\(\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx\) [590]
Optimal result
Integrand size = 35, antiderivative size = 473 \[
\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (19 a^2 A b+8 A b^3+63 a^3 B-14 a b^2 B\right ) \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \left (8 A b^2+a^2 (25 A-63 B)+2 a b (3 A-7 B)\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^3 d \sqrt {\sec (c+d x)}}+\frac {2 \left (25 a^2 A-4 A b^2+7 a b B\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 a^2 d}+\frac {2 (A b+7 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}
\]
[Out]
2/105*(25*A*a^2-4*A*b^2+7*B*a*b)*sec(d*x+c)^(3/2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a^2/d+2/35*(A*b+7*B*a)*sec
(d*x+c)^(5/2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/a/d+2/7*A*sec(d*x+c)^(7/2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d
+2/105*(a-b)*(19*A*a^2*b+8*A*b^3+63*B*a^3-14*B*a*b^2)*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^4/d/sec(d*x+c)^(1/2)+2/105*(a-b)*(8*A*b^2+a^2*(25*A-63*B)+2*a*b*(3*A-7*B))*csc(d*x+c)*El
lipticF((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^3/d/sec(d*x+c)^(1/2)
Rubi [A] (verified)
Time = 1.93 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3040, 3078, 3134, 3077, 2895,
3073} \[
\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {2 \left (25 a^2 A+7 a b B-4 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{105 a^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (a^2 (25 A-63 B)+2 a b (3 A-7 B)+8 A b^2\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{105 a^3 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \left (63 a^3 B+19 a^2 A b-14 a b^2 B+8 A b^3\right ) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{105 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (7 a B+A b) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{35 a d}+\frac {2 A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}{7 d}
\]
[In]
Int[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]
[Out]
(2*(a - b)*Sqrt[a + b]*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*Elliptic
E[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)])/(105*a^4*d*Sqrt[Sec[c + d*x]]) + (2*(a - b)*Sqrt[a + b]*
(8*A*b^2 + a^2*(25*A - 63*B) + 2*a*b*(3*A - 7*B))*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)])/(105*a^3*d*Sqrt[Sec[c + d*x]]) + (2*(25*a^2*A - 4*A*b^2 + 7*a*b*B)*Sqrt[a +
b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c
+ d*x]^(5/2)*Sin[c + d*x])/(35*a*d) + (2*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*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 3040
Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p, Int[(a + b*Sin[e + f*x])^m*((
c + d*Sin[e + f*x])^n/(g*Sin[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d
, 0] && !IntegerQ[p] && !(IntegerQ[m] && IntegerQ[n])
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 3078
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 - A*b)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e
+ f*x])^n/(f*(m + 1)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c +
d*Sin[e + f*x])^(n - 1)*Simp[c*(a*A - b*B)*(m + 1) + d*n*(A*b - a*B) + (d*(a*A - b*B)*(m + 1) - c*(A*b - a*B)*
(m + 2))*Sin[e + f*x] - d*(A*b - a*B)*(m + n + 2)*Sin[e + f*x]^2, 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] && LtQ[m, -1] && GtQ[n, 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])))
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)} (A+B \cos (c+d x))}{\cos ^{\frac {9}{2}}(c+d x)} \, dx \\ & = \frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{7} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} (A b+7 a B)+\frac {1}{2} (5 a A+7 b B) \cos (c+d x)+2 A b \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 (A b+7 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} \left (25 a^2 A-4 A b^2+7 a b B\right )+\frac {1}{4} a (23 A b+21 a B) \cos (c+d x)+\frac {1}{2} b (A b+7 a B) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{35 a} \\ & = \frac {2 \left (25 a^2 A-4 A b^2+7 a b B\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 a^2 d}+\frac {2 (A b+7 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} \left (19 a^2 A b+8 A b^3+63 a^3 B-14 a b^2 B\right )+\frac {1}{8} a \left (25 a^2 A+2 A b^2+49 a b B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{105 a^2} \\ & = \frac {2 \left (25 a^2 A-4 A b^2+7 a b B\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 a^2 d}+\frac {2 (A b+7 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {\left ((a-b) \left (8 A b^2+a^2 (25 A-63 B)+2 a b (3 A-7 B)\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}{105 a^2}-\frac {\left (\left (-19 a^2 A b-8 A b^3-63 a^3 B+14 a b^2 B\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}{105 a^2} \\ & = \frac {2 (a-b) \sqrt {a+b} \left (19 a^2 A b+8 A b^3+63 a^3 B-14 a b^2 B\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}}}{105 a^4 d \sqrt {\sec (c+d x)}}+\frac {2 (a-b) \sqrt {a+b} \left (8 A b^2+a^2 (25 A-63 B)+2 a b (3 A-7 B)\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}}}{105 a^3 d \sqrt {\sec (c+d x)}}+\frac {2 \left (25 a^2 A-4 A b^2+7 a b B\right ) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 a^2 d}+\frac {2 (A b+7 a B) \sqrt {a+b \cos (c+d x)} \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 a d}+\frac {2 A \sqrt {a+b \cos (c+d x)} \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d} \\
