\(\int \sqrt {a+b \cos (c+d x)} (A+C \cos ^2(c+d x)) \sec ^{\frac {11}{2}}(c+d x) \, dx\) [1411]
Optimal result
Integrand size = 37, antiderivative size = 544 \[
\int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (16 A b^4+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^5 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (12 a A b^2+16 A b^3+6 a^2 b (6 A+7 C)+21 a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 b \left (8 A b^2+a^2 (13 A+21 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 a^3 d}-\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 a^2 d}+\frac {2 A b \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}
\]
[Out]
2/315*b*(8*A*b^2+a^2*(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^2-7*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*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+6*a^2
*b^2*(4*A+7*C)-21*a^4*(7*A+9*C))*csc(d*x+c)*EllipticE((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)*(12*a*A*b^2+16*A*b^3+6*a^2*b*(6*A+7*C)+21*a^3*(7*A+9*C))*csc(d*x+c)*Elliptic
F((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 = 2.07 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {4306, 3127, 3134, 3077, 2895,
3073} \[
\int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=-\frac {2 \left (6 A b^2-7 a^2 (7 A+9 C)\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{315 a^2 d}+\frac {2 b \left (a^2 (13 A+21 C)+8 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{315 a^3 d}-\frac {2 (a-b) \sqrt {a+b} \left (-21 a^4 (7 A+9 C)+6 a^2 b^2 (4 A+7 C)+16 A b^4\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{315 a^5 d \sqrt {\sec (c+d x)}}-\frac {2 (a-b) \sqrt {a+b} \left (21 a^3 (7 A+9 C)+6 a^2 b (6 A+7 C)+12 a A b^2+16 A b^3\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{315 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 A \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{9 d}+\frac {2 A b \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{63 a d}
\]
[In]
Int[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]
[Out]
(-2*(a - b)*Sqrt[a + b]*(16*A*b^4 + 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)*Sq
rt[a + b]*(12*a*A*b^2 + 16*A*b^3 + 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*E
llipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - S
ec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(315*a^4*d*Sqrt[Sec[c + d*x]]) + (2*b*(8*A*b^2 +
a^2*(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 - 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*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 3127
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Si
n[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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n
+ 2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(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, 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+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 {A b}{2}+\frac {1}{2} a (7 A+9 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 \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+7 a^2 (7 A+9 C)\right )+\frac {1}{4} a b (47 A+63 C) \cos (c+d x)+A b^2 \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^2-7 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 \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} b \left (8 A b^2+a^2 (13 A+21 C)\right )+\frac {1}{8} a \left (2 A b^2+21 a^2 (7 A+9 C)\right ) \cos (c+d x)-\frac {1}{4} b \left (6 A b^2-7 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 b \left (8 A b^2+a^2 (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^2-7 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 \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+6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)\right )-\frac {3}{16} a b \left (4 A b^2-3 a^2 (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 b \left (8 A b^2+a^2 (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^2-7 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 \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 ((a-b) \left (12 a A b^2+16 A b^3+6 a^2 b (6 A+7 C)+21 a^3 (7 A+9 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 {\left (\left (16 A b^4+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 {2 (a-b) \sqrt {a+b} \left (16 A b^4+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 (12 a A b^2+16 A b^3+6 a^2 b (6 A+7 C)+21 a^3 (7 A+9 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 b \left (8 A b^2+a^2 (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^2-7 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 \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. \(3619\) vs. \(2(544)=1088\).
Time = 22.97 (sec) , antiderivative size = 3619, normalized size of antiderivative = 6.65
\[
\int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Result too large to show}
\]
[In]
Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + 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 + 189*a^4*C - 42*a^2*b^2
*C)*Sin[c + d*x])/(315*a^4) + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] - 6*A*b^2*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] + 21*a^2*b*C*Sin[c + d*x
]))/(315*a^3) + (2*A*b*Sec[c + d*x]^2*Tan[c + d*x])/(63*a) + (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]]) - (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]]) - (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]]) - (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]])/(1
5*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 - 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 + Co
s[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a
^2*b*(6*A + 7*C) + 21*a^3*(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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 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]^2*Tan[(c + d*x)/2]))/(315*a^4*d*Sq
rt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a
+ b)*(-16*A*b^4 - 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)] + 2*a*(a +
b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*S
qrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] +
(16*A*b^4 + 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]^2*
Tan[(c + d*x)/2]))/(315*a^4*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Se
c[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(-16*A*b^4 - 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)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqr
t[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Ta
n[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 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]^2*Tan[(c + d*x)/2]))/(315*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]
^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((16*A*b^4 + 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)/2 - ((a + b)*(-16*A*b^4 - 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)]*((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*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[(
a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((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 - 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)]*(-((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 + C
os[c + d*x]))] + (a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*
x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((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]))] - b*(16*A*b^4 + 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]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (16*A*b^4 + 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]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (16*A*b^4 + 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]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(12*a*A*b^2 -
16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(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]^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 - 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]^2*Sqrt[1 - ((-
a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(315*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec
[(c + d*x)/2]^2]) + ((-2*(a + b)*(-16*A*b^4 - 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)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(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]))]*EllipticF[ArcSin[Tan[(c + d
*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 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]^2*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]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d
*x)/2]^2*Sec[c + d*x]])))
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(5582\) vs. \(2(492)=984\).
Time = 30.00 (sec) , antiderivative size = 5583, normalized size of antiderivative =
10.26
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| method | result | size |
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| parts |
\(\text {Expression too large to display}\) |
\(5583\) |
| default |
\(\text {Expression too large to display}\) |
\(5651\) |
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[In]
int((A+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]
\[
\int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x }
\]
[In]
integrate((A+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 + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(11/2), x)
Sympy [F(-1)]
Timed out. \[
\int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\text {Timed out}
\]
[In]
integrate((A+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]
\[
\int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x }
\]
[In]
integrate((A+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 + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(11/2), x)
Giac [F]
\[
\int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {11}{2}} \,d x }
\]
[In]
integrate((A+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 + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(11/2), x)
Mupad [F(-1)]
Timed out. \[
\int \sqrt {a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {11}{2}}(c+d x) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x
\]
[In]
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(1/2),x)
[Out]
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(11/2)*(a + b*cos(c + d*x))^(1/2), x)