3.3.55 \(\int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^3} \, dx\) [255]

Optimal. Leaf size=132 \[ \frac {18 e^3 (e \cos (c+d x))^{5/2}}{5 a^3 d}+\frac {6 e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^3 d \sqrt {e \cos (c+d x)}}+\frac {6 e^5 \sqrt {e \cos (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{a d (a+a \sin (c+d x))^2} \]

[Out]

18/5*e^3*(e*cos(d*x+c))^(5/2)/a^3/d+4*e*(e*cos(d*x+c))^(9/2)/a/d/(a+a*sin(d*x+c))^2+6*e^6*(cos(1/2*d*x+1/2*c)^
2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/a^3/d/(e*cos(d*x+c))^(1/2)+
6*e^5*sin(d*x+c)*(e*cos(d*x+c))^(1/2)/a^3/d

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Rubi [A]
time = 0.11, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2759, 2761, 2715, 2721, 2720} \begin {gather*} \frac {6 e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^3 d \sqrt {e \cos (c+d x)}}+\frac {6 e^5 \sin (c+d x) \sqrt {e \cos (c+d x)}}{a^3 d}+\frac {18 e^3 (e \cos (c+d x))^{5/2}}{5 a^3 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{a d (a \sin (c+d x)+a)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^3,x]

[Out]

(18*e^3*(e*Cos[c + d*x])^(5/2))/(5*a^3*d) + (6*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(a^3*d*Sqrt[e
*Cos[c + d*x]]) + (6*e^5*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(a^3*d) + (4*e*(e*Cos[c + d*x])^(9/2))/(a*d*(a + a
*Sin[c + d*x])^2)

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 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

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))^{11/2}}{(a+a \sin (c+d x))^3} \, dx &=\frac {4 e (e \cos (c+d x))^{9/2}}{a d (a+a \sin (c+d x))^2}+\frac {\left (9 e^2\right ) \int \frac {(e \cos (c+d x))^{7/2}}{a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac {18 e^3 (e \cos (c+d x))^{5/2}}{5 a^3 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{a d (a+a \sin (c+d x))^2}+\frac {\left (9 e^4\right ) \int (e \cos (c+d x))^{3/2} \, dx}{a^3}\\ &=\frac {18 e^3 (e \cos (c+d x))^{5/2}}{5 a^3 d}+\frac {6 e^5 \sqrt {e \cos (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{a d (a+a \sin (c+d x))^2}+\frac {\left (3 e^6\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{a^3}\\ &=\frac {18 e^3 (e \cos (c+d x))^{5/2}}{5 a^3 d}+\frac {6 e^5 \sqrt {e \cos (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{a d (a+a \sin (c+d x))^2}+\frac {\left (3 e^6 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{a^3 \sqrt {e \cos (c+d x)}}\\ &=\frac {18 e^3 (e \cos (c+d x))^{5/2}}{5 a^3 d}+\frac {6 e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a^3 d \sqrt {e \cos (c+d x)}}+\frac {6 e^5 \sqrt {e \cos (c+d x)} \sin (c+d x)}{a^3 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{a d (a+a \sin (c+d x))^2}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.14, size = 66, normalized size = 0.50 \begin {gather*} -\frac {2 \sqrt [4]{2} (e \cos (c+d x))^{13/2} \, _2F_1\left (\frac {3}{4},\frac {13}{4};\frac {17}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{13 a^3 d e (1+\sin (c+d x))^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^3,x]

[Out]

(-2*2^(1/4)*(e*Cos[c + d*x])^(13/2)*Hypergeometric2F1[3/4, 13/4, 17/4, (1 - Sin[c + d*x])/2])/(13*a^3*d*e*(1 +
Sin[c + d*x])^(13/4))

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Maple [A]
time = 2.62, size = 181, normalized size = 1.37

method	result	size
default	\(-\frac {2 e^{6} \left (-8 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+34 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-19 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*cos(d*x+c))^(11/2)/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-2/5/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^6*(-8*sin(1/2*d*x+1/2*c)^7-20*sin(1/2*d*x+1/
2*c)^4*cos(1/2*d*x+1/2*c)+12*sin(1/2*d*x+1/2*c)^5+15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(
1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+10*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+34*sin(1/2*d*x+1/2*c)^3-
19*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))^(11/2)/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

e^(11/2)*integrate(cos(d*x + c)^(11/2)/(a*sin(d*x + c) + a)^3, x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 95, normalized size = 0.72 \begin {gather*} \frac {-15 i \, \sqrt {2} e^{\frac {11}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 i \, \sqrt {2} e^{\frac {11}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (\cos \left (d x + c\right )^{2} e^{\frac {11}{2}} + 5 \, e^{\frac {11}{2}} \sin \left (d x + c\right ) - 20 \, e^{\frac {11}{2}}\right )} \sqrt {\cos \left (d x + c\right )}}{5 \, a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))^(11/2)/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/5*(-15*I*sqrt(2)*e^(11/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*I*sqrt(2)*e^(11/2)*
weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*(cos(d*x + c)^2*e^(11/2) + 5*e^(11/2)*sin(d*x +
c) - 20*e^(11/2))*sqrt(cos(d*x + c)))/(a^3*d)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*cos(d*x+c))**(11/2)/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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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))^(11/2)/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^(11/2)*e^(11/2)/(a*sin(d*x + c) + a)^3, 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 )}^{11/2}}{{\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((e*cos(c + d*x))^(11/2)/(a + a*sin(c + d*x))^3,x)

[Out]

int((e*cos(c + d*x))^(11/2)/(a + a*sin(c + d*x))^3, x)

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