\end{align*}
Mathematica [A] (warning: unable to verify)
Time = 15.83 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.10
\[
\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\frac {2 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \left (-2 (a+b) \left (19 a^2 A b+8 A b^3+63 a^3 B-14 a b^2 B\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right )+2 a (a+b) \left (8 A b^2-2 a b (3 A+7 B)+a^2 (25 A+63 B)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )-\left (19 a^2 A b+8 A b^3+63 a^3 B-14 a b^2 B\right ) \cos (c+d x) (a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{105 a^3 d \sqrt {a+b \cos (c+d x)} \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )}}+\frac {\sqrt {a+b \cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \left (19 a^2 A b+8 A b^3+63 a^3 B-14 a b^2 B\right ) \sin (c+d x)}{105 a^3}+\frac {2 \sec ^2(c+d x) (A b \sin (c+d x)+7 a B \sin (c+d x))}{35 a}+\frac {2 \sec (c+d x) \left (25 a^2 A \sin (c+d x)-4 A b^2 \sin (c+d x)+7 a b B \sin (c+d x)\right )}{105 a^2}+\frac {2}{7} A \sec ^2(c+d x) \tan (c+d x)\right )}{d}
\]
[In]
Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]
[Out]
(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*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)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B))*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)] - (19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c
+ d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^3*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[a + b*Cos
[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Sin[c + d*x])/(105*a^3) + (2*
Sec[c + d*x]^2*(A*b*Sin[c + d*x] + 7*a*B*Sin[c + d*x]))/(35*a) + (2*Sec[c + d*x]*(25*a^2*A*Sin[c + d*x] - 4*A*
b^2*Sin[c + d*x] + 7*a*b*B*Sin[c + d*x]))/(105*a^2) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/7))/d
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(4627\) vs. \(2(427)=854\).
Time = 29.53 (sec) , antiderivative size = 4628, normalized size of antiderivative =
9.78
| | |
| method | result | size |
| | |
| parts |
\(\text {Expression too large to display}\) |
\(4628\) |
| default |
\(\text {Expression too large to display}\) |
\(4673\) |
| | |
|
|
|
|
[In]
int((A+B*cos(d*x+c))*sec(d*x+c)^(9/2)*(a+cos(d*x+c)*b)^(1/2),x,method=_RETURNVERBOSE)
[Out]
2/105*A/d*sec(d*x+c)^(9/2)/(1+cos(d*x+c))/(a+cos(d*x+c)*b)^(1/2)*(25*a^4*cos(d*x+c)^4*sin(d*x+c)+15*a^4*cos(d*
x+c)^2*sin(d*x+c)-a^2*b^2*cos(d*x+c)^4*sin(d*x+c)+8*cos(d*x+c)^5*b^4*sin(d*x+c)+16*EllipticE(cot(d*x+c)-csc(d*
x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4
*cos(d*x+c)^5-50*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))
^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^4*cos(d*x+c)^5+8*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/
2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^4*cos(d*x+c)^4-25*Ellipt
icF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*a^4*cos(d*x+c)^4+8*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(
1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^3*cos(d*x+c)^4-19*EllipticF(cot(d*x+c)-csc(d*
x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^3
*b*cos(d*x+c)^4-2*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b)
)^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^2*b^2*cos(d*x+c)^4-8*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b)
)^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^3*cos(d*x+c)^4+38
*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+
c)/(1+cos(d*x+c)))^(1/2)*a^3*b*cos(d*x+c)^5+38*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d
*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^2*b^2*cos(d*x+c)^5+16*EllipticE(cot(d
*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)
))^(1/2)*a*b^3*cos(d*x+c)^5-38*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(
d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^3*b*cos(d*x+c)^5-4*EllipticF(cot(d*x+c)-csc(d*x+c),(-
(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^2*b^2*co
s(d*x+c)^5-16*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1
/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^3*cos(d*x+c)^5+19*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/
2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^3*b*cos(d*x+c)^4+19*Elli
pticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1
+cos(d*x+c)))^(1/2)*a^2*b^2*cos(d*x+c)^4+8*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1
+cos(d*x+c))/(a+b))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*b^4-25*cos(d*x+c)^6*(cos(d*x+c
)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/
(a+b))^(1/2))*a^4-19*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1
/2)*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a^3*b-2*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/
2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b^2
