3.391 \(\int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=294 \[ \frac {2 a^3 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{1287 d}-\frac {4 a^2 (10439 A+9230 B+8368 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{45045 d}+\frac {2 a (10439 A+9230 B+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac {2 a (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d} \]

[Out]

2/15015*a*(10439*A+9230*B+8368*C)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/143*a*(13*B+5*C)*cos(d*x+c)^3*(a+a*cos
(d*x+c))^(3/2)*sin(d*x+c)/d+2/13*C*cos(d*x+c)^3*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/d+2/6435*a^3*(10439*A+9230*B
+8368*C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/9009*a^3*(2717*A+2522*B+2224*C)*cos(d*x+c)^3*sin(d*x+c)/d/(a+a*
cos(d*x+c))^(1/2)-4/45045*a^2*(10439*A+9230*B+8368*C)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d+2/1287*a^2*(143*A+18
2*B+136*C)*cos(d*x+c)^3*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.96, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {3045, 2976, 2981, 2759, 2751, 2646} \[ \frac {2 a^3 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \cos (c+d x)+a}}{1287 d}+\frac {2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt {a \cos (c+d x)+a}}-\frac {4 a^2 (10439 A+9230 B+8368 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{45045 d}+\frac {2 a (10439 A+9230 B+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac {2 a (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(2*a^3*(10439*A + 9230*B + 8368*C)*Sin[c + d*x])/(6435*d*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(2717*A + 2522*B +
2224*C)*Cos[c + d*x]^3*Sin[c + d*x])/(9009*d*Sqrt[a + a*Cos[c + d*x]]) - (4*a^2*(10439*A + 9230*B + 8368*C)*S
qrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(45045*d) + (2*a^2*(143*A + 182*B + 136*C)*Cos[c + d*x]^3*Sqrt[a + a*Cos
[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*a*(10439*A + 9230*B + 8368*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/
(15015*d) + (2*a*(13*B + 5*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*C*Cos[c + d
*x]^3*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)

Rule 2646

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 2751

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*Sin
[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 2759

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 2976

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)))*S
in[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 2981

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)*Sqr
t[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 3045

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*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {2 \int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left (\frac {1}{2} a (13 A+6 C)+\frac {1}{2} a (13 B+5 C) \cos (c+d x)\right ) \, dx}{13 a}\\ &=\frac {2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {4 \int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac {1}{4} a^2 (143 A+78 B+96 C)+\frac {1}{4} a^2 (143 A+182 B+136 C) \cos (c+d x)\right ) \, dx}{143 a}\\ &=\frac {2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {8 \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \left (\frac {3}{8} a^3 (715 A+598 B+560 C)+\frac {1}{8} a^3 (2717 A+2522 B+2224 C) \cos (c+d x)\right ) \, dx}{1287 a}\\ &=\frac {2 a^3 (2717 A+2522 B+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (a^2 (10439 A+9230 B+8368 C)\right ) \int \cos ^2(c+d x) \sqrt {a+a \cos (c+d x)} \, dx}{3003}\\ &=\frac {2 a^3 (2717 A+2522 B+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (10439 A+9230 B+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac {2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {(2 a (10439 A+9230 B+8368 C)) \int \left (\frac {3 a}{2}-a \cos (c+d x)\right ) \sqrt {a+a \cos (c+d x)} \, dx}{15015}\\ &=\frac {2 a^3 (2717 A+2522 B+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a^2 (10439 A+9230 B+8368 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{45045 d}+\frac {2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (10439 A+9230 B+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac {2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (a^2 (10439 A+9230 B+8368 C)\right ) \int \sqrt {a+a \cos (c+d x)} \, dx}{6435}\\ &=\frac {2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (2717 A+2522 B+2224 C) \cos ^3(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}-\frac {4 a^2 (10439 A+9230 B+8368 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{45045 d}+\frac {2 a^2 (143 A+182 B+136 C) \cos ^3(c+d x) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{1287 d}+\frac {2 a (10439 A+9230 B+8368 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{15015 d}+\frac {2 a (13 B+5 C) \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 C \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{13 d}\\ \end {align*}

