\(\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^{\frac {11}{2}}(c+d x) \, dx\) [1502]
Optimal result
Integrand size = 45, antiderivative size = 592 \[
\int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^5 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (16 A b^3+12 a b^2 (A-2 B)+6 a^2 b (6 A-3 B+7 C)+3 a^3 (49 A-25 B+63 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^3 d}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 (A b+9 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}
\]
[Out]
2/315*(8*A*b^3+75*B*a^3-12*B*a*b^2+a^2*b*(13*A+21*C))*sec(d*x+c)^(3/2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a^3/d
-2/315*(6*A*b^2-9*B*a*b-7*a^2*(7*A+9*C))*sec(d*x+c)^(5/2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a^2/d+2/63*(A*b+9*
B*a)*sec(d*x+c)^(7/2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a/d+2/9*A*sec(d*x+c)^(9/2)*sin(d*x+c)*(a+b*cos(d*x+c))
^(1/2)/d-2/315*(a-b)*(16*A*b^4-57*B*a^3*b-24*B*a*b^3+6*a^2*b^2*(4*A+7*C)-21*a^4*(7*A+9*C))*csc(d*x+c)*Elliptic
E((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*cos(d*x+c)^(1/2)*(a*(1
-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^5/d/sec(d*x+c)^(1/2)-2/315*(a-b)*(16*A*b^3+12*a*b^2
*(A-2*B)+6*a^2*b*(6*A-3*B+7*C)+3*a^3*(49*A-25*B+63*C))*csc(d*x+c)*EllipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)
/cos(d*x+c)^(1/2),((-a-b)/(a-b))^(1/2))*(a+b)^(1/2)*cos(d*x+c)^(1/2)*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(
d*x+c))/(a-b))^(1/2)/a^4/d/sec(d*x+c)^(1/2)
Rubi [A] (verified)
Time = 2.78 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4306, 3126, 3134, 3077, 2895,
3073} \[
\int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right ) \sqrt {a+b \cos (c+d x)}}{315 a^2 d}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right ) \sqrt {a+b \cos (c+d x)}}{315 a^3 d}-\frac {2 (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (3 a^3 (49 A-25 B+63 C)+6 a^2 b (6 A-3 B+7 C)+12 a b^2 (A-2 B)+16 A b^3\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{315 a^4 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \left (-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{315 a^5 d \sqrt {\sec (c+d x)}}+\frac {2 (9 a B+A b) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{63 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{9 d}
\]
[In]
Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]
[Out]
(-2*(a - b)*Sqrt[a + b]*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Sqrt
[Cos[c + d*x]]*Csc[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a
+ b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^5*d*Sqrt[Sec[
c + d*x]]) - (2*(a - b)*Sqrt[a + b]*(16*A*b^3 + 12*a*b^2*(A - 2*B) + 6*a^2*b*(6*A - 3*B + 7*C) + 3*a^3*(49*A -
25*B + 63*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos
[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(
315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b^3 + 75*a^3*B - 12*a*b^2*B + a^2*b*(13*A + 21*C))*Sqrt[a + b*Cos[c +
d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*a^3*d) - (2*(6*A*b^2 - 9*a*b*B - 7*a^2*(7*A + 9*C))*Sqrt[a + b*Cos
[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*a^2*d) + (2*(A*b + 9*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*
x]^(7/2)*Sin[c + d*x])/(63*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)
Rule 2895
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(
Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqrt[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]
*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2]], -(a + b)/(a - b)], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Rule 3073
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A*(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e +
f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e +
f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ
[A, B] && PosQ[(c + d)/b]
Rule 3077
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*s
in[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(A - B)/(a - b), Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e
+ f*x]]), x], x] - Dist[(A*b - a*B)/(a - b), Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin
[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2
- d^2, 0] && NeQ[A, B]
Rule 3126
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/(d*(n + 1)
*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) +
(c*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*
c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n +
1)))*Sin[e + f*x]^2, 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] && GtQ[m, 0] && LtQ[n, -1]
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 4306
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]
Rubi steps \begin{align*}
