\(\int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt {\sec (c+d x)}} \, dx\) [1472]

Optimal result

Integrand size = 43, antiderivative size = 401 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \]

[Out]

2/693*b*(99*A*b^2+143*B*a*b+24*C*a^2+81*C*b^2)*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/495*(242*B*a^2*b+77*B*b^3+24*a^
3*C+33*a*b^2*(9*A+7*C))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/99*(11*B*b+6*C*a)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/sec(
d*x+c)^(3/2)+2/11*C*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/231*(77*B*a^3+165*B*a*b^2+33*a^2*b*(7*A
+5*C)+5*b^3*(11*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/15*(27*B*a^2*b+7*B*b^3+3*a^3*(5*A+3*C)+3*a*b^2*(9*A+7*
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))*cos(d*x+c)^(1/2)*sec
(d*x+c)^(1/2)/d+2/231*(77*B*a^3+165*B*a*b^2+33*a^2*b*(7*A+5*C)+5*b^3*(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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4306, 3128, 3112, 3102, 2827, 2719, 2715, 2720} \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 b \sin (c+d x) \left (24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right )}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right )}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right )}{15 d}+\frac {2 (6 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{99 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac {3}{2}}(c+d x)} \]

[In]

Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]

[Out]

(2*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2,
2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sqrt[C
os[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*b*(99*A*b^2 + 143*a*b*B + 24*a^2*C + 8
1*b^2*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*(242*a^2*b*B + 77*b^3*B + 24*a^3*C + 33*a*b^2*(9*A + 7*
C))*Sin[c + d*x])/(495*d*Sec[c + d*x]^(3/2)) + (2*(11*b*B + 6*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(99*d*
Sec[c + d*x]^(3/2)) + (2*C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(3/2)) + (2*(77*a^3*B + 165
*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2719

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

Rule 2720

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

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

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 3112

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +
b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!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])))

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 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{11} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac {1}{2} a (11 A+3 C)+\frac {1}{2} (11 A b+11 a B+9 b C) \cos (c+d x)+\frac {1}{2} (11 b B+6 a C) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{99} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac {3}{4} a (33 a A+11 b B+15 a C)+\frac {1}{4} \left (198 a A b+99 a^2 B+77 b^2 B+150 a b C\right ) \cos (c+d x)+\frac {1}{4} \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{693} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \left (\frac {21}{8} a^2 (33 a A+11 b B+15 a C)+\frac {9}{8} \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \cos (c+d x)+\frac {7}{8} \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \left (\frac {231}{16} \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right )+\frac {45}{16} \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{3465} \\ & = \frac {2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{15} \left (\left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{77} \left (\left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} \left (\left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 b \left (99 A b^2+143 a b B+24 a^2 C+81 b^2 C\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (242 a^2 b B+77 b^3 B+24 a^3 C+33 a b^2 (9 A+7 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (11 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.32 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (3696 \left (27 a^2 b B+7 b^3 B+3 a^3 (5 A+3 C)+3 a b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+240 \left (77 a^3 B+165 a b^2 B+33 a^2 b (7 A+5 C)+5 b^3 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {\left (154 \left (108 a^2 b B+43 b^3 B+36 a^3 C+3 a b^2 (36 A+43 C)\right ) \cos (c+d x)+5 \left (1848 a^3 B+5148 a b^2 B+396 a^2 b (14 A+13 C)+3 b^3 (572 A+531 C)+36 b \left (11 A b^2+33 a b B+33 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+154 b^2 (b B+3 a C) \cos (3 (c+d x))+63 b^3 C \cos (4 (c+d x))\right )\right ) \sin (2 (c+d x))}{\sqrt {\cos (c+d x)}}\right )}{27720 d} \]

[In]

Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]

[Out]

(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(3696*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*
EllipticE[(c + d*x)/2, 2] + 240*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*EllipticF
[(c + d*x)/2, 2] + ((154*(108*a^2*b*B + 43*b^3*B + 36*a^3*C + 3*a*b^2*(36*A + 43*C))*Cos[c + d*x] + 5*(1848*a^
3*B + 5148*a*b^2*B + 396*a^2*b*(14*A + 13*C) + 3*b^3*(572*A + 531*C) + 36*b*(11*A*b^2 + 33*a*b*B + 33*a^2*C +
16*b^2*C)*Cos[2*(c + d*x)] + 154*b^2*(b*B + 3*a*C)*Cos[3*(c + d*x)] + 63*b^3*C*Cos[4*(c + d*x)]))*Sin[2*(c + d
*x)])/Sqrt[Cos[c + d*x]]))/(27720*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1081\) vs. \(2(421)=842\).

