3.260 \(\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx\)

Optimal. Leaf size=217 \[ \frac{73 \sin (c+d x) \sqrt{\cos (c+d x)}}{1024 a^3 d (a \cos (c+d x)+a)^{3/2}}+\frac{33 \sin (c+d x) \sqrt{\cos (c+d x)}}{256 a^2 d (a \cos (c+d x)+a)^{5/2}}+\frac{45 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{32 a d (a \cos (c+d x)+a)^{7/2}} \]

[Out]

(45*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(1024*Sqrt[2]*a^(9/2
)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x]
)/(32*a*d*(a + a*Cos[c + d*x])^(7/2)) + (33*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(256*a^2*d*(a + a*Cos[c + d*x])^(
5/2)) + (73*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2))

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Rubi [A]  time = 0.570841, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2765, 2977, 2978, 12, 2782, 205} \[ \frac{73 \sin (c+d x) \sqrt{\cos (c+d x)}}{1024 a^3 d (a \cos (c+d x)+a)^{3/2}}+\frac{33 \sin (c+d x) \sqrt{\cos (c+d x)}}{256 a^2 d (a \cos (c+d x)+a)^{5/2}}+\frac{45 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{32 a d (a \cos (c+d x)+a)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(9/2),x]

[Out]

(45*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])])/(1024*Sqrt[2]*a^(9/2
)*d) - (Cos[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) - (5*Sqrt[Cos[c + d*x]]*Sin[c + d*x]
)/(32*a*d*(a + a*Cos[c + d*x])^(7/2)) + (33*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(256*a^2*d*(a + a*Cos[c + d*x])^(
5/2)) + (73*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1024*a^3*d*(a + a*Cos[c + d*x])^(3/2))

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 2978

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*
x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, 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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2782

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[(-2*a)/f, Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c
+ d*Sin[e + f*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]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{\int \frac{\sqrt{\cos (c+d x)} \left (\frac{3 a}{2}-6 a \cos (c+d x)\right )}{(a+a \cos (c+d x))^{7/2}} \, dx}{8 a^2}\\ &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{5 \sqrt{\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}-\frac{\int \frac{\frac{15 a^2}{4}-21 a^2 \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx}{48 a^4}\\ &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{5 \sqrt{\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac{33 \sqrt{\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}-\frac{\int \frac{\frac{21 a^3}{8}-\frac{99}{4} a^3 \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx}{192 a^6}\\ &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{5 \sqrt{\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac{33 \sqrt{\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac{73 \sqrt{\cos (c+d x)} \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2}}-\frac{\int -\frac{135 a^4}{16 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{384 a^8}\\ &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{5 \sqrt{\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac{33 \sqrt{\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac{73 \sqrt{\cos (c+d x)} \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2}}+\frac{45 \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{2048 a^4}\\ &=-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{5 \sqrt{\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac{33 \sqrt{\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac{73 \sqrt{\cos (c+d x)} \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2}}-\frac{45 \operatorname{Subst}\left (\int \frac{1}{2 a^2+a x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{1024 a^3 d}\\ &=\frac{45 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{1024 \sqrt{2} a^{9/2} d}-\frac{\cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{5 \sqrt{\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac{33 \sqrt{\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac{73 \sqrt{\cos (c+d x)} \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 2.13667, size = 158, normalized size = 0.73 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left ((2466 \cos (c+d x)+1072 \cos (2 (c+d x))+702 \cos (3 (c+d x))+73 \cos (4 (c+d x))+999) \sqrt{2-2 \sec (c+d x)}+5760 \cos ^8\left (\frac{1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sqrt{\sin ^2\left (\frac{1}{2} (c+d x)\right ) (-\sec (c+d x))}\right )\right )}{65536 \sqrt{2} a^4 d \sqrt{\cos (c+d x)-1} \sqrt{a (\cos (c+d x)+1)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(9/2),x]

[Out]

(Sec[(c + d*x)/2]^6*(5760*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^8 + (999 + 2466*C
os[c + d*x] + 1072*Cos[2*(c + d*x)] + 702*Cos[3*(c + d*x)] + 73*Cos[4*(c + d*x)])*Sqrt[2 - 2*Sec[c + d*x]])*Ta
n[(c + d*x)/2])/(65536*Sqrt[2]*a^4*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])])

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Maple [A]  time = 0.387, size = 346, normalized size = 1.6 \begin{align*} -{\frac{\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{6}}{2048\,d{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{13}} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( 73\,\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+45\,\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +278\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+135\,\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -156\,\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+135\,\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -150\,\sqrt{2}\cos \left ( dx+c \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+45\,\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \sin \left ( dx+c \right ) -45\,\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(5/2)/(a+cos(d*x+c)*a)^(9/2),x)

[Out]

-1/2048/d*2^(1/2)/a^5*cos(d*x+c)^(5/2)*(-1+cos(d*x+c))^6*(a*(1+cos(d*x+c)))^(1/2)*(73*2^(1/2)*(cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*cos(d*x+c)^4+45*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^3*sin(d*x+c)+278*cos(d*x+c)^3*
2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+135*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^2*sin(d*x+c)-156*2
^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^2+135*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)*sin(d*
x+c)-150*2^(1/2)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+45*arcsin((-1+cos(d*x+c))/sin(d*x+c))*sin(d*x+c)
-45*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))/(cos(d*x+c)/(1+cos(d*x+c)))^(5/2)/sin(d*x+c)^13

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^(5/2)/(a*cos(d*x + c) + a)^(9/2), x)

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Fricas [A]  time = 2.23223, size = 680, normalized size = 3.13 \begin{align*} \frac{45 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{a} \arctan \left (\frac{\sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \,{\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) + 2 \,{\left (73 \, \cos \left (d x + c\right )^{3} + 351 \, \cos \left (d x + c\right )^{2} + 195 \, \cos \left (d x + c\right ) + 45\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2048 \,{\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(9/2),x, algorithm="fricas")

[Out]

1/2048*(45*sqrt(2)*(cos(d*x + c)^5 + 5*cos(d*x + c)^4 + 10*cos(d*x + c)^3 + 10*cos(d*x + c)^2 + 5*cos(d*x + c)
+ 1)*sqrt(a)*arctan(1/2*sqrt(2)*sqrt(a*cos(d*x + c) + a)*sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x +
c)^2 + a*cos(d*x + c))) + 2*(73*cos(d*x + c)^3 + 351*cos(d*x + c)^2 + 195*cos(d*x + c) + 45)*sqrt(a*cos(d*x +
c) + a)*sqrt(cos(d*x + c))*sin(d*x + c))/(a^5*d*cos(d*x + c)^5 + 5*a^5*d*cos(d*x + c)^4 + 10*a^5*d*cos(d*x +
c)^3 + 10*a^5*d*cos(d*x + c)^2 + 5*a^5*d*cos(d*x + c) + a^5*d)

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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)**(5/2)/(a+a*cos(d*x+c))**(9/2),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{5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(9/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^(5/2)/(a*cos(d*x + c) + a)^(9/2), x)