3.1089 \(\int \frac {(a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=379 \[ \frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{315 d}+\frac {2 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d}+\frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{21 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{15 d}-\frac {2 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d}-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}} \]

[Out]

2/15*(60*a^3*b*B+36*a*b^3*B-15*a^4*(A-C)+18*a^2*b^2*(5*A+3*C)+b^4*(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))/d+2/21*(21*a^4*B+42*a^2*b^2*B+5*b^4*B+28*a^3*b*(3*A+C)+4*
a*b^3*(7*A+5*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))/d+2/315
*b^2*(162*a*b*B-a^2*(315*A-123*C)+7*b^2*(9*A+7*C))*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2*A*(a+b*cos(d*x+c))^4*sin(d*
x+c)/d/cos(d*x+c)^(1/2)+2/63*b*(117*a^2*b*B+15*b^3*B-a^3*(126*A-62*C)+12*a*b^2*(7*A+5*C))*sin(d*x+c)*cos(d*x+c
)^(1/2)/d-2/21*b*(21*A*a-3*B*b-5*C*a)*(a+b*cos(d*x+c))^2*sin(d*x+c)*cos(d*x+c)^(1/2)/d-2/9*b*(9*A-C)*(a+b*cos(
d*x+c))^3*sin(d*x+c)*cos(d*x+c)^(1/2)/d

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Rubi [A]  time = 1.27, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3047, 3049, 3033, 3023, 2748, 2641, 2639} \[ \frac {2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{21 d}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+36 a b^3 B+b^4 (9 A+7 C)\right )}{15 d}+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 (-(315 A-123 C))+162 a b B+7 b^2 (9 A+7 C)\right )}{315 d}+\frac {2 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^3 (-(126 A-62 C))+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d}-\frac {2 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d}-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]

[Out]

(2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2
, 2])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*EllipticF[(c
+ d*x)/2, 2])/(21*d) + (2*b*(117*a^2*b*B + 15*b^3*B - a^3*(126*A - 62*C) + 12*a*b^2*(7*A + 5*C))*Sqrt[Cos[c +
d*x]]*Sin[c + d*x])/(63*d) + (2*b^2*(162*a*b*B - a^2*(315*A - 123*C) + 7*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*S
in[c + d*x])/(315*d) - (2*b*(21*a*A - 3*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(
21*d) - (2*b*(9*A - C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*A*(a + b*Cos[c + d*x
])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])

Rule 2639

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

Rule 2641

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

Rule 2748

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 3023

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 3033

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 3047

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

Rule 3049

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*} \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+2 \int \frac {(a+b \cos (c+d x))^3 \left (\frac {1}{2} (8 A b+a B)+\frac {1}{2} (b B-a (A-C)) \cos (c+d x)-\frac {1}{2} b (9 A-C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b (9 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {4}{9} \int \frac {(a+b \cos (c+d x))^2 \left (\frac {1}{4} a (63 A b+9 a B+b C)+\frac {1}{4} \left (18 a b B-9 a^2 (A-C)+b^2 (9 A+7 C)\right ) \cos (c+d x)-\frac {3}{4} b (21 a A-3 b B-5 a C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b (21 a A-3 b B-5 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}-\frac {2 b (9 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {8}{63} \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{8} a \left (378 a A b+63 a^2 B+9 b^2 B+22 a b C\right )+\frac {1}{8} \left (189 a^2 b B+45 b^3 B-63 a^3 (A-C)+a b^2 (189 A+131 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}-\frac {2 b (21 a A-3 b B-5 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}-\frac {2 b (9 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {16}{315} \int \frac {\frac {5}{16} a^2 \left (378 a A b+63 a^2 B+9 b^2 B+22 a b C\right )+\frac {21}{16} \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \cos (c+d x)+\frac {15}{16} b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{63 d}+\frac {2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}-\frac {2 b (21 a A-3 b B-5 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}-\frac {2 b (9 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {32}{945} \int \frac {\frac {45}{32} \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right )+\frac {63}{32} \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{63 d}+\frac {2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}-\frac {2 b (21 a A-3 b B-5 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}-\frac {2 b (9 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {1}{21} \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{63 d}+\frac {2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}-\frac {2 b (21 a A-3 b B-5 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}-\frac {2 b (9 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 4.36, size = 275, normalized size = 0.73 \[ \frac {10 F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )-14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^4 (A-C)-60 a^3 b B-18 a^2 b^2 (5 A+3 C)-36 a b^3 B-b^4 (9 A+7 C)\right )+\frac {1}{12} \sqrt {\cos (c+d x)} \left (35 \left (72 a^4 A \tan (c+d x)+b^4 C \sin (4 (c+d x))\right )+14 b^2 \sin (2 (c+d x)) \left (108 a^2 C+72 a b B+18 A b^2+19 b^2 C\right )+30 b \sin (c+d x) \left (112 a^3 C+168 a^2 b B+4 a b^2 (28 A+23 C)+23 b^3 B\right )+90 b^3 (4 a C+b B) \sin (3 (c+d x))\right )}{105 d} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]