Time = 12.84 (sec) , antiderivative size = 1082, normalized size of antiderivative = 2.70

method	result	size
default	\(\text {Expression too large to display}\)	\(1082\)
parts	\(\text {Expression too large to display}\)	\(1220\)

[In]

int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/3465*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*C*b^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1
/2*c)^12+(-12320*B*b^3-36960*C*a*b^2-50400*C*b^3)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^3+23760*B
*a*b^2+24640*B*b^3+23760*C*a^2*b+73920*C*a*b^2+56880*C*b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-16632*A*
a*b^2-11880*A*b^3-16632*B*a^2*b-35640*B*a*b^2-22792*B*b^3-5544*C*a^3-35640*C*a^2*b-68376*C*a*b^2-34920*C*b^3)*
sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(13860*A*a^2*b+16632*A*a*b^2+9240*A*b^3+4620*B*a^3+16632*B*a^2*b+27720
*B*a*b^2+10472*B*b^3+5544*C*a^3+27720*C*a^2*b+31416*C*a*b^2+13860*C*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*
c)+(-6930*A*a^2*b-4158*A*a*b^2-2640*A*b^3-2310*B*a^3-4158*B*a^2*b-7920*B*a*b^2-1848*B*b^3-1386*C*a^3-7920*C*a^
2*b-5544*C*a*b^2-2790*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+3465*A*a^2*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)
*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+825*A*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)
*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3465*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2
*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3-6237*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(
2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2+1155*B*a^3*(sin(1/2*d*x+1/2*c)^2)^
(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+2475*B*a*b^2*(sin(1/2*d*x+1/2*c)^
2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-6237*B*(sin(1/2*d*x+1/2*c)^2)^
(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b-1617*B*(sin(1/2*d*x+1/2*c)^
2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^3+2475*a^2*b*C*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+675*C*b^3*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-2079*C*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3-4851*C*(sin(1/2*d*x+1/
2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2)/(-2*sin(1/2*d*x+1/
2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.16 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {15 \, \sqrt {2} {\left (77 i \, B a^{3} + 33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 165 i \, B a b^{2} + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-77 i \, B a^{3} - 33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b - 165 i \, B a b^{2} - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-3 i \, {\left (5 \, A + 3 \, C\right )} a^{3} - 27 i \, B a^{2} b - 3 i \, {\left (9 \, A + 7 \, C\right )} a b^{2} - 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (3 i \, {\left (5 \, A + 3 \, C\right )} a^{3} + 27 i \, B a^{2} b + 3 i \, {\left (9 \, A + 7 \, C\right )} a b^{2} + 7 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (315 \, C b^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + 45 \, {\left (33 \, C a^{2} b + 33 \, B a b^{2} + {\left (11 \, A + 9 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (9 \, C a^{3} + 27 \, B a^{2} b + 3 \, {\left (9 \, A + 7 \, C\right )} a b^{2} + 7 \, B b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (77 \, B a^{3} + 33 \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 165 \, B a b^{2} + 5 \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \]

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

-1/3465*(15*sqrt(2)*(77*I*B*a^3 + 33*I*(7*A + 5*C)*a^2*b + 165*I*B*a*b^2 + 5*I*(11*A + 9*C)*b^3)*weierstrassPI
nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-77*I*B*a^3 - 33*I*(7*A + 5*C)*a^2*b - 165*I*B*a*b^
2 - 5*I*(11*A + 9*C)*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)*(-3*I*(5*A +
3*C)*a^3 - 27*I*B*a^2*b - 3*I*(9*A + 7*C)*a*b^2 - 7*I*B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0
, cos(d*x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(3*I*(5*A + 3*C)*a^3 + 27*I*B*a^2*b + 3*I*(9*A + 7*C)*a*b^2 +
7*I*B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(315*C*b^3*co
s(d*x + c)^5 + 385*(3*C*a*b^2 + B*b^3)*cos(d*x + c)^4 + 45*(33*C*a^2*b + 33*B*a*b^2 + (11*A + 9*C)*b^3)*cos(d*
x + c)^3 + 77*(9*C*a^3 + 27*B*a^2*b + 3*(9*A + 7*C)*a*b^2 + 7*B*b^3)*cos(d*x + c)^2 + 15*(77*B*a^3 + 33*(7*A +
5*C)*a^2*b + 165*B*a*b^2 + 5*(11*A + 9*C)*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d

Sympy [F]

\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {\left (a + b \cos {\left (c + d x \right )}\right )^{3} \left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right )}{\sqrt {\sec {\left (c + d x \right )}}}\, dx \]

[In]

integrate((a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/sec(d*x+c)**(1/2),x)

[Out]

Integral((a + b*cos(c + d*x))**3*(A + B*cos(c + d*x) + C*cos(c + d*x)**2)/sqrt(sec(c + d*x)), x)

Maxima [F]

\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/sqrt(sec(d*x + c)), x)

Giac [F]

\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/sqrt(sec(d*x + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]

[In]

int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/cos(c + d*x))^(1/2),x)

[Out]

int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/cos(c + d*x))^(1/2), x)