\(\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} (A+C \cos ^2(c+d x)) \, dx\) [83]

Optimal result

Integrand size = 35, antiderivative size = 225 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a^2 (143 A+112 C) \sin (c+d x)}{165 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (33 A+28 C) \cos ^3(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (143 A+112 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1155 d}+\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d}+\frac {2 (143 A+112 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{385 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d} \]

[Out]

2/385*(143*A+112*C)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/11*C*cos(d*x+c)^3*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/
d+2/165*a^2*(143*A+112*C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/231*a^2*(33*A+28*C)*cos(d*x+c)^3*sin(d*x+c)/d/
(a+a*cos(d*x+c))^(1/2)-4/1155*a*(143*A+112*C)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d+2/33*a*C*cos(d*x+c)^3*sin(d*
x+c)*(a+a*cos(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3125, 3055, 3060, 2838, 2830, 2725} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a^2 (33 A+28 C) \sin (c+d x) \cos ^3(c+d x)}{231 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (143 A+112 C) \sin (c+d x)}{165 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (143 A+112 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{385 d}-\frac {4 a (143 A+112 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{1155 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac {2 a C \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{33 d} \]

[In]

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

[Out]

(2*a^2*(143*A + 112*C)*Sin[c + d*x])/(165*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(33*A + 28*C)*Cos[c + d*x]^3*Si
n[c + d*x])/(231*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(143*A + 112*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11
55*d) + (2*a*C*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(33*d) + (2*(143*A + 112*C)*(a + a*Cos[c
+ d*x])^(3/2)*Sin[c + d*x])/(385*d) + (2*C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)

Rule 2725

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x
]])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2830

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[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m
, -2^(-1)]

Rule 2838

Int[sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-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*(b*(m + 1) -
a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)]

Rule 3055

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)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x
])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*
x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))
*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 3060

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

Rule 3125

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Si
mp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^
(-1)] && NeQ[m + n + 2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {2 \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (11 A+6 C)+\frac {3}{2} a C \cos (c+d x)\right ) \, dx}{11 a} \\ & = \frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {4 \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {9}{4} a^2 (11 A+8 C)+\frac {3}{4} a^2 (33 A+28 C) \cos (c+d x)\right ) \, dx}{99 a} \\ & = \frac {2 a^2 (33 A+28 C) \cos ^3(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {1}{77} (a (143 A+112 C)) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {2 a^2 (33 A+28 C) \cos ^3(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d}+\frac {2 (143 A+112 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{385 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {1}{385} (2 (143 A+112 C)) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {2 a^2 (33 A+28 C) \cos ^3(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (143 A+112 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1155 d}+\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d}+\frac {2 (143 A+112 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{385 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {1}{165} (a (143 A+112 C)) \int \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {2 a^2 (143 A+112 C) \sin (c+d x)}{165 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (33 A+28 C) \cos ^3(c+d x) \sin (c+d x)}{231 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (143 A+112 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1155 d}+\frac {2 a C \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d}+\frac {2 (143 A+112 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{385 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.51 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (21736 A+18494 C+2 (5566 A+5789 C) \cos (c+d x)+8 (429 A+581 C) \cos (2 (c+d x))+660 A \cos (3 (c+d x))+1645 C \cos (3 (c+d x))+490 C \cos (4 (c+d x))+105 C \cos (5 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{9240 d} \]

[In]

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

[Out]

(a*Sqrt[a*(1 + Cos[c + d*x])]*(21736*A + 18494*C + 2*(5566*A + 5789*C)*Cos[c + d*x] + 8*(429*A + 581*C)*Cos[2*
(c + d*x)] + 660*A*Cos[3*(c + d*x)] + 1645*C*Cos[3*(c + d*x)] + 490*C*Cos[4*(c + d*x)] + 105*C*Cos[5*(c + d*x)
])*Tan[(c + d*x)/2])/(9240*d)

Maple [A] (verified)