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Mathematica [A]  time = 1.73, size = 180, normalized size = 0.61 \[ \frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (4 (445588 A+454285 B+453146 C) \cos (c+d x)+(581152 A+676000 B+746519 C) \cos (2 (c+d x))+148720 A \cos (3 (c+d x))+20020 A \cos (4 (c+d x))+3233516 A+225550 B \cos (3 (c+d x))+58240 B \cos (4 (c+d x))+8190 B \cos (5 (c+d x))+2980640 B+287060 C \cos (3 (c+d x))+94010 C \cos (4 (c+d x))+23940 C \cos (5 (c+d x))+3465 C \cos (6 (c+d x))+2798182 C)}{720720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(3233516*A + 2980640*B + 2798182*C + 4*(445588*A + 454285*B + 453146*C)*Cos[c
+ d*x] + (581152*A + 676000*B + 746519*C)*Cos[2*(c + d*x)] + 148720*A*Cos[3*(c + d*x)] + 225550*B*Cos[3*(c + d
*x)] + 287060*C*Cos[3*(c + d*x)] + 20020*A*Cos[4*(c + d*x)] + 58240*B*Cos[4*(c + d*x)] + 94010*C*Cos[4*(c + d*
x)] + 8190*B*Cos[5*(c + d*x)] + 23940*C*Cos[5*(c + d*x)] + 3465*C*Cos[6*(c + d*x)])*Tan[(c + d*x)/2])/(720720*
d)

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fricas [A]  time = 0.67, size = 172, normalized size = 0.59 \[ \frac {2 \, {\left (3465 \, C a^{2} \cos \left (d x + c\right )^{6} + 315 \, {\left (13 \, B + 38 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 35 \, {\left (143 \, A + 416 \, B + 523 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (3718 \, A + 4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 4 \, {\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (10439 \, A + 9230 \, B + 8368 \, C\right )} a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

2/45045*(3465*C*a^2*cos(d*x + c)^6 + 315*(13*B + 38*C)*a^2*cos(d*x + c)^5 + 35*(143*A + 416*B + 523*C)*a^2*cos
(d*x + c)^4 + 5*(3718*A + 4615*B + 4184*C)*a^2*cos(d*x + c)^3 + 3*(10439*A + 9230*B + 8368*C)*a^2*cos(d*x + c)
^2 + 4*(10439*A + 9230*B + 8368*C)*a^2*cos(d*x + c) + 8*(10439*A + 9230*B + 8368*C)*a^2)*sqrt(a*cos(d*x + c) +
a)*sin(d*x + c)/(d*cos(d*x + c) + d)

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giac [A]  time = 5.97, size = 462, normalized size = 1.57 \[ \frac {1}{1441440} \, \sqrt {2} {\left (\frac {3465 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {13}{2} \, d x + \frac {13}{2} \, c\right )}{d} + \frac {4095 \, {\left (2 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right )}{d} + \frac {10010 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a^{2} \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 )}{d} + \frac {12870 \, {\left (10 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 13 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 15 \, C a^{2} \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 )}{d} + \frac {9009 \, {\left (48 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 50 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 51 \, C a^{2} \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 )}{d} + \frac {15015 \, {\left (80 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 76 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 71 \, C a^{2} \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 )}{d} + \frac {180180 \, {\left (14 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 15 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 14 \, C a^{2} \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 )}{d} + \frac {180180 \, {\left (12 \, A a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 8 \, B a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, C a^{2} \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 )}{d}\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/1441440*sqrt(2)*(3465*C*a^2*sgn(cos(1/2*d*x + 1/2*c))*sin(13/2*d*x + 13/2*c)/d + 4095*(2*B*a^2*sgn(cos(1/2*d
*x + 1/2*c)) + 5*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(11/2*d*x + 11/2*c)/d + 10010*(2*A*a^2*sgn(cos(1/2*d*x +
1/2*c)) + 5*B*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 7*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(9/2*d*x + 9/2*c)/d + 1287
0*(10*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 13*B*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 15*C*a^2*sgn(cos(1/2*d*x + 1/2*c)
))*sin(7/2*d*x + 7/2*c)/d + 9009*(48*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 50*B*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 51
*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(5/2*d*x + 5/2*c)/d + 15015*(80*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 76*B*a^
2*sgn(cos(1/2*d*x + 1/2*c)) + 71*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(3/2*d*x + 3/2*c)/d + 180180*(14*A*a^2*sg
n(cos(1/2*d*x + 1/2*c)) + 15*B*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 14*C*a^2*sgn(cos(1/2*d*x + 1/2*c)))*sin(1/2*d*x
+ 1/2*c)/d + 180180*(12*A*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 8*B*a^2*sgn(cos(1/2*d*x + 1/2*c)) + 7*C*a^2*sgn(cos
(1/2*d*x + 1/2*c)))*sin(1/2*d*x + 1/2*c)/d)*sqrt(a)

