\(\int \frac {\cos ^{\frac {7}{2}}(c+d x) (A+C \cos ^2(c+d x))}{a+b \cos (c+d x)} \, dx\) [704]

Optimal result

Integrand size = 35, antiderivative size = 299 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {2 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^5 d}-\frac {2 a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 b^6 d}+\frac {2 a^4 \left (A b^2+a^2 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^6 (a+b) d}-\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d} \]

[Out]

2/15*(15*a^4*C+3*a^2*b^2*(5*A+3*C)+b^4*(9*A+7*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(si
n(1/2*d*x+1/2*c),2^(1/2))/b^5/d-2/21*a*(21*a^4*C+7*a^2*b^2*(3*A+C)+b^4*(7*A+5*C))*(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^6/d+2*a^4*(A*b^2+C*a^2)*(cos(1/2*d*x+1/2*c)^2)^(1/
2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/b^6/(a+b)/d+2/45*(9*a^2*C+b^2*(9*A+7*C)
)*cos(d*x+c)^(3/2)*sin(d*x+c)/b^3/d-2/7*a*C*cos(d*x+c)^(5/2)*sin(d*x+c)/b^2/d+2/9*C*cos(d*x+c)^(7/2)*sin(d*x+c
)/b/d-2/21*a*(7*A*b^2+7*C*a^2+5*C*b^2)*sin(d*x+c)*cos(d*x+c)^(1/2)/b^4/d

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3129, 3128, 3138, 2719, 3081, 2720, 2884} \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=-\frac {2 a \left (7 a^2 C+7 A b^2+5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 b^3 d}+\frac {2 a^4 \left (a^2 C+A b^2\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^6 d (a+b)}-\frac {2 a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 b^6 d}+\frac {2 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^5 d}-\frac {2 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b^2 d}+\frac {2 C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 b d} \]

[In]

Int[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]

[Out]

(2*(15*a^4*C + 3*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*b^5*d) - (2*a*(21*a^4*C
+ 7*a^2*b^2*(3*A + C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*b^6*d) + (2*a^4*(A*b^2 + a^2*C)*Ellip
ticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b^6*(a + b)*d) - (2*a*(7*A*b^2 + 7*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*
Sin[c + d*x])/(21*b^4*d) + (2*(9*a^2*C + b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*b^3*d) - (2*a*C
*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b^2*d) + (2*C*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*b*d)

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2884

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

Rule 3081

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

Rule 3128

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

Rule 3129

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

Rule 3138

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {2 \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {7 a C}{2}+\frac {1}{2} b (9 A+7 C) \cos (c+d x)-\frac {9}{2} a C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{9 b} \\ & = -\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {4 \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\frac {45 a^2 C}{4}+a b C \cos (c+d x)+\frac {7}{4} \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{63 b^2} \\ & = \frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {8 \int \frac {\sqrt {\cos (c+d x)} \left (\frac {21}{8} a \left (9 a^2 C+b^2 (9 A+7 C)\right )+\frac {3}{8} b \left (63 A b^2-12 a^2 C+49 b^2 C\right ) \cos (c+d x)-\frac {45}{8} a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{315 b^3} \\ & = -\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {16 \int \frac {-\frac {45}{16} a^2 \left (7 A b^2+7 a^2 C+5 b^2 C\right )+\frac {9}{4} a b \left (7 A b^2+7 a^2 C+6 b^2 C\right ) \cos (c+d x)+\frac {63}{16} \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{945 b^4} \\ & = -\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}-\frac {16 \int \frac {\frac {45}{16} a^2 b \left (7 A b^2+7 a^2 C+5 b^2 C\right )+\frac {45}{16} a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{945 b^5}+\frac {\left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 b^5} \\ & = \frac {2 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^5 d}-\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d}+\frac {\left (a^4 \left (A b^2+a^2 C\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{b^6}-\frac {\left (a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 b^6} \\ & = \frac {2 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^5 d}-\frac {2 a \left (21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 b^6 d}+\frac {2 a^4 \left (A b^2+a^2 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^6 (a+b) d}-\frac {2 a \left (7 A b^2+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^4 d}+\frac {2 \left (9 a^2 C+b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 b^3 d}-\frac {2 a C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b^2 d}+\frac {2 C \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.35 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {\sqrt {\cos (c+d x)} \left (7 b \left (36 A b^2+36 a^2 C+43 b^2 C\right ) \cos (c+d x)-5 \left (84 a A b^2+84 a^3 C+78 a b^2 C+18 a b^2 C \cos (2 (c+d x))-7 b^3 C \cos (3 (c+d x))\right )\right ) \sin (c+d x)+6 \left (\frac {\left (35 a^4 C+7 b^4 (9 A+7 C)+a^2 b^2 (35 A+13 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {8 a \left (7 A b^2+7 a^2 C+6 b^2 C\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+\frac {7 \left (15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {\sin ^2(c+d x)}}\right )}{630 b^4 d} \]

