3.189 \(\int \frac{\cos ^{\frac{11}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx\)

Optimal. Leaf size=207 \[ -\frac{21 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac{231 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{63 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{77 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a^3 d}-\frac{21 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 a d (a \cos (c+d x)+a)^2} \]

[Out]

(231*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - (21*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) - (21*Sqrt[Cos[c + d*x]]
*Sin[c + d*x])/(2*a^3*d) + (77*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a^3*d) - (Cos[c + d*x]^(9/2)*Sin[c + d*x])
/(5*d*(a + a*Cos[c + d*x])^3) - (4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (63*Cos[c
+ d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))

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Rubi [A]  time = 0.335191, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2765, 2977, 2748, 2635, 2641, 2639} \[ -\frac{21 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac{231 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{63 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{77 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a^3 d}-\frac{21 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 a d (a \cos (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(231*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - (21*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) - (21*Sqrt[Cos[c + d*x]]
*Sin[c + d*x])/(2*a^3*d) + (77*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a^3*d) - (Cos[c + d*x]^(9/2)*Sin[c + d*x])
/(5*d*(a + a*Cos[c + d*x])^3) - (4*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^2) - (63*Cos[c
+ d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))

Rule 2765

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2977

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2635

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] && Integer
Q[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{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{\cos ^{\frac{11}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx &=-\frac{\cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left (\frac{9 a}{2}-\frac{15}{2} a \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{\cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{4 \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac{\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (42 a^2-\frac{105}{2} a^2 \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{15 a^4}\\ &=-\frac{\cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{4 \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac{63 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{\int \cos ^{\frac{3}{2}}(c+d x) \left (\frac{945 a^3}{4}-\frac{1155}{4} a^3 \cos (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{\cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{4 \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac{63 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{63 \int \cos ^{\frac{3}{2}}(c+d x) \, dx}{4 a^3}+\frac{77 \int \cos ^{\frac{5}{2}}(c+d x) \, dx}{4 a^3}\\ &=-\frac{21 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}+\frac{77 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{10 a^3 d}-\frac{\cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{4 \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac{63 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}-\frac{21 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{4 a^3}+\frac{231 \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=\frac{231 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{21 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac{21 \sqrt{\cos (c+d x)} \sin (c+d x)}{2 a^3 d}+\frac{77 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{10 a^3 d}-\frac{\cos ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{4 \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^2}-\frac{63 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 2.67888, size = 388, normalized size = 1.87 \[ \frac{2 \cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (-\sqrt{\cos (c+d x)} \left (\frac{1}{16} \sec \left (\frac{c}{2}\right ) \left (-770 \sin \left (c+\frac{d x}{2}\right )+840 \sin \left (c+\frac{3 d x}{2}\right )-150 \sin \left (2 c+\frac{3 d x}{2}\right )+238 \sin \left (2 c+\frac{5 d x}{2}\right )+40 \sin \left (3 c+\frac{5 d x}{2}\right )+5 \sin \left (3 c+\frac{7 d x}{2}\right )+5 \sin \left (4 c+\frac{7 d x}{2}\right )-\sin \left (4 c+\frac{9 d x}{2}\right )-\sin \left (5 c+\frac{9 d x}{2}\right )+1210 \sin \left (\frac{d x}{2}\right )\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right )+264 \cot (c)+198 \csc (c)\right )+\frac{42 i \sqrt{2} e^{-i (c+d x)} \left (11 \left (-1+e^{2 i c}\right ) \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},-e^{2 i (c+d x)}\right )+5 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{1}{2},\frac{5}{4},-e^{2 i (c+d x)}\right )+11 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^3 d (\cos (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[(c + d*x)/2]^6*(((42*I)*Sqrt[2]*(11*(1 + E^((2*I)*(c + d*x))) + 11*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)
*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sq
rt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^
((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - Sqrt[Cos[c + d*x]]*(264*Cot[c] + 198*Csc[c] + (S
ec[c/2]*Sec[(c + d*x)/2]^5*(1210*Sin[(d*x)/2] - 770*Sin[c + (d*x)/2] + 840*Sin[c + (3*d*x)/2] - 150*Sin[2*c +
(3*d*x)/2] + 238*Sin[2*c + (5*d*x)/2] + 40*Sin[3*c + (5*d*x)/2] + 5*Sin[3*c + (7*d*x)/2] + 5*Sin[4*c + (7*d*x)
/2] - Sin[4*c + (9*d*x)/2] - Sin[5*c + (9*d*x)/2]))/16)))/(5*a^3*d*(1 + Cos[c + d*x])^3)

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Maple [A]  time = 2.644, size = 296, normalized size = 1.4 \begin{align*} -{\frac{1}{20\,{a}^{3}d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 64\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{12}-288\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}-76\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-210\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-462\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1} \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +530\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}-248\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+19\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(64*cos(1/2*d*x+1/2*c)^12-288*cos(1/2*d*x+1/2*c)
^10-76*cos(1/2*d*x+1/2*c)^8-210*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1
/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^5-462*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)
*cos(1/2*d*x+1/2*c)^5*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+530*cos(1/2*d*x+1/2*c)^6-248*cos(1/2*d*x+1/2*c)^4+
19*cos(1/2*d*x+1/2*c)^2-1)/a^3/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)^5/sin(1
/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{11}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (d x + c\right )^{\frac{11}{2}}}{a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(d*x + c)^(11/2)/(a^3*cos(d*x + c)^3 + 3*a^3*cos(d*x + c)^2 + 3*a^3*cos(d*x + c) + a^3), 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(cos(d*x+c)**(11/2)/(a+a*cos(d*x+c))**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{11}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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