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maple [A]  time = 0.81, size = 176, normalized size = 0.60 \[ \frac {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (55440 C \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-32760 B -262080 C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20020 A +140140 B +520520 C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-77220 A -244530 B -566280 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (117117 A +225225 B +369369 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-90090 A -120120 B -150150 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45045 A +45045 B +45045 C \right ) \sqrt {2}}{45045 \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)

[Out]

8/45045*cos(1/2*d*x+1/2*c)*a^3*sin(1/2*d*x+1/2*c)*(55440*C*sin(1/2*d*x+1/2*c)^12+(-32760*B-262080*C)*sin(1/2*d
*x+1/2*c)^10+(20020*A+140140*B+520520*C)*sin(1/2*d*x+1/2*c)^8+(-77220*A-244530*B-566280*C)*sin(1/2*d*x+1/2*c)^
6+(117117*A+225225*B+369369*C)*sin(1/2*d*x+1/2*c)^4+(-90090*A-120120*B-150150*C)*sin(1/2*d*x+1/2*c)^2+45045*A+
45045*B+45045*C)*2^(1/2)/(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d

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maxima [A]  time = 0.73, size = 332, normalized size = 1.13 \[ \frac {572 \, {\left (35 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 225 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 756 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 2100 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 8190 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 130 \, {\left (63 \, \sqrt {2} a^{2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 385 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 1287 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 3465 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 8778 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 31878 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + {\left (3465 \, \sqrt {2} a^{2} \sin \left (\frac {13}{2} \, d x + \frac {13}{2} \, c\right ) + 20475 \, \sqrt {2} a^{2} \sin \left (\frac {11}{2} \, d x + \frac {11}{2} \, c\right ) + 70070 \, \sqrt {2} a^{2} \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 193050 \, \sqrt {2} a^{2} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 459459 \, \sqrt {2} a^{2} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1066065 \, \sqrt {2} a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3783780 \, \sqrt {2} a^{2} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{1441440 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

1/1441440*(572*(35*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 225*sqrt(2)*a^2*sin(7/2*d*x + 7/2*c) + 756*sqrt(2)*a^2*s
in(5/2*d*x + 5/2*c) + 2100*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 8190*sqrt(2)*a^2*sin(1/2*d*x + 1/2*c))*A*sqrt(a)
+ 130*(63*sqrt(2)*a^2*sin(11/2*d*x + 11/2*c) + 385*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 1287*sqrt(2)*a^2*sin(7/
2*d*x + 7/2*c) + 3465*sqrt(2)*a^2*sin(5/2*d*x + 5/2*c) + 8778*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 31878*sqrt(2)
*a^2*sin(1/2*d*x + 1/2*c))*B*sqrt(a) + (3465*sqrt(2)*a^2*sin(13/2*d*x + 13/2*c) + 20475*sqrt(2)*a^2*sin(11/2*d
*x + 11/2*c) + 70070*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 193050*sqrt(2)*a^2*sin(7/2*d*x + 7/2*c) + 459459*sqrt(
2)*a^2*sin(5/2*d*x + 5/2*c) + 1066065*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) + 3783780*sqrt(2)*a^2*sin(1/2*d*x + 1/2
*c))*C*sqrt(a))/d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^2*(a + a*cos(c + d*x))^(5/2)*(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))^(5/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*(a+a*cos(d*x+c))**(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

Timed out

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