*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-40*sin(1/2*d*x+1/2*c)^3+5*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((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

e^(9/2)*integrate(cos(d*x + c)^(9/2)/(a*sin(d*x + c) + a), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 104, normalized size = 1.03 \begin {gather*} \frac {21 i \, \sqrt {2} e^{\frac {9}{2}} {\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 i \, \sqrt {2} e^{\frac {9}{2}} {\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 (5 \, \cos \left (d x + c\right )^{3} e^{\frac {9}{2}} + 7 \, \cos \left (d x + c\right ) e^{\frac {9}{2}} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{35 \, a d} \end {gather*}

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)),x, algorithm="fricas")

[Out]

1/35*(21*I*sqrt(2)*e^(9/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) -
21*I*sqrt(2)*e^(9/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(5
*cos(d*x + c)^3*e^(9/2) + 7*cos(d*x + c)*e^(9/2)*sin(d*x + c))*sqrt(cos(d*x + c)))/(a*d)

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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((e*cos(d*x+c))**(9/2)/(a+a*sin(d*x+c)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8857 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((e*cos(d*x+c))^(9/2)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^(9/2)*e^(9/2)/(a*sin(d*x + c) + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}}{a+a\,\sin \left (c+d\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x)),x)

[Out]

int((e*cos(c + d*x))^(9/2)/(a + a*sin(c + d*x)), x)

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