[In]

Integrate[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]

[Out]

(Sqrt[Cos[c + d*x]]*(7*b*(36*A*b^2 + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] - 5*(84*a*A*b^2 + 84*a^3*C + 78*a*b^2*C
+ 18*a*b^2*C*Cos[2*(c + d*x)] - 7*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x] + 6*(((35*a^4*C + 7*b^4*(9*A + 7*C) +
a^2*b^2*(35*A + 13*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(7*A*b^2 + 7*a^2*C + 6*b^2*C
)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (7*(15*a^4*C +
3*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*Ellip
ticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[
c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2])))/(630*b^4*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1553\) vs. \(2(357)=714\).

Time = 20.90 (sec) , antiderivative size = 1554, normalized size of antiderivative = 5.20

method	result	size
default	\(\text {Expression too large to display}\)	\(1554\)

[In]

int(cos(d*x+c)^(7/2)*(A+C*cos(d*x+c)^2)/(a+cos(d*x+c)*b),x,method=_RETURNVERBOSE)

[Out]

-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-189*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2
*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3*b^3-315*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin
(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^5*b+189*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*s
in(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b^4+75*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(
2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^5+315*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)
*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))*a^4*b^2+315*C*(sin(1/2*d*x
+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^4*b^2-315*A*(sin(1/2
*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^4*b^2+(-1120*C*a
*b^5+1120*C*b^6)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+(-720*C*a^2*b^4+2960*C*a*b^5-2240*C*b^6)*sin(1/2*d*x
+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-504*A*a*b^5+504*A*b^6-504*C*a^3*b^3+1584*C*a^2*b^4-3152*C*a*b^5+2072*C*b^6)*sin
(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(-420*A*a^2*b^4+924*A*a*b^5-504*A*b^6-420*C*a^4*b^2+924*C*a^3*b^3-1344*C*
a^2*b^4+1792*C*a*b^5-952*C*b^6)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(210*A*a^2*b^4-336*A*a*b^5+126*A*b^6+2
10*C*a^4*b^2-336*C*a^3*b^3+366*C*a^2*b^4-408*C*a*b^5+168*C*b^6)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+315*C*
(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^5*b-105*
C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^4*b^2-
147*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^
5-75*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^2
*b^4-105*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))
*a^2*b^4+315*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1
/2))*a^2*b^4-189*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),
2^(1/2))*a*b^5+105*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c
),2^(1/2))*a^3*b^3+105*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1
/2*c),2^(1/2))*a*b^5+315*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x
+1/2*c),2^(1/2))*a^3*b^3-315*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2
*d*x+1/2*c),2^(1/2))*a^3*b^3+147*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos
(1/2*d*x+1/2*c),2^(1/2))*b^6+315*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticPi(co
s(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))*a^6+189*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*E
llipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^6-315*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*E
llipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^6)/b^6/(a-b)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1
/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)^(7/2)*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**(7/2)*(A+C*cos(d*x+c)**2)/(a+b*cos(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]

[In]

integrate(cos(d*x+c)^(7/2)*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)),x, algorithm="maxima")

[Out]

integrate((C*cos(d*x + c)^2 + A)*cos(d*x + c)^(7/2)/(b*cos(d*x + c) + a), x)

Giac [F]

\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]

[In]

integrate(cos(d*x+c)^(7/2)*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)),x, algorithm="giac")

[Out]

integrate((C*cos(d*x + c)^2 + A)*cos(d*x + c)^(7/2)/(b*cos(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \]

[In]

int((cos(c + d*x)^(7/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x)),x)

[Out]

int((cos(c + d*x)^(7/2)*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x)), x)