\(\int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx\) [395]

Optimal result

Integrand size = 25, antiderivative size = 237 \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {35 \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{1024 \sqrt {2} a^{9/2} d}-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}}+\frac {853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}} \]

[Out]

-1/8*sin(d*x+c)/d/(a+a*cos(d*x+c))^(9/2)/sec(d*x+c)^(5/2)-19/96*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^(7/2)/sec(d*x+
c)^(3/2)-187/768*sin(d*x+c)/a^2/d/(a+a*cos(d*x+c))^(5/2)/sec(d*x+c)^(1/2)+853/3072*sin(d*x+c)/a^3/d/(a+a*cos(d
*x+c))^(3/2)/sec(d*x+c)^(1/2)+35/2048*arctan(1/2*sin(d*x+c)*a^(1/2)*2^(1/2)/cos(d*x+c)^(1/2)/(a+a*cos(d*x+c))^
(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/a^(9/2)/d*2^(1/2)

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4307, 2844, 3056, 3057, 12, 2861, 211} \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {35 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{1024 \sqrt {2} a^{9/2} d}+\frac {853 \sin (c+d x)}{3072 a^3 d \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac {187 \sin (c+d x)}{768 a^2 d \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac {19 \sin (c+d x)}{96 a d \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac {\sin (c+d x)}{8 d \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}} \]

[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 12

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

Rule 211

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

Rule 2844

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 2861

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 3056

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 3057

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 4307

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 \arctan \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] (verified)

Time = 6.06 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.67 \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \cos ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\frac {1}{1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}} \sqrt {1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )} \left (\frac {35}{128} \arcsin \left (\frac {\sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{\sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )}}\right )+\frac {93 \sin \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{128 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )}}-\frac {163 \sin ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{192 \cos ^2\left (\frac {1}{2} (c+d x)\right )^{3/2}}+\frac {25 \sin ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{48 \cos ^2\left (\frac {1}{2} (c+d x)\right )^{5/2}}-\frac {\sin ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}}{8 \cos ^2\left (\frac {1}{2} (c+d x)\right )^{7/2}}\right )}{d (a (1+\cos (c+d x)))^{9/2}} \]

[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*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((35*ArcSin[S
in[c/2 + (d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]])/128 + (93*Sin[c/2 + (d*x)/2]*Sqrt[1 - Sec[(c + d*x)/2]^2*Sin[c/2
+ (d*x)/2]^2])/(128*Sqrt[Cos[(c + d*x)/2]^2]) - (163*Sin[c/2 + (d*x)/2]^3*Sqrt[1 - Sec[(c + d*x)/2]^2*Sin[c/2
+ (d*x)/2]^2])/(192*(Cos[(c + d*x)/2]^2)^(3/2)) + (25*Sin[c/2 + (d*x)/2]^5*Sqrt[1 - Sec[(c + d*x)/2]^2*Sin[c/2
+ (d*x)/2]^2])/(48*(Cos[(c + d*x)/2]^2)^(5/2)) - (Sin[c/2 + (d*x)/2]^7*Sqrt[1 - Sec[(c + d*x)/2]^2*Sin[c/2 +
(d*x)/2]^2])/(8*(Cos[(c + d*x)/2]^2)^(7/2))))/(d*(a*(1 + Cos[c + d*x]))^(9/2))

Maple [A] (verified)

Time = 4.17 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {\sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (853 \sin \left (d x +c \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+819 \tan \left (d x +c \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-105 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) \cos \left (d x +c \right )+455 \tan \left (d x +c \right ) \sec \left (d x +c \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-420 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )+105 \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-630 \sec \left (d x +c \right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )-420 \left (\sec ^{2}\left (d x +c \right )\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )-105 \left (\sec ^{3}\left (d x +c \right )\right ) \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {2}}{6144 d \left (1+\cos \left (d x +c \right )\right )^{5} \sec \left (d x +c \right )^{\frac {7}{2}} \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, a^{5}}\)	\(314\)

[In]

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

[Out]

1/6144/d*(a*(1+cos(d*x+c)))^(1/2)/(1+cos(d*x+c))^5/sec(d*x+c)^(7/2)/(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*(853*sin
(d*x+c)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+819*tan(d*x+c)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-105
*arcsin(cot(d*x+c)-csc(d*x+c))*cos(d*x+c)+455*tan(d*x+c)*sec(d*x+c)*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-
420*arcsin(cot(d*x+c)-csc(d*x+c))+105*tan(d*x+c)*sec(d*x+c)^2*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)-630*se
c(d*x+c)*arcsin(cot(d*x+c)-csc(d*x+c))-420*sec(d*x+c)^2*arcsin(cot(d*x+c)-csc(d*x+c))-105*sec(d*x+c)^3*arcsin(
cot(d*x+c)-csc(d*x+c)))*2^(1/2)/a^5

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {105 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {2 \, {\left (853 \, \cos \left (d x + c\right )^{4} + 819 \, \cos \left (d x + c\right )^{3} + 455 \, \cos \left (d x + c\right )^{2} + 105 \, \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{6144 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {9}{2}} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {1}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \]

[In]

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

[Out]

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