\(\int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx\) [1102]

Optimal result

Integrand size = 43, antiderivative size = 318 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d}+\frac {2 b^2 \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a^4 (a+b) d}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}} \]

[Out]

2/5*(5*A*b^3-3*B*a^3-5*B*a*b^2+a^2*b*(3*A+5*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(
1/2*d*x+1/2*c),2^(1/2))/a^4/d+2/21*(7*A*b^2-7*B*a*b+a^2*(5*A+7*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/
2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/a^3/d+2*b^2*(A*b^2-a*(B*b-C*a))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/
2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/a^4/(a+b)/d+2/7*A*sin(d*x+c)/a/d/cos(d*x+c)^(7/2
)-2/5*(A*b-B*a)*sin(d*x+c)/a^2/d/cos(d*x+c)^(5/2)+2/21*(7*A*b^2-7*B*a*b+a^2*(5*A+7*C))*sin(d*x+c)/a^3/d/cos(d*
x+c)^(3/2)-2/5*(5*A*b^3-3*B*a^3-5*B*a*b^2+a^2*b*(3*A+5*C))*sin(d*x+c)/a^4/d/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3134, 3138, 2719, 3081, 2720, 2884} \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\frac {2 b^2 \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a^4 d (a+b)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d}+\frac {2 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{5 a^4 d}-\frac {2 \sin (c+d x) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{5 a^4 d \sqrt {\cos (c+d x)}}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)} \]

[In]

Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]

[Out]

(2*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*a^4*d) + (2*(7*A*b^2 - 7*
a*b*B + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*a^3*d) + (2*b^2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*
b)/(a + b), (c + d*x)/2, 2])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(7*a*d*Cos[c + d*x]^(7/2)) - (2*(A*b - a*B)*
Sin[c + d*x])/(5*a^2*d*Cos[c + d*x]^(5/2)) + (2*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*a^3*d*
Cos[c + d*x]^(3/2)) - (2*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))*Sin[c + d*x])/(5*a^4*d*Sqrt[Cos[c
+ d*x]])

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 3081

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3134

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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3138

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \int \frac {-\frac {7}{2} (A b-a B)+\frac {1}{2} a (5 A+7 C) \cos (c+d x)+\frac {5}{2} A b \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{7 a} \\ & = \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 \int \frac {\frac {5}{4} \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right )+\frac {1}{4} a (4 A b+21 a B) \cos (c+d x)-\frac {21}{4} b (A b-a B) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{35 a^2} \\ & = \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {-\frac {21}{8} \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right )-\frac {1}{8} a \left (28 A b^2-28 a b B-5 a^2 (5 A+7 C)\right ) \cos (c+d x)+\frac {5}{8} b \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{105 a^3} \\ & = \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}}+\frac {16 \int \frac {\frac {5}{16} \left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )+\frac {1}{16} a \left (140 A b^3-63 a^3 B-140 a b^2 B+4 a^2 b (22 A+35 C)\right ) \cos (c+d x)+\frac {21}{16} b \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{105 a^4} \\ & = \frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}}-\frac {16 \int \frac {-\frac {5}{16} b \left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )-\frac {5}{16} a b^2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{105 a^4 b}+\frac {\left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^4} \\ & = \frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}}+\frac {\left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a^3}+\frac {\left (b^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^4} \\ & = \frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^4 d}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^3 d}+\frac {2 b^2 \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a^4 (a+b) d}+\frac {2 A \sin (c+d x)}{7 a d \cos ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sin (c+d x)}{5 a^4 d \sqrt {\cos (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.67 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.31 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\frac {\frac {2 \left (315 A b^4-133 a^3 b B-315 a b^3 B+10 a^4 (5 A+7 C)+7 a^2 b^2 (19 A+45 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {4 a \left (140 A b^3-63 a^3 B-140 a b^2 B+4 a^2 b (22 A+35 C)\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{b (a+b)}-\frac {42 \left (-5 A b^3+3 a^3 B+5 a b^2 B-a^2 b (3 A+5 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b \sqrt {\sin ^2(c+d x)}}+\frac {2 \left (42 \left (a^2 (-A b+a B)+\left (-5 A b^3+3 a^3 B+5 a b^2 B-a^2 b (3 A+5 C)\right ) \cos ^2(c+d x)\right ) \sin (c+d x)+5 \left (a \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (2 (c+d x))+6 a^3 A \tan (c+d x)\right )\right )}{\cos ^{\frac {5}{2}}(c+d x)}}{210 a^4 d} \]

[In]

Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]

[Out]

((2*(315*A*b^4 - 133*a^3*b*B - 315*a*b^3*B + 10*a^4*(5*A + 7*C) + 7*a^2*b^2*(19*A + 45*C))*EllipticPi[(2*b)/(a
+ b), (c + d*x)/2, 2])/(a + b) + (4*a*(140*A*b^3 - 63*a^3*B - 140*a*b^2*B + 4*a^2*b*(22*A + 35*C))*((a + b)*E
llipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) - (42*(-5*A*b^3 + 3*a^3*B
+ 5*a*b^2*B - a^2*b*(3*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[Ar
cSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x
])/(a*b*Sqrt[Sin[c + d*x]^2]) + (2*(42*(a^2*(-(A*b) + a*B) + (-5*A*b^3 + 3*a^3*B + 5*a*b^2*B - a^2*b*(3*A + 5*
C))*Cos[c + d*x]^2)*Sin[c + d*x] + 5*(a*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*Sin[2*(c + d*x)] + 6*a^3*A*Tan[c
+ d*x])))/Cos[c + d*x]^(5/2))/(210*a^4*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(975\) vs. \(2(376)=752\).

Time = 5.49 (sec) , antiderivative size = 976, normalized size of antiderivative = 3.07

method	result	size
default	\(\text {Expression too large to display}\)	\(976\)

[In]

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A/a*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/
2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)
^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+
1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+
2/5*(-A*b+B*a)/a^2/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c
)^2*(24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/
2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+12*(2
*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+
1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(
1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*(A*b^2-B*a*
b+C*a^2)/a^3*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)
^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*
d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))-2*(A*b^2-B*a*b+C*a^2)/a^4*b/sin(1/2*d*x+1/2*c)^2/(2
*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*
d*x+1/2*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))
)-4*(A*b^2-B*a*b+C*a^2)*b^3/a^4/(-2*a*b+2*b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/
(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2)))/sin(1/
2*d*x+1/2*c)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \]

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\text {Timed out} \]

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)**(9/2)/(a+b*cos(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)*cos(d*x + c)^(9/2)), x)

Giac [F]

\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]

[In]

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(9/2)/(a+b*cos(d*x+c)),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)*cos(d*x + c)^(9/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^{9/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \]

[In]

int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*(a + b*cos(c + d*x))),x)

[Out]

int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/(cos(c + d*x)^(9/2)*(a + b*cos(c + d*x))), x)