\(\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [1021]
Optimal result
Integrand size = 43, antiderivative size = 518 \[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}
\]
[Out]
2/3465*(88*B*a^2*b+539*B*b^3-48*a^3*C-6*a*b^2*(33*A+34*C))*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^3/d+2/693*(99*A
*b^2-44*B*a*b+24*C*a^2+81*C*b^2)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b^3/d+2/99*(11*B*b-6*C*a)*cos(d*x+c)*(a+b*c
os(d*x+c))^(5/2)*sin(d*x+c)/b^2/d+2/11*C*cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b/d+2/3465*(88*B*a^3*b
+429*B*a*b^3-48*a^4*C-18*a^2*b^2*(11*A+8*C)+75*b^4*(11*A+9*C))*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^3/d+2/3465*
(88*B*a^4*b+363*B*a^2*b^3+1617*B*b^5-48*C*a^5-18*a^3*b^2*(11*A+6*C)+6*a*b^4*(451*A+348*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)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/b^4/
d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/3465*(a^2-b^2)*(88*B*a^3*b+429*B*a*b^3-48*a^4*C-18*a^2*b^2*(11*A+8*C)+75*b^
4*(11*A+9*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)*(b/(a+b))^(
1/2))*((a+b*cos(d*x+c))/(a+b))^(1/2)/b^4/d/(a+b*cos(d*x+c))^(1/2)
Rubi [A] (verified)
Time = 1.39 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.00, number of
steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {3128, 3102, 2832, 2831, 2742,
2740, 2734, 2732} \[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \left (24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2}}{693 b^3 d}+\frac {2 \sin (c+d x) \left (-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right ) (a+b \cos (c+d x))^{3/2}}{3465 b^3 d}+\frac {2 \sin (c+d x) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)}}{3465 b^3 d}-\frac {2 \left (a^2-b^2\right ) \left (-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{99 b^2 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}
\]
[In]
Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
[Out]
(2*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 18*a^3*b^2*(11*A + 6*C) + 6*a*b^4*(451*A + 348*C))*Sq
rt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^4*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])
- (2*(a^2 - b^2)*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[(a
+ b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(3465*b^4*d*Sqrt[a + b*Cos[c + d*x]]) + (2*
(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 18*a^2*b^2*(11*A + 8*C) + 75*b^4*(11*A + 9*C))*Sqrt[a + b*Cos[c + d*x]]
*Sin[c + d*x])/(3465*b^3*d) + (2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 6*a*b^2*(33*A + 34*C))*(a + b*Cos[c + d*
x])^(3/2)*Sin[c + d*x])/(3465*b^3*d) + (2*(99*A*b^2 - 44*a*b*B + 24*a^2*C + 81*b^2*C)*(a + b*Cos[c + d*x])^(5/
2)*Sin[c + d*x])/(693*b^3*d) + (2*(11*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(99*b
^2*d) + (2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*b*d)
Rule 2732
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 2734
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] && !GtQ[a + b, 0]
Rule 2740
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Rule 2742
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] && !GtQ[a + b, 0]
Rule 2831
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Rule 2832
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Rule 3102
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Rule 3128
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e
+ f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Rubi steps \begin{align*}
\text {integral}& = \frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {2 \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (2 a C+\frac {1}{2} b (11 A+9 C) \cos (c+d x)+\frac {1}{2} (11 b B-6 a C) \cos ^2(c+d x)\right ) \, dx}{11 b} \\ & = \frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {4 \int (a+b \cos (c+d x))^{3/2} \left (\frac {1}{2} a (11 b B-6 a C)+\frac {1}{4} b (77 b B-6 a C) \cos (c+d x)+\frac {1}{4} \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{99 b^2} \\ & = \frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {8 \int (a+b \cos (c+d x))^{3/2} \left (\frac {3}{8} b \left (165 A b^2-22 a b B+12 a^2 C+135 b^2 C\right )+\frac {1}{8} \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) \cos (c+d x)\right ) \, dx}{693 b^3} \\ & = \frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {16 \int \sqrt {a+b \cos (c+d x)} \left (-\frac {3}{16} b \left (22 a^2 b B-539 b^3 B-12 a^3 C-3 a b^2 (209 A+157 C)\right )+\frac {3}{16} \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{3465 b^3} \\ & = \frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {32 \int \frac {\frac {3}{32} b \left (22 a^3 b B+2046 a b^3 B-12 a^4 C+75 b^4 (11 A+9 C)+9 a^2 b^2 (187 A+141 C)\right )+\frac {3}{32} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{10395 b^3} \\ & = \frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}-\frac {\left (\left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{3465 b^4}+\frac {\left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{3465 b^4} \\ & = \frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d}+\frac {\left (\left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{3465 b^4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{3465 b^4 \sqrt {a+b \cos (c+d x)}} \\ & = \frac {2 \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-18 a^3 b^2 (11 A+6 C)+6 a b^4 (451 A+348 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3465 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-18 a^2 b^2 (11 A+8 C)+75 b^4 (11 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-6 a b^2 (33 A+34 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3465 b^3 d}+\frac {2 \left (99 A b^2-44 a b B+24 a^2 C+81 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{11 b d} \\
\end{align*}
Mathematica [A] (verified)
Time = 4.25 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.79
\[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (22 a^3 b B+2046 a b^3 B-12 a^4 C+75 b^4 (11 A+9 C)+9 a^2 b^2 (187 A+141 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (-88 a^4 b B-363 a^2 b^3 B-1617 b^5 B+48 a^5 C+18 a^3 b^2 (11 A+6 C)-6 a b^4 (451 A+348 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (2 \left (-352 a^3 b B+8844 a b^3 B+192 a^4 C+18 a^2 b^2 (44 A+27 C)+15 b^4 (506 A+435 C)\right ) \sin (c+d x)+b \left (4 \left (66 a^2 b B+1463 b^3 B-36 a^3 C+48 a b^2 (33 A+34 C)\right ) \sin (2 (c+d x))+5 b \left (\left (396 A b^2+440 a b B+12 a^2 C+513 b^2 C\right ) \sin (3 (c+d x))+7 b ((22 b B+24 a C) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right )\right )\right )}{27720 b^4 d \sqrt {a+b \cos (c+d x)}}
\]
[In]
Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
[Out]
(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(22*a^3*b*B + 2046*a*b^3*B - 12*a^4*C + 75*b^4*(11*A + 9*C) + 9*a^
2*b^2*(187*A + 141*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*
a^5*C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*
EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(2*(-352*a^3*b*B + 8844*a*b^3*B + 192*a^4*C +
18*a^2*b^2*(44*A + 27*C) + 15*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(4*(66*a^2*b*B + 1463*b^3*B - 36*a^3*C +
48*a*b^2*(33*A + 34*C))*Sin[2*(c + d*x)] + 5*b*((396*A*b^2 + 440*a*b*B + 12*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)
] + 7*b*((22*b*B + 24*a*C)*Sin[4*(c + d*x)] + 9*b*C*Sin[5*(c + d*x)])))))/(27720*b^4*d*Sqrt[a + b*Cos[c + d*x]
])
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(2602\) vs. \(2(544)=1088\).
Time = 25.13 (sec) , antiderivative size = 2603, normalized size of antiderivative =
5.03
| | |
| method | result | size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(2603\) |
| parts |
\(\text {Expression too large to display}\) |
\(2964\) |
| | |
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[In]
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)
[Out]
-2/3465*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*((-12320*B*b^6-23520*C*a*b^5-50400*C*b^6)*
sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^6+14960*B*a*b^5+24640*B*b^6+6960*C*a^2*b^4+47040*C*a*b^5+56
880*C*b^6)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-10296*A*a*b^5-11880*A*b^6-4664*B*a^2*b^4-22440*B*a*b^5-22
792*B*b^6+24*C*a^3*b^3-10440*C*a^2*b^4-43368*C*a*b^5-34920*C*b^6)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(356
4*A*a^2*b^4+10296*A*a*b^5+9240*A*b^6-44*B*a^3*b^3+4664*B*a^2*b^4+17248*B*a*b^5+10472*B*b^6+24*C*a^4*b^2-24*C*a
^3*b^3+7872*C*a^2*b^4+19848*C*a*b^5+13860*C*b^6)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-198*A*a^3*b^3-1782*
A*a^2*b^4-4224*A*a*b^5-2640*A*b^6+88*B*a^4*b^2+22*B*a^3*b^3-3102*B*a^2*b^4-4884*B*a*b^5-1848*B*b^6-48*C*a^5*b-
12*C*a^4*b^2-108*C*a^3*b^3-2196*C*a^2*b^4-4842*C*a*b^5-2790*C*b^6)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+675
*C*b^6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+
1/2*c),(-2*b/(a-b))^(1/2))+48*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/
2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6-1617*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/
2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^6+825*A*b^6*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^
(1/2))-48*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2
*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6+429*B*a*b^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+
(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+108*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(
