\(\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx\) [304]

Optimal result

Integrand size = 33, antiderivative size = 378 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=-\frac {2 \left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d} \]

[Out]

-2/315*(18*A*a*b-8*B*a^2-49*B*b^2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d+2/63*(9*A*b-4*B*a)*(a+b*cos(d*x+c))
^(5/2)*sin(d*x+c)/b^2/d+2/9*B*cos(d*x+c)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b/d-2/315*(18*A*a^2*b-75*A*b^3-8*B*
a^3-39*B*a*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^2/d-2/315*(18*A*a^3*b-246*A*a*b^3-8*B*a^4-33*B*a^2*b^2-147
*B*b^4)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*
(a+b*cos(d*x+c))^(1/2)/b^3/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/315*(a^2-b^2)*(18*A*a^2*b-75*A*b^3-8*B*a^3-39*B*
a*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*(
(a+b*cos(d*x+c))/(a+b))^(1/2)/b^3/d/(a+b*cos(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3069, 3102, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=-\frac {2 \left (-8 a^2 B+18 a A b-49 b^2 B\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac {2 \left (-8 a^3 B+18 a^2 A b-39 a b^2 B-75 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{315 b^2 d}+\frac {2 \left (a^2-b^2\right ) \left (-8 a^3 B+18 a^2 A b-39 a b^2 B-75 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 B+18 a^3 A b-33 a^2 b^2 B-246 a A b^3-147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 (9 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac {2 B \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d} \]

[In]

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

[Out]

(-2*(18*a^3*A*b - 246*a*A*b^3 - 8*a^4*B - 33*a^2*b^2*B - 147*b^4*B)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*
x)/2, (2*b)/(a + b)])/(315*b^3*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(18*a^2*A*b - 75*A*b^3 -
8*a^3*B - 39*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(315*b^3*d*Sq
rt[a + b*Cos[c + d*x]]) - (2*(18*a^2*A*b - 75*A*b^3 - 8*a^3*B - 39*a*b^2*B)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d
*x])/(315*b^2*d) - (2*(18*a*A*b - 8*a^2*B - 49*b^2*B)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(315*b^2*d) + (
2*(9*A*b - 4*a*B)*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(63*b^2*d) + (2*B*Cos[c + d*x]*(a + b*Cos[c + d*x])
^(5/2)*Sin[c + d*x])/(9*b*d)

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

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

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

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

Rule 2832

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[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 3069

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {2 \int (a+b \cos (c+d x))^{3/2} \left (a B+\frac {7}{2} b B \cos (c+d x)+\frac {1}{2} (9 A b-4 a B) \cos ^2(c+d x)\right ) \, dx}{9 b} \\ & = \frac {2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {4 \int (a+b \cos (c+d x))^{3/2} \left (\frac {3}{4} b (15 A b-2 a B)-\frac {1}{4} \left (18 a A b-8 a^2 B-49 b^2 B\right ) \cos (c+d x)\right ) \, dx}{63 b^2} \\ & = -\frac {2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {8 \int \sqrt {a+b \cos (c+d x)} \left (\frac {3}{8} b \left (57 a A b-2 a^2 B+49 b^2 B\right )-\frac {3}{8} \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \cos (c+d x)\right ) \, dx}{315 b^2} \\ & = -\frac {2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {16 \int \frac {\frac {3}{16} b \left (153 a^2 A b+75 A b^3+2 a^3 B+186 a b^2 B\right )-\frac {3}{16} \left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{945 b^2} \\ & = -\frac {2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}+\frac {\left (\left (a^2-b^2\right ) \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right )\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{315 b^3}-\frac {\left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{315 b^3} \\ & = -\frac {2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d}-\frac {\left (\left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{315 b^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (\left (a^2-b^2\right ) \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{315 b^3 \sqrt {a+b \cos (c+d x)}} \\ & = -\frac {2 \left (18 a^3 A b-246 a A b^3-8 a^4 B-33 a^2 b^2 B-147 b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (18 a^2 A b-75 A b^3-8 a^3 B-39 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}-\frac {2 \left (18 a A b-8 a^2 B-49 b^2 B\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}+\frac {2 (9 A b-4 a B) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 B \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.77 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (153 a^2 A b+75 A b^3+2 a^3 B+186 a b^2 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-18 a^3 A b+246 a A b^3+8 a^4 B+33 a^2 b^2 B+147 b^4 B\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (72 a^2 A b+690 A b^3-32 a^3 B+804 a b^2 B\right ) \sin (c+d x)+b \left (2 \left (144 a A b+6 a^2 B+133 b^2 B\right ) \sin (2 (c+d x))+5 b (2 (9 A b+10 a B) \sin (3 (c+d x))+7 b B \sin (4 (c+d x)))\right )\right )}{1260 b^3 d \sqrt {a+b \cos (c+d x)}} \]

[In]

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

[Out]

(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(153*a^2*A*b + 75*A*b^3 + 2*a^3*B + 186*a*b^2*B)*EllipticF[(c + d*x
)/2, (2*b)/(a + b)] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*((a + b)*EllipticE[(c +
d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((72*a^2*A*b + 69
0*A*b^3 - 32*a^3*B + 804*a*b^2*B)*Sin[c + d*x] + b*(2*(144*a*A*b + 6*a^2*B + 133*b^2*B)*Sin[2*(c + d*x)] + 5*b
*(2*(9*A*b + 10*a*B)*Sin[3*(c + d*x)] + 7*b*B*Sin[4*(c + d*x)]))))/(1260*b^3*d*Sqrt[a + b*Cos[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1634\) vs. \(2(408)=816\).

