3.196 \(\int \frac {(a+a \cos (c+d x))^{5/2} (A+C \cos ^2(c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=266 \[ \frac {2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (32 A+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]

[Out]

10/99*a*A*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/11*A*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/d/cos(
d*x+c)^(11/2)+2/693*a^3*(232*A+297*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(1/2)+2/693*a^3*(568*A+75
9*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(1/2)+4/693*a^3*(568*A+759*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2
)/(a+a*cos(d*x+c))^(1/2)+2/231*a^2*(32*A+33*C)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(7/2)

________________________________________________________________________________________

Rubi [A]  time = 0.80, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {3044, 2975, 2980, 2772, 2771} \[ \frac {2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (32 A+33 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]

[Out]

(2*a^3*(232*A + 297*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^3*(568*A + 759
*C)*Sin[c + d*x])/(693*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a^3*(568*A + 759*C)*Sin[c + d*x])/(
693*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(32*A + 33*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x
])/(231*d*Cos[c + d*x]^(7/2)) + (10*a*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (
2*A*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))

Rule 2771

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> Sim
p[(-2*b^2*Cos[e + f*x])/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b,
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2772

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[((b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x]
+ Dist[((2*n + 3)*(b*c - a*d))/(2*b*(n + 1)*(c^2 - d^2)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]

Rule 2975

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^2*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*S
in[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])
^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n +
1) - B*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, 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 2980

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n + 1)
*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rule 3044

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^
m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n +
2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b
*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0
])

Rubi steps

\begin {align*} \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \cos (c+d x))^{5/2} \left (\frac {5 a A}{2}+\frac {1}{2} a (4 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \cos (c+d x))^{3/2} \left (\frac {3}{4} a^2 (32 A+33 C)+\frac {1}{4} a^2 (56 A+99 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac {2 a^2 (32 A+33 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {8 \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {5}{8} a^3 (232 A+297 C)+\frac {1}{8} a^3 (776 A+1089 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac {2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (32 A+33 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {1}{231} \left (a^2 (568 A+759 C)\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (32 A+33 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {1}{693} \left (2 a^2 (568 A+759 C)\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (32 A+33 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {10 a A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.99, size = 149, normalized size = 0.56 \[ \frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cos (c+d x)+1)} (2 (5014 A+4983 C) \cos (c+d x)+52 (71 A+66 C) \cos (2 (c+d x))+3692 A \cos (3 (c+d x))+568 A \cos (4 (c+d x))+568 A \cos (5 (c+d x))+3628 A+4587 C \cos (3 (c+d x))+759 C \cos (4 (c+d x))+759 C \cos (5 (c+d x))+2673 C)}{2772 d \cos ^{\frac {11}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(3628*A + 2673*C + 2*(5014*A + 4983*C)*Cos[c + d*x] + 52*(71*A + 66*C)*Cos[2*(
c + d*x)] + 3692*A*Cos[3*(c + d*x)] + 4587*C*Cos[3*(c + d*x)] + 568*A*Cos[4*(c + d*x)] + 759*C*Cos[4*(c + d*x)
] + 568*A*Cos[5*(c + d*x)] + 759*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(2772*d*Cos[c + d*x]^(11/2))

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 148, normalized size = 0.56 \[ \frac {2 \, {\left (2 \, {\left (568 \, A + 759 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + {\left (568 \, A + 759 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 6 \, {\left (71 \, A + 66 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (355 \, A + 99 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 224 \, A a^{2} \cos \left (d x + c\right ) + 63 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{693 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="fricas")

[Out]

2/693*(2*(568*A + 759*C)*a^2*cos(d*x + c)^5 + (568*A + 759*C)*a^2*cos(d*x + c)^4 + 6*(71*A + 66*C)*a^2*cos(d*x
+ c)^3 + (355*A + 99*C)*a^2*cos(d*x + c)^2 + 224*A*a^2*cos(d*x + c) + 63*A*a^2)*sqrt(a*cos(d*x + c) + a)*sqrt
(cos(d*x + c))*sin(d*x + c)/(d*cos(d*x + c)^7 + d*cos(d*x + c)^6)

________________________________________________________________________________________

giac [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((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.40, size = 146, normalized size = 0.55 \[ -\frac {2 a^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (1136 A \left (\cos ^{5}\left (d x +c \right )\right )+1518 C \left (\cos ^{5}\left (d x +c \right )\right )+568 A \left (\cos ^{4}\left (d x +c \right )\right )+759 C \left (\cos ^{4}\left (d x +c \right )\right )+426 A \left (\cos ^{3}\left (d x +c \right )\right )+396 C \left (\cos ^{3}\left (d x +c \right )\right )+355 A \left (\cos ^{2}\left (d x +c \right )\right )+99 C \left (\cos ^{2}\left (d x +c \right )\right )+224 A \cos \left (d x +c \right )+63 A \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{693 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x)