a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^3+2088*C*(
sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(
-2*b/(a-b))^(1/2))*a^2*b^4+96*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/
2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2+2706*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*si
n(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^4+20160*C*b^6*cos
(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12-2706*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+
b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^5+363*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*
b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^3+198*A
*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c)
,(-2*b/(a-b))^(1/2))*a^4*b^2-2088*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))
^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^5-363*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*s
in(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^4+1617*B*(sin(1/
2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(
a-b))^(1/2))*a*b^5+88*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ellip
ticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b-341*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1
/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^3-88*B*(sin(1/2*d*x+1/2*c)^2
)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a
^4*b^2-198*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/
2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2+198*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(
a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^3-819*a^2*C*(sin(1/2*d*x+1/2*c)^2)^(1
/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^4+4
8*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2
*c),(-2*b/(a-b))^(1/2))*a^5*b-1023*A*a^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(
a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^4-108*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b
)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b^2-88*B*(sin(1
/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/
(a-b))^(1/2))*a^5*b)/b^4/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-2*b
*sin(1/2*d*x+1/2*c)^2+a+b)^(1/2)/d
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.29 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.57
\[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")
[Out]
1/10395*(sqrt(2)*(-96*I*C*a^6 + 176*I*B*a^5*b - 36*I*(11*A + 5*C)*a^4*b^2 + 660*I*B*a^3*b^3 + 3*I*(121*A + 123
*C)*a^2*b^4 - 2904*I*B*a*b^5 - 225*I*(11*A + 9*C)*b^6)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8
/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(2)*(96*I*C*a^6 - 176*I*
B*a^5*b + 36*I*(11*A + 5*C)*a^4*b^2 - 660*I*B*a^3*b^3 - 3*I*(121*A + 123*C)*a^2*b^4 + 2904*I*B*a*b^5 + 225*I*(
11*A + 9*C)*b^6)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*co
s(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) - 3*sqrt(2)*(48*I*C*a^5*b - 88*I*B*a^4*b^2 + 18*I*(11*A + 6*C)*a^3*b
^3 - 363*I*B*a^2*b^4 - 6*I*(451*A + 348*C)*a*b^5 - 1617*I*B*b^6)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b
^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3
*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) - 3*sqrt(2)*(-48*I*C*a^5*b + 88*I*B*a^4*b^2 - 18*I*(11*A +
6*C)*a^3*b^3 + 363*I*B*a^2*b^4 + 6*I*(451*A + 348*C)*a*b^5 + 1617*I*B*b^6)*sqrt(b)*weierstrassZeta(4/3*(4*a^2
- 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2
)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) + 6*(315*C*b^6*cos(d*x + c)^4 + 24*C*a^4*b^2 - 44
*B*a^3*b^3 + 3*(33*A + 19*C)*a^2*b^4 + 968*B*a*b^5 + 75*(11*A + 9*C)*b^6 + 35*(12*C*a*b^5 + 11*B*b^6)*cos(d*x
+ c)^3 + 5*(3*C*a^2*b^4 + 110*B*a*b^5 + 9*(11*A + 9*C)*b^6)*cos(d*x + c)^2 - (18*C*a^3*b^3 - 33*B*a^2*b^4 - 6*
(132*A + 101*C)*a*b^5 - 539*B*b^6)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^5*d)
Sympy [F(-1)]
Timed out. \[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out}
\]
[In]
integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)
[Out]
Timed out
Maxima [F]
\[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \,d x }
\]
[In]
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/2)*cos(d*x + c)^2, x)
Giac [F]
\[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \,d x }
\]
[In]
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")
[Out]
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/2)*cos(d*x + c)^2, x)
Mupad [F(-1)]
Timed out. \[
\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x
\]
[In]
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
[Out]
int(cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)