Time = 17.54 (sec) , antiderivative size = 1635, normalized size of antiderivative = 4.33

method	result	size
default	\(\text {Expression too large to display}\)	\(1635\)
parts	\(\text {Expression too large to display}\)	\(1824\)

[In]

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

[Out]

-2/315*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*B*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2
*c)^10*b^5+(720*A*b^5+1360*B*a*b^4+2240*B*b^5)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-936*A*a*b^4-1080*A*b^
5-424*B*a^2*b^3-2040*B*a*b^4-2072*B*b^5)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(324*A*a^2*b^3+936*A*a*b^4+84
0*A*b^5-4*B*a^3*b^2+424*B*a^2*b^3+1568*B*a*b^4+952*B*b^5)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-18*A*a^3*b
^2-162*A*a^2*b^3-384*A*a*b^4-240*A*b^5+8*B*a^4*b+2*B*a^3*b^2-282*B*a^2*b^3-444*B*a*b^4-168*B*b^5)*sin(1/2*d*x+
1/2*c)^2*cos(1/2*d*x+1/2*c)+18*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1
/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b-93*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1
/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^3+75*A*b^5*(sin(1/2*
d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-
b))^(1/2))-18*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos
(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b+18*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(
a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^2+246*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*
(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^3-2
46*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/
2*c),(-2*b/(a-b))^(1/2))*a*b^4-8*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^
(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5-31*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1
/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^2+39*B*a*(sin(1/2*d*
x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b)
)^(1/2))*b^4+8*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(co
s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5-8*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+
b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^4*b+33*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b
/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b^2-33*B*(
sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(
-2*b/(a-b))^(1/2))*a^2*b^3+147*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1
/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^4-147*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(
1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^5)/b^3/(-2*sin(1/2*d*x+
1/2*c)^4*b+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(-2*b*sin(1/2*d*x+1/2*c)^2+a+b)^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.17 (sec) , antiderivative size = 639, normalized size of antiderivative = 1.69 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\frac {\sqrt {2} {\left (16 i \, B a^{5} - 36 i \, A a^{4} b + 60 i \, B a^{3} b^{2} + 33 i \, A a^{2} b^{3} - 264 i \, B a b^{4} - 225 i \, A b^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (-16 i \, B a^{5} + 36 i \, A a^{4} b - 60 i \, B a^{3} b^{2} - 33 i \, A a^{2} b^{3} + 264 i \, B a b^{4} + 225 i \, A b^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-8 i \, B a^{4} b + 18 i \, A a^{3} b^{2} - 33 i \, B a^{2} b^{3} - 246 i \, A a b^{4} - 147 i \, B b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (8 i \, B a^{4} b - 18 i \, A a^{3} b^{2} + 33 i \, B a^{2} b^{3} + 246 i \, A a b^{4} + 147 i \, B b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left (35 \, B b^{5} \cos \left (d x + c\right )^{3} - 4 \, B a^{3} b^{2} + 9 \, A a^{2} b^{3} + 88 \, B a b^{4} + 75 \, A b^{5} + 5 \, {\left (10 \, B a b^{4} + 9 \, A b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, B a^{2} b^{3} + 72 \, A a b^{4} + 49 \, B b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{945 \, b^{4} d} \]

[In]

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

[Out]

1/945*(sqrt(2)*(16*I*B*a^5 - 36*I*A*a^4*b + 60*I*B*a^3*b^2 + 33*I*A*a^2*b^3 - 264*I*B*a*b^4 - 225*I*A*b^5)*sqr
t(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*s
in(d*x + c) + 2*a)/b) + sqrt(2)*(-16*I*B*a^5 + 36*I*A*a^4*b - 60*I*B*a^3*b^2 - 33*I*A*a^2*b^3 + 264*I*B*a*b^4
+ 225*I*A*b^5)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(
d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) - 3*sqrt(2)*(-8*I*B*a^4*b + 18*I*A*a^3*b^2 - 33*I*B*a^2*b^3 - 246*I*A*
a*b^4 - 147*I*B*b^5)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrass
PInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*
a)/b)) - 3*sqrt(2)*(8*I*B*a^4*b - 18*I*A*a^3*b^2 + 33*I*B*a^2*b^3 + 246*I*A*a*b^4 + 147*I*B*b^5)*sqrt(b)*weier
strassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2,
-8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) + 6*(35*B*b^5*cos(d*x + c)^
3 - 4*B*a^3*b^2 + 9*A*a^2*b^3 + 88*B*a*b^4 + 75*A*b^5 + 5*(10*B*a*b^4 + 9*A*b^5)*cos(d*x + c)^2 + (3*B*a^2*b^3
+ 72*A*a*b^4 + 49*B*b^5)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^4*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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