\(\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [382]

Optimal result

Integrand size = 43, antiderivative size = 243 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d} \]

[Out]

2/1155*(429*A+374*B+336*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+2/495*a^2*(429*A+374*B+336*C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/693*a^2*(99*A+110*B+84*C)*cos(d*x
+c)^3*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)-4/3465*a*(429*A+374*B+336*C)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d+2/9
9*a*(11*B+3*C)*cos(d*x+c)^3*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3124, 3055, 3060, 2838, 2830, 2725} \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a^2 (99 A+110 B+84 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (429 A+374 B+336 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac {4 a (429 A+374 B+336 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3465 d}+\frac {2 a (11 B+3 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{99 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d} \]

[In]

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

[Out]

(2*a^2*(429*A + 374*B + 336*C)*Sin[c + d*x])/(495*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(99*A + 110*B + 84*C)*C
os[c + d*x]^3*Sin[c + d*x])/(693*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a*(429*A + 374*B + 336*C)*Sqrt[a + a*Cos[c +
d*x]]*Sin[c + d*x])/(3465*d) + (2*a*(11*B + 3*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(99*d)
+ (2*(429*A + 374*B + 336*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(1155*d) + (2*C*Cos[c + d*x]^3*(a + a*C
os[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 3124

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/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f
*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*
B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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 {1}{2} a (11 B+3 C) \cos (c+d x)\right ) \, dx}{11 a} \\ & = \frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 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 {3}{4} a^2 (33 A+22 B+24 C)+\frac {1}{4} a^2 (99 A+110 B+84 C) \cos (c+d x)\right ) \, dx}{99 a} \\ & = \frac {2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 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}{231} (a (429 A+374 B+336 C)) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d}+\frac {(2 (429 A+374 B+336 C)) \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 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 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}{495} (a (429 A+374 B+336 C)) \int \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (99 A+110 B+84 C) \cos ^3(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a (429 A+374 B+336 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{3465 d}+\frac {2 a (11 B+3 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{99 d}+\frac {2 (429 A+374 B+336 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{1155 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 = 1.17 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.60 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (65208 A+59158 B+55482 C+(33396 A+35156 B+34734 C) \cos (c+d x)+8 (1287 A+1507 B+1743 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+3740 B \cos (3 (c+d x))+4935 C \cos (3 (c+d x))+770 B \cos (4 (c+d x))+1470 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{27720 d} \]

[In]

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

[Out]

(a*Sqrt[a*(1 + Cos[c + d*x])]*(65208*A + 59158*B + 55482*C + (33396*A + 35156*B + 34734*C)*Cos[c + d*x] + 8*(1
287*A + 1507*B + 1743*C)*Cos[2*(c + d*x)] + 1980*A*Cos[3*(c + d*x)] + 3740*B*Cos[3*(c + d*x)] + 4935*C*Cos[3*(
c + d*x)] + 770*B*Cos[4*(c + d*x)] + 1470*C*Cos[4*(c + d*x)] + 315*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(2772
0*d)

Maple [A] (verified)

Time = 7.44 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.63

method	result	size
default	\(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (-5040 C \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3080 B +18480 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1980 A -9900 B -27720 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (5544 A +12474 B +22176 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-5775 A -8085 B -10395 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 A +3465 B +3465 C \right ) \sqrt {2}}{3465 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(154\)
parts	\(\frac {4 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (60 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+19 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+38\right ) \sqrt {2}}{105 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {4 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (280 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-220 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+114 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+47 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+94\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {4 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (240 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-320 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+200 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+23 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+46\right ) \sqrt {2}}{165 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(299\)

[In]

int(cos(d*x+c)^2*(a+cos(d*x+c)*a)^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),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*C*sin(1/2*d*x+1/2*c)^10+(3080*B+18480*C)*sin(1/2*d*x+1
/2*c)^8+(-1980*A-9900*B-27720*C)*sin(1/2*d*x+1/2*c)^6+(5544*A+12474*B+22176*C)*sin(1/2*d*x+1/2*c)^4+(-5775*A-8
085*B-10395*C)*sin(1/2*d*x+1/2*c)^2+3465*A+3465*B+3465*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 = 137, normalized size of antiderivative = 0.56 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (315 \, C a \cos \left (d x + c\right )^{5} + 35 \, {\left (11 \, B + 21 \, C\right )} a \cos \left (d x + c\right )^{4} + 5 \, {\left (99 \, A + 187 \, B + 168 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{2} + 4 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right ) + 8 \, {\left (429 \, A + 374 \, B + 336 \, C\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]

[In]

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

[Out]

2/3465*(315*C*a*cos(d*x + c)^5 + 35*(11*B + 21*C)*a*cos(d*x + c)^4 + 5*(99*A + 187*B + 168*C)*a*cos(d*x + c)^3
+ 3*(429*A + 374*B + 336*C)*a*cos(d*x + c)^2 + 4*(429*A + 374*B + 336*C)*a*cos(d*x + c) + 8*(429*A + 374*B +
336*C)*a)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.48 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.04 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {132 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 63 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 175 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 735 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 22 \, {\left (35 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 378 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1050 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3780 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + 21 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 55 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 165 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 429 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 990 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3630 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{55440 \, d} \]

[In]

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

[Out]

1/55440*(132*(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) + 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*sq
rt(2)*a*sin(1/2*d*x + 1/2*c))*B*sqrt(a) + 21*(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 = 2.89 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.20 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (315 \, 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 \, {\left (2 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, C 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 (4 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, B 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 ) + 693 \, {\left (12 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 12 \, B 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 ) + 2310 \, {\left (10 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 10 \, B 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 ) + 6930 \, {\left (14 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 12 \, B 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}}{55440 \, d} \]

[In]

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

[Out]

1/55440*sqrt(2)*(315*C*a*sgn(cos(1/2*d*x + 1/2*c))*sin(11/2*d*x + 11/2*c) + 385*(2*B*a*sgn(cos(1/2*d*x + 1/2*c
)) + 3*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(9/2*d*x + 9/2*c) + 495*(4*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 6*B*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) + 693*(12*A*a*sgn(cos(1/2*d*x +
1/2*c)) + 12*B*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) + 2310*(10
*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 10*B*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) + 6930*(14*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 12*B*a*sgn(cos(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. \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]

[In]

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

[Out]

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