\text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \\ & = \frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{9} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} (A b+9 a B)+\frac {1}{2} (7 a A+9 b B+9 a C) \cos (c+d x)+\frac {3}{2} b (2 A+3 C) \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 (A b+9 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} \left (-6 A b^2+9 a b B+7 a^2 (7 A+9 C)\right )+\frac {1}{4} a (47 A b+45 a B+63 b C) \cos (c+d x)+b (A b+9 a B) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{63 a} \\ & = -\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 (A b+9 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{8} \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right )+\frac {1}{8} a \left (2 A b^2+207 a b B+21 a^2 (7 A+9 C)\right ) \cos (c+d x)-\frac {1}{4} b \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{315 a^2} \\ & = \frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^3 d}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 (A b+9 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3}{16} \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right )-\frac {3}{16} a \left (4 A b^3-75 a^3 B-6 a b^2 B-3 a^2 b (37 A+49 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{945 a^3} \\ & = \frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^3 d}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 (A b+9 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}-\frac {\left (\left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{315 a^3}-\frac {\left ((a-b) \left (16 A b^3+12 a b^2 (A-2 B)+6 a^2 b (6 A-3 B+7 C)+3 a^3 (49 A-25 B+63 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{315 a^3} \\ & = -\frac {2 (a-b) \sqrt {a+b} \left (16 A b^4-57 a^3 b B-24 a b^3 B+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^5 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (16 A b^3+12 a b^2 (A-2 B)+6 a^2 b (6 A-3 B+7 C)+3 a^3 (49 A-25 B+63 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 \left (8 A b^3+75 a^3 B-12 a b^2 B+a^2 b (13 A+21 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^3 d}-\frac {2 \left (6 A b^2-9 a b B-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 (A b+9 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \\
\end{align*}
Mathematica [B] (warning: unable to verify)
Leaf count is larger than twice the leaf count of optimal. \(4669\) vs. \(2(592)=1184\).
Time = 27.47 (sec) , antiderivative size = 4669, normalized size of antiderivative = 7.89
\[
\int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Result too large to show}
\]
[In]
Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]
[Out]
(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4*A - 24*a^2*A*b^2 - 16*A*b^4 + 57*a^3*b*B + 24*a*b^3*
B + 189*a^4*C - 42*a^2*b^2*C)*Sin[c + d*x])/(315*a^4) + (2*Sec[c + d*x]^3*(A*b*Sin[c + d*x] + 9*a*B*Sin[c + d*
x]))/(63*a) + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] - 6*A*b^2*Sin[c + d*x] + 9*a*b*B*Sin[c + d*x] + 63*a^2*
C*Sin[c + d*x]))/(315*a^2) + (2*Sec[c + d*x]*(13*a^2*A*b*Sin[c + d*x] + 8*A*b^3*Sin[c + d*x] + 75*a^3*B*Sin[c
+ d*x] - 12*a*b^2*B*Sin[c + d*x] + 21*a^2*b*C*Sin[c + d*x]))/(315*a^3) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))
/d + (2*((-7*a*A)/(15*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*A*b^2)/(105*a*Sqrt[a + b*Cos[c + d*x]]
*Sqrt[Sec[c + d*x]]) + (16*A*b^4)/(315*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (19*b*B)/(105*Sqrt[a
+ b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*b^3*B)/(105*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*
a*C)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*C)/(15*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d
*x]]) - (4*A*b*Sqrt[Sec[c + d*x]])/(35*Sqrt[a + b*Cos[c + d*x]]) + (4*A*b^3*Sqrt[Sec[c + d*x]])/(63*a^2*Sqrt[a
+ b*Cos[c + d*x]]) + (16*A*b^5*Sqrt[Sec[c + d*x]])/(315*a^4*Sqrt[a + b*Cos[c + d*x]]) + (5*a*B*Sqrt[Sec[c + d
*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (17*b^2*B*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[a + b*Cos[c + d*x]]) - (8*b^4*
B*Sqrt[Sec[c + d*x]])/(105*a^3*Sqrt[a + b*Cos[c + d*x]]) - (2*b*C*Sqrt[Sec[c + d*x]])/(15*Sqrt[a + b*Cos[c + d
*x]]) + (2*b^3*C*Sqrt[Sec[c + d*x]])/(15*a^2*Sqrt[a + b*Cos[c + d*x]]) - (7*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c +
d*x]])/(15*Sqrt[a + b*Cos[c + d*x]]) + (8*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a^2*Sqrt[a + b*Cos[c
+ d*x]]) + (16*A*b^5*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*a^4*Sqrt[a + b*Cos[c + d*x]]) - (19*b^2*B*Cos[
2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[a + b*Cos[c + d*x]]) - (8*b^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x
]])/(105*a^3*Sqrt[a + b*Cos[c + d*x]]) - (3*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x
]]) + (2*b^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*a^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2
*Sec[c + d*x]]*(-2*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*
Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin
[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b
*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(Cos[c
+ d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (16*A*
b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*
Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*a^4*d*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*((b*Sqrt[
Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*
A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + C
os[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(a + b)*(-16*A*b^3
