3.395 \(\int \frac{1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=237 \[ \frac{853 \sin (c+d x)}{3072 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x)}{768 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{35 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{1024 \sqrt{2} a^{9/2} d}-\frac{19 \sin (c+d x)}{96 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac{\sin (c+d x)}{8 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}} \]

[Out]

(35*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sq
rt[Sec[c + d*x]])/(1024*Sqrt[2]*a^(9/2)*d) - Sin[c + d*x]/(8*d*(a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2))
- (19*Sin[c + d*x])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)) - (187*Sin[c + d*x])/(768*a^2*d*(a
+ a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (853*Sin[c + d*x])/(3072*a^3*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[S
ec[c + d*x]])

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Rubi [A]  time = 0.636186, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {4222, 2765, 2977, 2978, 12, 2782, 205} \[ \frac{853 \sin (c+d x)}{3072 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x)}{768 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{35 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{1024 \sqrt{2} a^{9/2} d}-\frac{19 \sin (c+d x)}{96 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac{\sin (c+d x)}{8 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)),x]

[Out]

(35*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sq
rt[Sec[c + d*x]])/(1024*Sqrt[2]*a^(9/2)*d) - Sin[c + d*x]/(8*d*(a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2))
- (19*Sin[c + d*x])/(96*a*d*(a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)) - (187*Sin[c + d*x])/(768*a^2*d*(a
+ a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]) + (853*Sin[c + d*x])/(3072*a^3*d*(a + a*Cos[c + d*x])^(3/2)*Sqrt[S
ec[c + d*x]])

Rule 4222

Int[(csc[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sin[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
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 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{1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx\\ &=-\frac{\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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{\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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{\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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{\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}+\frac{853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int -\frac{105 a^4}{16 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{384 a^8}\\ &=-\frac{\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}+\frac{853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}}+\frac{\left (35 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{2048 a^4}\\ &=-\frac{\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}+\frac{853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}}-\frac{\left (35 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \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 ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{1024 \sqrt{2} a^{9/2} d}-\frac{\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac{5}{2}}(c+d x)}-\frac{19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}}+\frac{853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 6.09238, size = 395, normalized size = 1.67 \[ \frac{2 \sin \left (\frac{c}{2}+\frac{d x}{2}\right ) \sqrt{\frac{1}{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}} \sqrt{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )} \cos ^9\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{9/2} \left (\frac{1}{8} \left (\frac{1}{1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}+\frac{7}{6 \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{35}{24 \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{35}{16 \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^4}\right )+\frac{35 \sqrt{\cos ^2\left (\frac{1}{2} (c+d x)\right )} \csc \left (\frac{c}{2}+\frac{d x}{2}\right ) \sin ^{-1}\left (\frac{\sin \left (\frac{c}{2}+\frac{d x}{2}\right )}{\sqrt{\cos ^2\left (\frac{1}{2} (c+d x)\right )}}\right )}{128 \left (1-\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{9/2}}\right )}{d \sqrt{\cos ^2\left (\frac{1}{2} (c+d x)\right )} (a (\cos (c+d x)+1))^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)),x]

[Out]

(2*Cos[c/2 + (d*x)/2]^9*Sin[c/2 + (d*x)/2]*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/
2]^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 - Sec[(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 = 0.431, size = 354, normalized size = 1.5 \begin{align*} -{\frac{\sqrt{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{8}\cos \left ( dx+c \right ) }{6144\,d{a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{17}}\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}+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 ) -34\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{2}\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 ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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 ) \cos \left ( dx+c \right ) \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 ) }}}+105\,\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \sin \left ( dx+c \right ) -105\,\sqrt{2}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{7}{2}}} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+cos(d*x+c)*a)^(9/2)/sec(d*x+c)^(7/2),x)

[Out]

-1/6144/d*2^(1/2)/a^5*(a*(1+cos(d*x+c)))^(1/2)*(-1+cos(d*x+c))^8*cos(d*x+c)*(853*2^(1/2)*(cos(d*x+c)/(1+cos(d*
x+c)))^(1/2)*cos(d*x+c)^4+105*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^3*sin(d*x+c)-34*cos(d*x+c)^3*2^(1/
2)*(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)^2*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*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)*sin(d*x+c)-
350*2^(1/2)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+105*arcsin((-1+cos(d*x+c))/sin(d*x+c))*sin(d*x+c)-105
*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2))/(1/cos(d*x+c))^(7/2)/(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)/sin(d*x+c)^
17

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{9}{2}} \sec \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(1/((a*cos(d*x + c) + a)^(9/2)*sec(d*x + c)^(7/2)), x)

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Fricas [A]  time = 2.36078, size = 655, normalized size = 2.76 \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{\cos \left (d x + c\right )}}{\sqrt{a} \sin \left (d x + c\right )}\right ) - \frac{2 \,{\left (853 \, \cos \left (d x + c\right )^{4} + 819 \, \cos \left (d x + c\right )^{3} + 455 \, \cos \left (d x + c\right )^{2} + 105 \, \cos \left (d x + c\right )\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \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(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(7/2),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(sqrt(2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) - 2*(853*co
s(d*x + c)^4 + 819*cos(d*x + c)^3 + 455*cos(d*x + c)^2 + 105*cos(d*x + c))*sqrt(a*cos(d*x + c) + a)*sin(d*x +
c)/sqrt(cos(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(1/(a+a*cos(d*x+c))**(9/2)/sec(d*x+c)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{9}{2}} \sec \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(7/2),x, algorithm="giac")

[Out]

integrate(1/((a*cos(d*x + c) + a)^(9/2)*sec(d*x + c)^(7/2)), x)