3.259 \(\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx\)

Optimal. Leaf size=217 \[ \frac{853 \sin (c+d x) \sqrt{\cos (c+d x)}}{3072 a^3 d (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x) \sqrt{\cos (c+d x)}}{768 a^2 d (a \cos (c+d x)+a)^{5/2}}+\frac{35 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{19 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 a d (a \cos (c+d x)+a)^{7/2}} \]

[Out]

(35*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]^(5/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) - (19*Cos[c + d*x]^(3/2)*Sin[c + d*x
])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)) - (187*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(768*a^2*d*(a + a*Cos[c + d*x])
^(5/2)) + (853*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3072*a^3*d*(a + a*Cos[c + d*x])^(3/2))

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Rubi [A]  time = 0.556924, 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} \[ \frac{853 \sin (c+d x) \sqrt{\cos (c+d x)}}{3072 a^3 d (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x) \sqrt{\cos (c+d x)}}{768 a^2 d (a \cos (c+d x)+a)^{5/2}}+\frac{35 \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{19 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 a d (a \cos (c+d x)+a)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(9/2),x]

[Out]

(35*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]^(5/2)*Sin[c + d*x])/(8*d*(a + a*Cos[c + d*x])^(9/2)) - (19*Cos[c + d*x]^(3/2)*Sin[c + d*x
])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)) - (187*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(768*a^2*d*(a + a*Cos[c + d*x])
^(5/2)) + (853*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3072*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{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx &=-\frac{\cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{5 a}{2}-7 a \cos (c+d x)\right )}{(a+a \cos (c+d x))^{7/2}} \, dx}{8 a^2}\\ &=-\frac{\cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{19 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2}}-\frac{\int \frac{\sqrt{\cos (c+d x)} \left (\frac{57 a^2}{4}-\frac{65}{2} a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{48 a^4}\\ &=-\frac{\cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{19 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2}}-\frac{187 \sqrt{\cos (c+d x)} \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2}}-\frac{\int \frac{\frac{187 a^3}{8}-\frac{333}{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{5}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{19 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2}}-\frac{187 \sqrt{\cos (c+d x)} \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac{853 \sqrt{\cos (c+d x)} \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2}}-\frac{\int -\frac{105 a^4}{16 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{384 a^8}\\ &=-\frac{\cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{19 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2}}-\frac{187 \sqrt{\cos (c+d x)} \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac{853 \sqrt{\cos (c+d x)} \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2}}+\frac{35 \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{2048 a^4}\\ &=-\frac{\cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{19 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2}}-\frac{187 \sqrt{\cos (c+d x)} \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac{853 \sqrt{\cos (c+d x)} \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2}}-\frac{35 \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{35 \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{5}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac{19 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2}}-\frac{187 \sqrt{\cos (c+d x)} \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac{853 \sqrt{\cos (c+d x)} \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 6.03563, size = 347, normalized size = 1.6 \[ \frac{2 \sin \left (\frac{c}{2}+\frac{d x}{2}\right ) \cos ^9\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{9/2} \left (\frac{1}{8} \left (\frac{1}{1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}+\frac{7}{6 \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{35}{24 \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{35}{16 \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^4}\right )+\frac{35 \sqrt{\cos ^2\left (\frac{1}{2} (c+d x)\right )} \csc \left (\frac{c}{2}+\frac{d x}{2}\right ) \sin ^{-1}\left (\frac{\sin \left (\frac{c}{2}+\frac{d x}{2}\right )}{\sqrt{\cos ^2\left (\frac{1}{2} (c+d x)\right )}}\right )}{128 \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{9/2}}\right )}{d \sqrt{\cos ^2\left (\frac{1}{2} (c+d x)\right )} (a (\cos (c+d x)+1))^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(9/2),x]

[Out]

(2*Cos[c/2 + (d*x)/2]^9*Sin[c/2 + (d*x)/2]*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^(9/2)*((35*ArcSin[Sin
[c/2 + (d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Sqrt[Cos[(c + d*x)/2]^2]*Csc[c/2 + (d*x)/2])/(128*(1 - Sec[(c + d*x)
/2]^2*Sin[c/2 + (d*x)/2]^2)^(9/2)) + (35/(16*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^4) + 35/(24*(1 - Se
c[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^3) + 7/(6*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^2) + (1 - Sec[(
c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^(-1))/8))/(d*Sqrt[Cos[(c + d*x)/2]^2]*(a*(1 + Cos[c + d*x]))^(9/2))

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Maple [A]  time = 0.386, size = 346, normalized size = 1.6 \begin{align*}{\frac{\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{7}}{6144\,d{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{15}} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{7}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( 853\,\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \left ( \cos \left ( dx+c \right ) \right ) ^{4}-34\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+105\,\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 ) -364\,\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+315\,\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 ) -350\,\sqrt{2}\cos \left ( dx+c \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+315\,\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 ) -105\,\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+105\,\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \sin \left ( dx+c \right ) \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(7/2)/(a+cos(d*x+c)*a)^(9/2),x)

[Out]

1/6144/d*2^(1/2)/a^5*cos(d*x+c)^(7/2)*(-1+cos(d*x+c))^7*(a*(1+cos(d*x+c)))^(1/2)*(853*2^(1/2)*(cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*cos(d*x+c)^4-34*cos(d*x+c)^3*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+105*arcsin((-1+cos(d*
x+c))/sin(d*x+c))*cos(d*x+c)^3*sin(d*x+c)-364*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^2+315*arcsi
n((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^2*sin(d*x+c)-350*2^(1/2)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)
+315*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)*sin(d*x+c)-105*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+10
5*arcsin((-1+cos(d*x+c))/sin(d*x+c))*sin(d*x+c))/(cos(d*x+c)/(1+cos(d*x+c)))^(7/2)/sin(d*x+c)^15

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{\frac{7}{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)^(7/2)/(a+a*cos(d*x+c))^(9/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^(7/2)/(a*cos(d*x + c) + a)^(9/2), x)

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Fricas [A]  time = 2.27238, size = 684, normalized size = 3.15 \begin{align*} \frac{105 \, \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 (853 \, \cos \left (d x + c\right )^{3} + 819 \, \cos \left (d x + c\right )^{2} + 455 \, \cos \left (d x + c\right ) + 105\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{6144 \,{\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)^(7/2)/(a+a*cos(d*x+c))^(9/2),x, algorithm="fricas")

[Out]

1/6144*(105*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*(853*cos(d*x + c)^3 + 819*cos(d*x + c)^2 + 455*cos(d*x + c) + 105)*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)**(7/2)/(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{7}{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)^(7/2)/(a+a*cos(d*x+c))^(9/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^(7/2)/(a*cos(d*x + c) + a)^(9/2), x)