[Out]

(-14*(-60*a^3*b*B - 36*a*b^3*B + 15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*EllipticE[(c + d*x
)/2, 2] + 10*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*EllipticF[(c + d*x
)/2, 2] + (Sqrt[Cos[c + d*x]]*(30*b*(168*a^2*b*B + 23*b^3*B + 112*a^3*C + 4*a*b^2*(28*A + 23*C))*Sin[c + d*x]
+ 14*b^2*(18*A*b^2 + 72*a*b*B + 108*a^2*C + 19*b^2*C)*Sin[2*(c + d*x)] + 90*b^3*(b*B + 4*a*C)*Sin[3*(c + d*x)]
+ 35*(b^4*C*Sin[4*(c + d*x)] + 72*a^4*A*Tan[c + d*x])))/12)/(105*d)

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{4} \cos \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

integral((C*b^4*cos(d*x + c)^6 + (4*C*a*b^3 + B*b^4)*cos(d*x + c)^5 + A*a^4 + (6*C*a^2*b^2 + 4*B*a*b^3 + A*b^4
)*cos(d*x + c)^4 + 2*(2*C*a^3*b + 3*B*a^2*b^2 + 2*A*a*b^3)*cos(d*x + c)^3 + (C*a^4 + 4*B*a^3*b + 6*A*a^2*b^2)*
cos(d*x + c)^2 + (B*a^4 + 4*A*a^3*b)*cos(d*x + c))/cos(d*x + c)^(3/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(3/2), x)

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maple [B]  time = 3.89, size = 1652, normalized size = 4.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x)

[Out]

-2/315*(-1120*C*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)
^10+80*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*b^3*(9*B*b+36*C*a+28*C*b)*sin(1/2*d*x+1/2*c)^8*cos
(1/2*d*x+1/2*c)-8*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*b^2*(63*A*b^2+252*B*a*b+135*B*b^2+378*C
*a^2+540*C*a*b+259*C*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+56*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*
c)^2)^(1/2)*b*(30*A*a*b^2+9*A*b^3+45*B*a^2*b+36*B*a*b^2+15*B*b^3+30*C*a^3+54*C*a^2*b+60*C*a*b^2+17*C*b^3)*sin(
1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-6*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(105*A*a^4+140*A*a*
b^3+21*A*b^4+210*B*a^2*b^2+84*B*a*b^3+40*B*b^4+140*C*a^3*b+126*C*a^2*b^2+160*C*a*b^3+28*C*b^4)*sin(1/2*d*x+1/2
*c)^2*cos(1/2*d*x+1/2*c)+315*A*(-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)^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^4-1890*A*(-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)^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^2-189*A*(-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)^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^4+1260*A*a^3*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin
(1/2*d*x+1/2*c)^2)^(1/2)+420*a*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(c
os(1/2*d*x+1/2*c),2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)-1260*B*(-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)^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*b-756*B*(-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)^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^3+315*a^4*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(
1/2*d*x+1/2*c)^2)^(1/2)+630*a^2*b^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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+75*B*b^4*(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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1
/2*d*x+1/2*c)^2)^(1/2)-315*C*(-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)^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^4-1134*C*(-2*sin(1/2*d*x+1/2*c)^4+si
n(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+
1/2*c),2^(1/2))*a^2*b^2-147*C*(-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)^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^4+420*a^3*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/
2*d*x+1/2*c)^2)^(1/2)+300*C*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))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2))/(-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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(3/2), x)

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mupad [B]  time = 3.97, size = 547, normalized size = 1.44 \[ \frac {2\,\left (B\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,B\,a^3\,b\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,B\,a^2\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,B\,a^2\,b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {2\,C\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,A\,a^3\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {4\,A\,a\,b^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {4\,C\,a^3\,b\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {12\,A\,a^2\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,A\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,b^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,b^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,B\,a\,b^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,C\,a\,b^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {12\,C\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2),x)

[Out]

(2*(B*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*B*a^3*b*ellipticE(c/2 + (d*x)/2, 2) + 2*B*a^2*b^2*ellipticF(c/2 + (d
*x)/2, 2) + 2*B*a^2*b^2*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (2*C*a^4*ellipticE(c/2 + (d*x)/2, 2))/d + (8*A*a
^3*b*ellipticF(c/2 + (d*x)/2, 2))/d + (4*A*a*b^3*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (
d*x)/2, 2))/3))/d + (4*C*a^3*b*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d
+ (12*A*a^2*b^2*ellipticE(c/2 + (d*x)/2, 2))/d + (2*A*a^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x
)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) - (2*A*b^4*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2,
7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*B*b^4*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeo
m([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (2*C*b^4*cos(c + d*x)^(11/2)*sin(c + d*x)*
hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (8*B*a*b^3*cos(c + d*x)^(7/2)*si
n(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (8*C*a*b^3*cos(c + d*x)
^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (12*C*a^2*b^2*
cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)**(3/2),x)

[Out]

Timed out

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