\(\int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx\) [82]

Optimal result

Integrand size = 33, antiderivative size = 234 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 a (187 A+168 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d} \]

[Out]

4/1155*(187*A+168*B)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+4/495*a^2*(187*A+168*B)*sin(d*x+c)/d/(a+a*cos(d*x+c))
^(1/2)+2/693*a^2*(187*A+168*B)*cos(d*x+c)^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/99*a^2*(11*A+12*B)*cos(d*x+c
)^4*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)-8/3465*a*(187*A+168*B)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d+2/11*a*B*co
s(d*x+c)^4*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3055, 3060, 2849, 2838, 2830, 2725} \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {2 a^2 (11 A+12 B) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (187 A+168 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt {a \cos (c+d x)+a}}+\frac {4 (187 A+168 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac {8 a (187 A+168 B) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {2 a B \sin (c+d x) \cos ^4(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d} \]

[In]

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

[Out]

(4*a^2*(187*A + 168*B)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(187*A + 168*B)*Cos[c + d*x]^3*
Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(11*A + 12*B)*Cos[c + d*x]^4*Sin[c + d*x])/(99*d*Sqrt[
a + a*Cos[c + d*x]]) - (8*a*(187*A + 168*B)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3465*d) + (2*a*B*Cos[c + d
*x]^4*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(11*d) + (4*(187*A + 168*B)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*
x])/(1155*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 2849

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {2}{11} \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {1}{2} a (11 A+8 B)+\frac {1}{2} a (11 A+12 B) \cos (c+d x)\right ) \, dx \\ & = \frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{99} (a (187 A+168 B)) \int \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {1}{231} (2 a (187 A+168 B)) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {(4 (187 A+168 B)) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx}{1155} \\ & = \frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 a (187 A+168 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {1}{495} (2 a (187 A+168 B)) \int \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (187 A+168 B) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (11 A+12 B) \cos ^4(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}-\frac {8 a (187 A+168 B) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a B \cos ^4(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{11 d}+\frac {4 (187 A+168 B) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.53 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (59158 A+55482 B+(35156 A+34734 B) \cos (c+d x)+8 (1507 A+1743 B) \cos (2 (c+d x))+3740 A \cos (3 (c+d x))+4935 B \cos (3 (c+d x))+770 A \cos (4 (c+d x))+1470 B \cos (4 (c+d x))+315 B \cos (5 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{27720 d} \]

[In]

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

[Out]

(a*Sqrt[a*(1 + Cos[c + d*x])]*(59158*A + 55482*B + (35156*A + 34734*B)*Cos[c + d*x] + 8*(1507*A + 1743*B)*Cos[
2*(c + d*x)] + 3740*A*Cos[3*(c + d*x)] + 4935*B*Cos[3*(c + d*x)] + 770*A*Cos[4*(c + d*x)] + 1470*B*Cos[4*(c +
d*x)] + 315*B*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(27720*d)

Maple [A] (verified)

Time = 2.55 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-5040 B \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3080 A +18480 B \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-9900 A -27720 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (12474 A +22176 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-8085 A -10395 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 A +3465 B \right ) \sqrt {2}}{3465 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(142\)
parts	\(\frac {4 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (280 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-220 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+114 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+47 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+94\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {4 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (240 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-320 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+200 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+23 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+46\right ) \sqrt {2}}{165 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(213\)

[In]

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

[Out]

4/3465*cos(1/2*d*x+1/2*c)*a^2*sin(1/2*d*x+1/2*c)*(-5040*B*sin(1/2*d*x+1/2*c)^10+(3080*A+18480*B)*sin(1/2*d*x+1
/2*c)^8+(-9900*A-27720*B)*sin(1/2*d*x+1/2*c)^6+(12474*A+22176*B)*sin(1/2*d*x+1/2*c)^4+(-8085*A-10395*B)*sin(1/
2*d*x+1/2*c)^2+3465*A+3465*B)*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 = 125, normalized size of antiderivative = 0.53 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {2 \, {\left (315 \, B a \cos \left (d x + c\right )^{5} + 35 \, {\left (11 \, A + 21 \, B\right )} a \cos \left (d x + c\right )^{4} + 5 \, {\left (187 \, A + 168 \, B\right )} a \cos \left (d x + c\right )^{3} + 6 \, {\left (187 \, A + 168 \, B\right )} a \cos \left (d x + c\right )^{2} + 8 \, {\left (187 \, A + 168 \, B\right )} a \cos \left (d x + c\right ) + 16 \, {\left (187 \, A + 168 \, B\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]

[In]

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

[Out]

2/3465*(315*B*a*cos(d*x + c)^5 + 35*(11*A + 21*B)*a*cos(d*x + c)^4 + 5*(187*A + 168*B)*a*cos(d*x + c)^3 + 6*(1
87*A + 168*B)*a*cos(d*x + c)^2 + 8*(187*A + 168*B)*a*cos(d*x + c) + 16*(187*A + 168*B)*a)*sqrt(a*cos(d*x + c)
+ a)*sin(d*x + c)/(d*cos(d*x + c) + d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.79 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {22 \, {\left (35 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 378 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1050 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3780 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 21 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 55 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 165 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 429 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 990 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3630 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a}}{55440 \, d} \]

[In]

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

[Out]

1/55440*(22*(35*sqrt(2)*a*sin(9/2*d*x + 9/2*c) + 135*sqrt(2)*a*sin(7/2*d*x + 7/2*c) + 378*sqrt(2)*a*sin(5/2*d*
x + 5/2*c) + 1050*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 3780*sqrt(2)*a*sin(1/2*d*x + 1/2*c))*A*sqrt(a) + 21*(15*sqr
t(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*s
qrt(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))*B*sq
rt(a))/d

Giac [A] (verification not implemented)

none

Time = 2.30 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00 \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {\sqrt {2} {\left (315 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, {\left (2 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 495 \, {\left (6 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 693 \, {\left (12 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 13 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2310 \, {\left (10 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 9 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 6930 \, {\left (12 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 11 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{55440 \, d} \]

[In]

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

[Out]

1/55440*sqrt(2)*(315*B*a*sgn(cos(1/2*d*x + 1/2*c))*sin(11/2*d*x + 11/2*c) + 385*(2*A*a*sgn(cos(1/2*d*x + 1/2*c
)) + 3*B*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(9/2*d*x + 9/2*c) + 495*(6*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 7*B*a*sgn(
cos(1/2*d*x + 1/2*c)))*sin(7/2*d*x + 7/2*c) + 693*(12*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 13*B*a*sgn(cos(1/2*d*x +
1/2*c)))*sin(5/2*d*x + 5/2*c) + 2310*(10*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 9*B*a*sgn(cos(1/2*d*x + 1/2*c)))*sin
(3/2*d*x + 3/2*c) + 6930*(12*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 11*B*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. \[ \int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^3\,\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]

[In]

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

[Out]

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