3.1337 \(\int \frac{(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^{\frac{11}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx\)

Optimal. Leaf size=305 \[ \frac{2 (19 A-3 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A-93 B+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A-129 B+273 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}-\frac{2 (A-9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}} \]

[Out]

-((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sq
rt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A - 129*B + 273*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x]
)/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A - 93*B + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a +
a*Cos[c + d*x]]) + (2*(19*A - 3*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) -
(2*(A - 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c
+ d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.13197, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4221, 3043, 2984, 12, 2782, 205} \[ \frac{2 (19 A-3 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A-93 B+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A-129 B+273 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}-\frac{2 (A-9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]],x]

[Out]

-((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])]*Sq
rt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(257*A - 129*B + 273*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x]
)/(315*d*Sqrt[a + a*Cos[c + d*x]]) - (2*(29*A - 93*B + 21*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a +
a*Cos[c + d*x]]) + (2*(19*A - 3*B + 21*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Cos[c + d*x]]) -
(2*(A - 9*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Cos[c + d*x]]) + (2*A*Sec[c + d*x]^(9/2)*Sin[c
+ d*x])/(9*d*Sqrt[a + a*Cos[c + d*x]])

Rule 4221

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 3043

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

Rule 2984

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*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^(n + 1))/(f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin
[e + f*x])^(n + 1)*Simp[A*(a*d*m + b*c*(n + 1)) - B*(a*c*m + b*d*(n + 1)) + b*(B*c - A*d)*(m + n + 2)*Sin[e +
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && LtQ[n, -1] && (IntegerQ[n] || EqQ[m + 1/2, 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{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{11}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{11}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx\\ &=\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{2} a (A-9 B)+\frac{1}{2} a (8 A+9 C) \cos (c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{9 a}\\ &=-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{4} a^2 (19 A-3 B+21 C)-\frac{3}{2} a^2 (A-9 B) \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{63 a^2}\\ &=\frac{2 (19 A-3 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{3}{8} a^3 (29 A-93 B+21 C)+\frac{3}{2} a^3 (19 A-3 B+21 C) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{315 a^3}\\ &=-\frac{2 (29 A-93 B+21 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (16 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{16} a^4 (257 A-129 B+273 C)-\frac{3}{8} a^4 (29 A-93 B+21 C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{945 a^4}\\ &=\frac{2 (257 A-129 B+273 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (29 A-93 B+21 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (32 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int -\frac{945 a^5 (A-B+C)}{32 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{945 a^5}\\ &=\frac{2 (257 A-129 B+273 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (29 A-93 B+21 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}-\left ((A-B+C) \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\\ &=\frac{2 (257 A-129 B+273 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (29 A-93 B+21 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (2 a (A-B+C) \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 )}{d}\\ &=-\frac{\sqrt{2} (A-B+C) \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)}}{\sqrt{a} d}+\frac{2 (257 A-129 B+273 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (29 A-93 B+21 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt{a+a \cos (c+d x)}}+\frac{2 (19 A-3 B+21 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}-\frac{2 (A-9 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 34.5876, size = 7123, normalized size = 23.35 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]],x]

[Out]

Result too large to show

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Maple [B]  time = 0.21, size = 1131, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2)/(a+a*cos(d*x+c))^(1/2),x)

[Out]

1/315/d*2^(1/2)/a*(315*A*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))+45*B*2^(1/2)*cos
(d*x+c)*sin(d*x+c)-129*B*2^(1/2)*cos(d*x+c)^4*sin(d*x+c)-1575*B*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*a
rcsin((-1+cos(d*x+c))/sin(d*x+c))-1575*B*cos(d*x+c)^4*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))
/sin(d*x+c))-3150*B*cos(d*x+c)^3*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))-3150*B*c
os(d*x+c)^2*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))-315*B*cos(d*x+c)^5*(cos(d*x+c
)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))+93*B*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)-9*B*2^(1/2)*co
s(d*x+c)^2*sin(d*x+c)+315*A*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^5+
315*C*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^5+1575*A*(cos(d*x+c)/(1+
cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^4+1575*C*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*ar
csin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^4+3150*A*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/
sin(d*x+c))*cos(d*x+c)^3+3150*C*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c
)^3+3150*A*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^2+3150*C*(cos(d*x+c
)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)^2+1575*A*(cos(d*x+c)/(1+cos(d*x+c)))^(9/
2)*arcsin((-1+cos(d*x+c))/sin(d*x+c))*cos(d*x+c)+1575*C*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*x+c
))/sin(d*x+c))*cos(d*x+c)+257*A*2^(1/2)*cos(d*x+c)^4*sin(d*x+c)+273*C*2^(1/2)*cos(d*x+c)^4*sin(d*x+c)-29*A*2^(
1/2)*cos(d*x+c)^3*sin(d*x+c)-21*C*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)+57*A*2^(1/2)*cos(d*x+c)^2*sin(d*x+c)+63*C*2^
(1/2)*cos(d*x+c)^2*sin(d*x+c)-5*A*2^(1/2)*cos(d*x+c)*sin(d*x+c)+315*C*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin
((-1+cos(d*x+c))/sin(d*x+c))+35*A*2^(1/2)*sin(d*x+c)-315*B*(cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arcsin((-1+cos(d*
x+c))/sin(d*x+c)))*cos(d*x+c)*(1/cos(d*x+c))^(11/2)*(a*(1+cos(d*x+c)))^(1/2)*sin(d*x+c)^8/(-1+cos(d*x+c))^4/(1
+cos(d*x+c))^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2)/(a+a*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.90098, size = 579, normalized size = 1.9 \begin{align*} \frac{\frac{315 \, \sqrt{2}{\left ({\left (A - B + C\right )} a \cos \left (d x + c\right )^{5} +{\left (A - B + C\right )} a \cos \left (d x + c\right )^{4}\right )} \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 )}{\sqrt{a}} + \frac{2 \,{\left ({\left (257 \, A - 129 \, B + 273 \, C\right )} \cos \left (d x + c\right )^{4} -{\left (29 \, A - 93 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (19 \, A - 3 \, B + 21 \, C\right )} \cos \left (d x + c\right )^{2} - 5 \,{\left (A - 9 \, B\right )} \cos \left (d x + c\right ) + 35 \, A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{315 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2)/(a+a*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/315*(315*sqrt(2)*((A - B + C)*a*cos(d*x + c)^5 + (A - B + C)*a*cos(d*x + c)^4)*arctan(sqrt(2)*sqrt(a*cos(d*x
+ c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c)))/sqrt(a) + 2*((257*A - 129*B + 273*C)*cos(d*x + c)^4 - (2
9*A - 93*B + 21*C)*cos(d*x + c)^3 + 3*(19*A - 3*B + 21*C)*cos(d*x + c)^2 - 5*(A - 9*B)*cos(d*x + c) + 35*A)*sq
rt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4)

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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((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**(11/2)/(a+a*cos(d*x+c))**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2)/(a+a*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sec(d*x + c)^(11/2)/sqrt(a*cos(d*x + c) + a), x)