\(\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} (A+C \cos ^2(c+d x)) \, dx\) [84]

Optimal result

Integrand size = 33, antiderivative size = 174 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {8 a^2 (63 A+47 C) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (63 A+47 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 (63 A+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d} \]

[Out]

2/315*(63*A+22*C)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/9*C*cos(d*x+c)^2*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2
/21*C*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/a/d+8/315*a^2*(63*A+47*C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/315*a*
(63*A+47*C)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 174, 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 = {3125, 3047, 3102, 2830, 2726, 2725} \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {8 a^2 (63 A+47 C) \sin (c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}+\frac {2 (63 A+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac {2 a (63 A+47 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 d}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d} \]

[In]

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

[Out]

(8*a^2*(63*A + 47*C)*Sin[c + d*x])/(315*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a*(63*A + 47*C)*Sqrt[a + a*Cos[c + d*
x]]*Sin[c + d*x])/(315*d) + (2*(63*A + 22*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*d) + (2*C*Cos[c + d
*x]^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d) + (2*C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(21*a*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 2726

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
- 1)/(d*n)), x] + Dist[a*((2*n - 1)/n), Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[a^2 - b^2, 0] && IGtQ[n - 1/2, 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 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3102

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 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 ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (9 A+4 C)+\frac {3}{2} a C \cos (c+d x)\right ) \, dx}{9 a} \\ & = \frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 \int (a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (9 A+4 C) \cos (c+d x)+\frac {3}{2} a C \cos ^2(c+d x)\right ) \, dx}{9 a} \\ & = \frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}+\frac {4 \int (a+a \cos (c+d x))^{3/2} \left (\frac {15 a^2 C}{4}+\frac {1}{4} a^2 (63 A+22 C) \cos (c+d x)\right ) \, dx}{63 a^2} \\ & = \frac {2 (63 A+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}+\frac {1}{105} (63 A+47 C) \int (a+a \cos (c+d x))^{3/2} \, dx \\ & = \frac {2 a (63 A+47 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 (63 A+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d}+\frac {1}{315} (4 a (63 A+47 C)) \int \sqrt {a+a \cos (c+d x)} \, dx \\ & = \frac {8 a^2 (63 A+47 C) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (63 A+47 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 (63 A+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.53 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (3276 A+2689 C+2 (756 A+799 C) \cos (c+d x)+4 (63 A+137 C) \cos (2 (c+d x))+170 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{1260 d} \]

[In]

Integrate[Cos[c + d*x]*(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])]*(3276*A + 2689*C + 2*(756*A + 799*C)*Cos[c + d*x] + 4*(63*A + 137*C)*Cos[2*(c +
d*x)] + 170*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)

Maple [A] (verified)

Time = 5.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.68

method	result	size
default	\(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (280 C \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-900 C \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (126 A +1134 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-315 A -735 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 A +315 C \right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(118\)
parts	\(\frac {4 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+2\right ) \sqrt {2}}{5 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}+\frac {4 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (280 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-220 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+114 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+47 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+94\right ) \sqrt {2}}{315 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d}\)	\(172\)

[In]

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

[Out]

4/315*cos(1/2*d*x+1/2*c)*a^2*sin(1/2*d*x+1/2*c)*(280*C*sin(1/2*d*x+1/2*c)^8-900*C*sin(1/2*d*x+1/2*c)^6+(126*A+
1134*C)*sin(1/2*d*x+1/2*c)^4+(-315*A-735*C)*sin(1/2*d*x+1/2*c)^2+315*A+315*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 = 100, normalized size of antiderivative = 0.57 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (35 \, C a \cos \left (d x + c\right )^{4} + 85 \, C a \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 34 \, C\right )} a \cos \left (d x + c\right )^{2} + {\left (189 \, A + 136 \, C\right )} a \cos \left (d x + c\right ) + 2 \, {\left (189 \, A + 136 \, C\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]

[In]

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

[Out]

2/315*(35*C*a*cos(d*x + c)^4 + 85*C*a*cos(d*x + c)^3 + 3*(21*A + 34*C)*a*cos(d*x + c)^2 + (189*A + 136*C)*a*co
s(d*x + c) + 2*(189*A + 136*C)*a)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos(d*x + c) + d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)*(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 = 0.44 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.79 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {252 \, {\left (\sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 20 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + {\left (35 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 378 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1050 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3780 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{2520 \, d} \]

[In]

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

[Out]

1/2520*(252*(sqrt(2)*a*sin(5/2*d*x + 5/2*c) + 5*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 20*sqrt(2)*a*sin(1/2*d*x + 1/
2*c))*A*sqrt(a) + (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))*C*sqrt(a))/d

Giac [A] (verification not implemented)

none

Time = 0.77 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.02 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (35 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 126 \, {\left (2 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 210 \, {\left (6 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 1260 \, {\left (4 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \]

[In]

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

[Out]

1/2520*sqrt(2)*(35*C*a*sgn(cos(1/2*d*x + 1/2*c))*sin(9/2*d*x + 9/2*c) + 135*C*a*sgn(cos(1/2*d*x + 1/2*c))*sin(
7/2*d*x + 7/2*c) + 126*(2*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 3*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(5/2*d*x + 5/2*c
) + 210*(6*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 5*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(3/2*d*x + 3/2*c) + 1260*(4*A*a
*sgn(cos(1/2*d*x + 1/2*c)) + 3*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. \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]

[In]

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

[Out]

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