+ 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*x)
/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)
/(a + b)]*Sec[c + d*x] + (16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos
[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*a^4*(a + b*Cos[c + d*x])^(3/2)*(Sec[
(c + d*x)/2]^2)^(3/2)) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(-16*A*b^4 + 57*a
^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(
a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c
+ d*x)/2]^2 + a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63
*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a +
b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4
*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(105*a
^4*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + ((-2*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B -
6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((
a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(a +
b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[
Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c
+ d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*
A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d
*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*a^4*Sqrt[a + b*Cos[c + d*x]]*(Sec[
(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((16*
A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x]
)*Sec[(c + d*x)/2]^6)/2 - ((a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A
+ 9*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(
a + b)]*Sec[(c + d*x)/2]^2*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])
))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*
C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a
+ b)]*Sec[(c + d*x)/2]^2*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*
x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - 2*(a + b)*(-16*
A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d
*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a
+ b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - b*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21
*a^4*(7*A + 9*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (16*A*b^4 - 57*a^3*b*B - 24*
a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan
[(c + d*x)/2] + 2*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*
x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (3*a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) -
6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]
*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-
(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + (a*(a + b)*(-16*A*b
^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*
x)/2]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*(-((b*Sec[(c + d*x)/2]^2*Sin[c
+ d*x])/(a + b)) + ((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((a + b*Cos[c
+ d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C)
+ 3*a^3*(49*A + 25*B + 63*C))*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c +
d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c +
d*x)/2]^2)/(a + b)]) - ((a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9
*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*
x)/2]^4*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2] + a*(a + b)*(-16*A*b^3 +
12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*x)/2
]], (-a + b)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(
a + b)]*Sec[c + d*x]*Tan[c + d*x]))/(315*a^4*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(8091\) vs. \(2(540)=1080\).
Time = 33.20 (sec) , antiderivative size = 8092, normalized size of antiderivative =
13.67
| | |
| method | result | size |
| | |
| parts |
\(\text {Expression too large to display}\) |
\(8092\) |
| default |
\(\text {Expression too large to display}\) |
\(8180\) |
| | |
|
|
|
|
[In]
int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2)*(a+b*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
[Out]
result too large to display
Fricas [F]
\[
\int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x }
\]
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2)*(a+b*cos(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
integral((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(11/2), x)
Sympy [F(-1)]
Timed out. \[
\int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out}
\]
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**(11/2)*(a+b*cos(d*x+c))**(1/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x }
\]
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2)*(a+b*cos(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(11/2), x)
Giac [F]
\[
\int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x }
\]
[In]
integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(11/2)*(a+b*cos(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(11/2), x)
Mupad [F(-1)]
Timed out. \[
\int \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,\sqrt {a+b\,\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x
\]
[In]
int((1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
[Out]
int((1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)