[Out]

-2/693/d*a^2*(-1+cos(d*x+c))*(1136*A*cos(d*x+c)^5+1518*C*cos(d*x+c)^5+568*A*cos(d*x+c)^4+759*C*cos(d*x+c)^4+42
6*A*cos(d*x+c)^3+396*C*cos(d*x+c)^3+355*A*cos(d*x+c)^2+99*C*cos(d*x+c)^2+224*A*cos(d*x+c)+63*A)*(a*(1+cos(d*x+
c)))^(1/2)/sin(d*x+c)/cos(d*x+c)^(11/2)

________________________________________________________________________________________

maxima [B]  time = 1.01, size = 579, normalized size = 2.18 \[ \frac {8 \, {\left (\frac {33 \, {\left (\frac {21 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {56 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {63 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {36 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {8 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )} C {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{{\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}} {\left (\frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}} + \frac {{\left (\frac {693 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2310 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {4620 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5478 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {3575 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {1300 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {200 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}}\right )} A {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{{\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (\frac {4 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {\sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 1\right )}}\right )}}{693 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

8/693*(33*(21*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 56*sqrt(2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c
) + 1)^3 + 63*sqrt(2)*a^(5/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 36*sqrt(2)*a^(5/2)*sin(d*x + c)^7/(cos(d*x
+ c) + 1)^7 + 8*sqrt(2)*a^(5/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)*C*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 +
1)^2/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(2*sin(d*x + c
)^2/(cos(d*x + c) + 1)^2 + sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1)) + (693*sqrt(2)*a^(5/2)*sin(d*x + c)/(cos(
d*x + c) + 1) - 2310*sqrt(2)*a^(5/2)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 4620*sqrt(2)*a^(5/2)*sin(d*x + c)^5
/(cos(d*x + c) + 1)^5 - 5478*sqrt(2)*a^(5/2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 3575*sqrt(2)*a^(5/2)*sin(d*
x + c)^9/(cos(d*x + c) + 1)^9 - 1300*sqrt(2)*a^(5/2)*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 200*sqrt(2)*a^(5/
2)*sin(d*x + c)^13/(cos(d*x + c) + 1)^13)*A*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^4/((sin(d*x + c)/(cos(d*
x + c) + 1) + 1)^(13/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 +
6*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + sin(d*x + c)^8/(cos(d*x + c)
+ 1)^8 + 1)))/d

________________________________________________________________________________________

mupad [B]  time = 8.35, size = 773, normalized size = 2.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(5/2))/cos(c + d*x)^(13/2),x)

[Out]

((a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((a^2*(568*A + 759*C)*4i)/(693*d) - (a^2*exp(c*
5i + d*x*5i)*(3*A + 5*C)*16i)/(3*d) + (a^2*exp(c*6i + d*x*6i)*(3*A + 5*C)*16i)/(3*d) + (a^2*exp(c*4i + d*x*4i)
*(32*A + 33*C)*8i)/(7*d) - (a^2*exp(c*7i + d*x*7i)*(32*A + 33*C)*8i)/(7*d) + (a^2*exp(c*2i + d*x*2i)*(71*A + 8
7*C)*16i)/(63*d) - (a^2*exp(c*9i + d*x*9i)*(71*A + 87*C)*16i)/(63*d) - (a^2*exp(c*11i + d*x*11i)*(568*A + 759*
C)*4i)/(693*d) - (C*a^2*exp(c*3i + d*x*3i)*20i)/(3*d) + (C*a^2*exp(c*8i + d*x*8i)*20i)/(3*d)))/((exp(- c*1i -
d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + exp(c*1i + d*x*1i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(
1/2) + 5*exp(c*2i + d*x*2i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 5*exp(c*3i + d*x*3i)*(exp(
- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 10*exp(c*4i + d*x*4i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i +
d*x*1i)/2)^(1/2) + 10*exp(c*5i + d*x*5i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 10*exp(c*6i +
d*x*6i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + 10*exp(c*7i + d*x*7i)*(exp(- c*1i - d*x*1i)/2
+ exp(c*1i + d*x*1i)/2)^(1/2) + 5*exp(c*8i + d*x*8i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) +
5*exp(c*9i + d*x*9i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + exp(c*10i + d*x*10i)*(exp(- c*1i
- d*x*1i)/2 + exp(c*1i + d*x*1i)/2)^(1/2) + exp(c*11i + d*x*11i)*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/
2)^(1/2))

________________________________________________________________________________________

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((a+a*cos(d*x+c))**(5/2)*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(13/2),x)

[Out]

Timed out

________________________________________________________________________________________