Time = 4.71 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-1680 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6160 C \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-660 A -9240 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1848 A +7392 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1925 A -3465 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1155 A +1155 C \right ) \sqrt {2}}{1155 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(137\)
parts	\(\frac {4 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (60 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+19 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+38\right ) \sqrt {2}}{105 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {4 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (240 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-320 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+200 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+23 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+46\right ) \sqrt {2}}{165 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(200\)

[In]

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

[Out]

4/1155*cos(1/2*d*x+1/2*c)*a^2*sin(1/2*d*x+1/2*c)*(-1680*C*sin(1/2*d*x+1/2*c)^10+6160*C*sin(1/2*d*x+1/2*c)^8+(-
660*A-9240*C)*sin(1/2*d*x+1/2*c)^6+(1848*A+7392*C)*sin(1/2*d*x+1/2*c)^4+(-1925*A-3465*C)*sin(1/2*d*x+1/2*c)^2+
1155*A+1155*C)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.53 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (105 \, C a \cos \left (d x + c\right )^{5} + 245 \, C a \cos \left (d x + c\right )^{4} + 5 \, {\left (33 \, A + 56 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (143 \, A + 112 \, C\right )} a \cos \left (d x + c\right )^{2} + 4 \, {\left (143 \, A + 112 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (143 \, A + 112 \, C\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{1155 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]

[In]

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

[Out]

2/1155*(105*C*a*cos(d*x + c)^5 + 245*C*a*cos(d*x + c)^4 + 5*(33*A + 56*C)*a*cos(d*x + c)^3 + 3*(143*A + 112*C)
*a*cos(d*x + c)^2 + 4*(143*A + 112*C)*a*cos(d*x + c) + 8*(143*A + 112*C)*a)*sqrt(a*cos(d*x + c) + a)*sin(d*x +
c)/(d*cos(d*x + c) + d)

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.76 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {44 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 63 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 175 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 735 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 7 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 55 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 165 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 429 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 990 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3630 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{18480 \, d} \]

[In]

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

[Out]

1/18480*(44*(15*sqrt(2)*a*sin(7/2*d*x + 7/2*c) + 63*sqrt(2)*a*sin(5/2*d*x + 5/2*c) + 175*sqrt(2)*a*sin(3/2*d*x
+ 3/2*c) + 735*sqrt(2)*a*sin(1/2*d*x + 1/2*c))*A*sqrt(a) + 7*(15*sqrt(2)*a*sin(11/2*d*x + 11/2*c) + 55*sqrt(2
)*a*sin(9/2*d*x + 9/2*c) + 165*sqrt(2)*a*sin(7/2*d*x + 7/2*c) + 429*sqrt(2)*a*sin(5/2*d*x + 5/2*c) + 990*sqrt(
2)*a*sin(3/2*d*x + 3/2*c) + 3630*sqrt(2)*a*sin(1/2*d*x + 1/2*c))*C*sqrt(a))/d

Giac [A] (verification not implemented)

none

Time = 1.87 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.97 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (105 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 165 \, {\left (4 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 231 \, {\left (12 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 13 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 770 \, {\left (10 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 9 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 2310 \, {\left (14 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 11 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{18480 \, d} \]

[In]

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

[Out]

1/18480*sqrt(2)*(105*C*a*sgn(cos(1/2*d*x + 1/2*c))*sin(11/2*d*x + 11/2*c) + 385*C*a*sgn(cos(1/2*d*x + 1/2*c))*
sin(9/2*d*x + 9/2*c) + 165*(4*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 7*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(7/2*d*x + 7
/2*c) + 231*(12*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 13*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(5/2*d*x + 5/2*c) + 770*(
10*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 9*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(3/2*d*x + 3/2*c) + 2310*(14*A*a*sgn(co
s(1/2*d*x + 1/2*c)) + 11*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(1/2*d*x + 1/2*c))*sqrt(a)/d

Mupad [F(-1)]

Timed out. \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]

[In]

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

[Out]

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