\(\int (a+b \cos (c+d x))^{3/2} (a^2-b^2 \cos ^2(c+d x)) \, dx\) [646]

Optimal result

Integrand size = 32, antiderivative size = 246 \[ \int (a+b \cos (c+d x))^{3/2} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\frac {4 a \left (73 a^2-41 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (41 a^4-66 a^2 b^2+25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 d \sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (41 a^2-25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {4 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}-\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \]

[Out]

4/35*a*b*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/d-2/7*b*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/d+2/105*b*(41*a^2-25*b^2)
*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/d+4/105*a*(73*a^2-41*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*E
llipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2
/105*(41*a^4-66*a^2*b^2+25*b^4)*(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)/d/(a+b*cos(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {3095, 2832, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \cos (c+d x))^{3/2} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\frac {2 b \left (41 a^2-25 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{105 d}+\frac {4 a \left (73 a^2-41 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (41 a^4-66 a^2 b^2+25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 d \sqrt {a+b \cos (c+d x)}}-\frac {2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac {4 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d} \]

[In]

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

[Out]

(4*a*(73*a^2 - 41*b^2)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(105*d*Sqrt[(a + b*Cos[
c + d*x])/(a + b)]) - (2*(41*a^4 - 66*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)
/2, (2*b)/(a + b)])/(105*d*Sqrt[a + b*Cos[c + d*x]]) + (2*b*(41*a^2 - 25*b^2)*Sqrt[a + b*Cos[c + d*x]]*Sin[c +
d*x])/(105*d) + (4*a*b*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(a + b*Cos[c + d*x])^(5/2)*Sin[
c + d*x])/(7*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 3095

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

Rubi steps \begin{align*} \text {integral}& = -\int (-a+b \cos (c+d x)) (a+b \cos (c+d x))^{5/2} \, dx \\ & = -\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}-\frac {2}{7} \int (a+b \cos (c+d x))^{3/2} \left (\frac {1}{2} \left (-7 a^2+5 b^2\right )-a b \cos (c+d x)\right ) \, dx \\ & = \frac {4 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}-\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}-\frac {4}{35} \int \sqrt {a+b \cos (c+d x)} \left (-\frac {1}{4} a \left (35 a^2-19 b^2\right )-\frac {1}{4} b \left (41 a^2-25 b^2\right ) \cos (c+d x)\right ) \, dx \\ & = \frac {2 b \left (41 a^2-25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {4 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}-\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}-\frac {8}{105} \int \frac {\frac {1}{8} \left (-105 a^4+16 a^2 b^2+25 b^4\right )-\frac {1}{4} a b \left (73 a^2-41 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 b \left (41 a^2-25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {4 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}-\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{105} \left (2 a \left (73 a^2-41 b^2\right )\right ) \int \sqrt {a+b \cos (c+d x)} \, dx-\frac {1}{105} \left (41 a^4-66 a^2 b^2+25 b^4\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 b \left (41 a^2-25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {4 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}-\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {\left (2 a \left (73 a^2-41 b^2\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{105 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (\left (41 a^4-66 a^2 b^2+25 b^4\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}{105 \sqrt {a+b \cos (c+d x)}} \\ & = \frac {4 a \left (73 a^2-41 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (41 a^4-66 a^2 b^2+25 b^4\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 )}{105 d \sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (41 a^2-25 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {4 a b (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}-\frac {2 b (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.15 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.86 \[ \int (a+b \cos (c+d x))^{3/2} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\frac {8 a \left (73 a^3+73 a^2 b-41 a b^2-41 b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-4 \left (41 a^4-66 a^2 b^2+25 b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-b \left (-128 a^3+178 a b^2+\left (-32 a^2 b+145 b^3\right ) \cos (c+d x)+78 a b^2 \cos (2 (c+d x))+15 b^3 \cos (3 (c+d x))\right ) \sin (c+d x)}{210 d \sqrt {a+b \cos (c+d x)}} \]

[In]

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

[Out]

(8*a*(73*a^3 + 73*a^2*b - 41*a*b^2 - 41*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(
a + b)] - 4*(41*a^4 - 66*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a
+ b)] - b*(-128*a^3 + 178*a*b^2 + (-32*a^2*b + 145*b^3)*Cos[c + d*x] + 78*a*b^2*Cos[2*(c + d*x)] + 15*b^3*Cos[
3*(c + d*x)])*Sin[c + d*x])/(210*d*Sqrt[a + b*Cos[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(823\) vs. \(2(280)=560\).

Time = 18.19 (sec) , antiderivative size = 824, normalized size of antiderivative = 3.35

method	result	size
default	\(\text {Expression too large to display}\)	\(824\)
parts	\(\text {Expression too large to display}\)	\(1277\)

[In]

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

[Out]

2/105*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*cos(1/2*d*x+1/2*c)^9*b^4+312*cos(1/2*d*
x+1/2*c)^7*a*b^3-600*cos(1/2*d*x+1/2*c)^7*b^4-32*cos(1/2*d*x+1/2*c)^5*a^2*b^2-624*cos(1/2*d*x+1/2*c)^5*a*b^3+6
40*cos(1/2*d*x+1/2*c)^5*b^4-64*cos(1/2*d*x+1/2*c)^3*a^3*b+48*cos(1/2*d*x+1/2*c)^3*a^2*b^2+440*cos(1/2*d*x+1/2*
c)^3*a*b^3-360*cos(1/2*d*x+1/2*c)^3*b^4+41*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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^4-66*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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^2+25*(sin(1/2*d*x+1/2*c)^2)^
(1/2)*((2*b*cos(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^4-146*(s
in(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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+146*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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+82*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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^2-82*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(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^3+64*cos(1/2*d*x+1/2*c)*a^3*
b-16*cos(1/2*d*x+1/2*c)*a^2*b^2-128*cos(1/2*d*x+1/2*c)*a*b^3+80*cos(1/2*d*x+1/2*c)*b^4)/(-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 = 0.12 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.93 \[ \int (a+b \cos (c+d x))^{3/2} \left (a^2-b^2 \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (-23 i \, a^{4} - 116 i \, a^{2} b^{2} + 75 i \, b^{4}\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 (23 i \, a^{4} + 116 i \, a^{2} b^{2} - 75 i \, b^{4}\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 ) - 6 \, \sqrt {2} {\left (-73 i \, a^{3} b + 41 i \, a b^{3}\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 \, \sqrt {2} {\left (73 i \, a^{3} b - 41 i \, a b^{3}\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 (15 \, b^{4} \cos \left (d x + c\right )^{2} + 24 \, a b^{3} \cos \left (d x + c\right ) - 32 \, a^{2} b^{2} + 25 \, b^{4}\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, b d} \]

[In]

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

[Out]

1/315*(sqrt(2)*(-23*I*a^4 - 116*I*a^2*b^2 + 75*I*b^4)*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) + sqrt(2)*(23*I*a^4 + 116*I*a^2
*b^2 - 75*I*b^4)*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*co
s(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) - 6*sqrt(2)*(-73*I*a^3*b + 41*I*a*b^3)*sqrt(b)*weierstrassZeta(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*sqrt(2)*(73*I*a^3*b - 41*I*a*b^3)*sqrt(
b)*weierstrassZeta(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*(15*b^4*cos(d*x
+ c)^2 + 24*a*b^3*cos(d*x + c) - 32*a^2*b^2 + 25*b^4)*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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