-8*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticF(cot(
d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^3+19*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)
*b)/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a^3*b+19*cos(d*x+c)^6*(c
os(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),
(-(a-b)/(a+b))^(1/2))*a^2*b^2+8*cos(d*x+c)^6*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c)
)/(a+b))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^3+25*a^4*cos(d*x+c)^3*sin(d*x+c)+15*a
^4*cos(d*x+c)*sin(d*x+c)+25*cos(d*x+c)^5*a^3*b*sin(d*x+c)+19*cos(d*x+c)^5*a^2*b^2*sin(d*x+c)-4*cos(d*x+c)^5*a*
b^3*sin(d*x+c)+18*a^3*b*cos(d*x+c)^2*sin(d*x+c)+18*a^3*b*cos(d*x+c)^3*sin(d*x+c)-a^2*b^2*cos(d*x+c)^3*sin(d*x+
c)+44*a^3*b*cos(d*x+c)^4*sin(d*x+c)+4*a*b^3*cos(d*x+c)^4*sin(d*x+c))/a^3+2/15*B/d*sec(d*x+c)^(9/2)/(1+cos(d*x+
c))/(a+cos(d*x+c)*b)^(1/2)*(-2*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)
))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*a*b^2*cos(d*x+c)^4-7*EllipticF(cot(d*x+c)-csc(d*x+c),(-
(a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*a^2*b*cos(
d*x+c)^4+2*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x
+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*a*b^2*cos(d*x+c)^4+9*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*
(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*a^2*b*cos(d*x+c)^4-18*cos(d*x+
c)^5*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(
d*x+c)/(1+cos(d*x+c)))^(1/2)*a^3+18*cos(d*x+c)^5*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+
cos(d*x+c)))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a^3+4*a^2*b*cos(d*x+c)^3*sin(d*x+c)-1
4*cos(d*x+c)^5*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(
1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^2*b+4*cos(d*x+c)^5*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/
2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^2+18*cos(d*x+c)^5*((a+
cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),(-
(a-b)/(a+b))^(1/2))*a^2*b-4*cos(d*x+c)^5*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+
c)))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a*b^2+9*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-
b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*a^3*cos(d*x+c
)^4-2*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b
)/(1+cos(d*x+c))/(a+b))^(1/2)*b^3*cos(d*x+c)^4-9*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(cos(d*
x+c)/(1+cos(d*x+c)))^(1/2)*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*a^3*cos(d*x+c)^4+9*cos(d*x+c)^6*Ellip
ticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+
cos(d*x+c)))^(1/2)*a^3-2*cos(d*x+c)^6*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/
(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^3-9*cos(d*x+c)^6*((a+cos(d*x+c)*b)/(1+cos(d*x+
c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a^3+3
*a^3*cos(d*x+c)^2*sin(d*x+c)-4*cos(d*x+c)^5*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d
*x+c)))^(1/2)*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*b^3+9*cos(d*x+c)^6*EllipticE(cot(d*x+c)-cs
c(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)
*a^2*b-2*cos(d*x+c)^6*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(
a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^2-7*cos(d*x+c)^6*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/
2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*a^2*b+2*cos(d*x+c)^
6*((a+cos(d*x+c)*b)/(1+cos(d*x+c))/(a+b))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*EllipticF(cot(d*x+c)-csc(d*x
+c),(-(a-b)/(a+b))^(1/2))*a*b^2+9*cos(d*x+c)^4*a^3*sin(d*x+c)-2*cos(d*x+c)^5*b^3*sin(d*x+c)+3*cos(d*x+c)^3*a^3
*sin(d*x+c)+4*cos(d*x+c)^4*a^2*b*sin(d*x+c)-cos(d*x+c)^4*a*b^2*sin(d*x+c)+cos(d*x+c)^5*a*b^2*sin(d*x+c)+9*cos(
d*x+c)^5*a^2*b*sin(d*x+c))/a^2
Fricas [F]
\[
\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x }
\]
[In]
integrate((A+B*cos(d*x+c))*sec(d*x+c)^(9/2)*(a+b*cos(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
integral((B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(9/2), x)
Sympy [F(-1)]
Timed out. \[
\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\text {Timed out}
\]
[In]
integrate((A+B*cos(d*x+c))*sec(d*x+c)**(9/2)*(a+b*cos(d*x+c))**(1/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x }
\]
[In]
integrate((A+B*cos(d*x+c))*sec(d*x+c)^(9/2)*(a+b*cos(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate((B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(9/2), x)
Giac [F]
\[
\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {9}{2}} \,d x }
\]
[In]
integrate((A+B*cos(d*x+c))*sec(d*x+c)^(9/2)*(a+b*cos(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate((B*cos(d*x + c) + A)*sqrt(b*cos(d*x + c) + a)*sec(d*x + c)^(9/2), x)
Mupad [F(-1)]
Timed out. \[
\int \sqrt {a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx=\int \left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,\sqrt {a+b\,\cos \left (c+d\,x\right )} \,d x
\]
[In]
int((A + B*cos(c + d*x))*(1/cos(c + d*x))^(9/2)*(a + b*cos(c + d*x))^(1/2),x)
[Out]
int((A + B*cos(c + d*x))*(1/cos(c + d*x))^(9/2)*(a + b*cos(c + d*x))^(1/2), x)