3.1081 \(\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt{\cos (c+d x)}} \, dx\)

Optimal. Leaf size=296 \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right )}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{315 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d} \]

[Out]

(2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*
a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(54*a^2*b*B
+ 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b*(63*A*b^2 + 99*a*b*
B + 24*a^2*C + 49*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(3*b*B + 2*a*C)*Sqrt[Cos[c + d*x]]*(a +
b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)

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Rubi [A]  time = 0.838873, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {3049, 3033, 3023, 2748, 2641, 2639} \[ \frac{2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )}{21 d}+\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right )}{15 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{315 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left (54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*
a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(54*a^2*b*B
+ 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*b*(63*A*b^2 + 99*a*b*
B + 24*a^2*C + 49*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(3*b*B + 2*a*C)*Sqrt[Cos[c + d*x]]*(a +
b*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)

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])))

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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx &=\frac{2 C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{2}{9} \int \frac{(a+b \cos (c+d x))^2 \left (\frac{1}{2} a (9 A+C)+\frac{1}{2} (9 A b+9 a B+7 b C) \cos (c+d x)+\frac{3}{2} (3 b B+2 a C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 (3 b B+2 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{4}{63} \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{4} a (63 a A+9 b B+13 a C)+\frac{1}{4} \left (126 a A b+63 a^2 B+45 b^2 B+86 a b C\right ) \cos (c+d x)+\frac{1}{4} \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (3 b B+2 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{8}{315} \int \frac{\frac{5}{8} a^2 (63 a A+9 b B+13 a C)+\frac{21}{8} \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \cos (c+d x)+\frac{15}{8} \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{63 d}+\frac{2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (3 b B+2 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{16}{945} \int \frac{\frac{45}{16} \left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right )+\frac{63}{16} \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{63 d}+\frac{2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (3 b B+2 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac{1}{21} \left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{2 \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 \left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{63 d}+\frac{2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac{2 (3 b B+2 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}+\frac{2 C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}\\ \end{align*}

Mathematica [A]  time = 2.01, size = 230, normalized size = 0.78 \[ \frac{60 F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )+84 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (7 b \cos (c+d x) \left (108 a^2 C+108 a b B+36 A b^2+43 b^2 C\right )+5 \left (252 a^2 b B+84 a^3 C+18 a b^2 (14 A+13 C)+18 b^2 (3 a C+b B) \cos (2 (c+d x))+78 b^3 B+7 b^3 C \cos (3 (c+d x))\right )\right )}{630 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(84*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 60*(21*a^2*b*B
+ 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(36*A*
b^2 + 108*a*b*B + 108*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(252*a^2*b*B + 78*b^3*B + 84*a^3*C + 18*a*b^2*(14*A +
13*C) + 18*b^2*(b*B + 3*a*C)*Cos[2*(c + d*x)] + 7*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)

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Maple [B]  time = 1.05, size = 975, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*C*b^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2
*c)^10+(720*B*b^3+2160*C*a*b^2+2240*C*b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-504*A*b^3-1512*B*a*b^2-10
80*B*b^3-1512*C*a^2*b-3240*C*a*b^2-2072*C*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(1260*A*a*b^2+504*A*b^3
+1260*B*a^2*b+1512*B*a*b^2+840*B*b^3+420*C*a^3+1512*C*a^2*b+2520*C*a*b^2+952*C*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1
/2*d*x+1/2*c)+(-630*A*a*b^2-126*A*b^3-630*B*a^2*b-378*B*a*b^2-240*B*b^3-210*C*a^3-378*C*a^2*b-720*C*a*b^2-168*
C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+315*A*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))+315*a*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))-945*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^2*b-189*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))*b^3+315*a^2*b*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))+75*b^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))-315*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^3-567*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*b^2+105*a^3*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))+225*C*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))-567*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^2*b-147*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))*b^3)/(-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., size = 0, normalized size = 0. \begin{align*} \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{\cos \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C b^{3} \cos \left (d x + c\right )^{5} +{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{\cos \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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