\(\int \frac {(a+a \cos (c+d x))^{3/2} (A+C \cos ^2(c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) [187]

Optimal result

Integrand size = 37, antiderivative size = 266 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]

[Out]

2/11*A*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d/cos(d*x+c)^(11/2)+2/231*a^2*(28*A+33*C)*sin(d*x+c)/d/cos(d*x+c)^(7/
2)/(a+a*cos(d*x+c))^(1/2)+2/385*a^2*(112*A+143*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)/(a+a*cos(d*x+c))^(1/2)+8/1155*
a^2*(112*A+143*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(1/2)+16/1155*a^2*(112*A+143*C)*sin(d*x+c)/d/
cos(d*x+c)^(1/2)/(a+a*cos(d*x+c))^(1/2)+2/33*a*A*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(9/2)

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {3123, 3054, 3059, 2851, 2850} \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {2 a A \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{33 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]

[In]

Int[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]

[Out]

(2*a^2*(28*A + 33*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (2*a^2*(112*A + 143*C
)*Sin[c + d*x])/(385*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sin[c + d*x])/(11
55*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (16*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Sqrt[Cos[c +
d*x]]*Sqrt[a + a*Cos[c + d*x]]) + (2*a*A*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(33*d*Cos[c + d*x]^(9/2)) + (
2*A*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2))

Rule 2850

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 2851

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 3054

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
*Sin[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 3059

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 3123

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*Si
n[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*} \text {integral}& = \frac {2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \cos (c+d x))^{3/2} \left (\frac {3 a A}{2}+\frac {1}{2} a (6 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx}{11 a} \\ & = \frac {2 a A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {3}{4} a^2 (28 A+33 C)+\frac {9}{4} a^2 (8 A+11 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{99 a} \\ & = \frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {1}{77} (a (112 A+143 C)) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {1}{385} (4 a (112 A+143 C)) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {(8 a (112 A+143 C)) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{1155} \\ & = \frac {2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{33 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.55 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (1652 A+1188 C+(4228 A+4147 C) \cos (c+d x)+2 (728 A+737 C) \cos (2 (c+d x))+1456 A \cos (3 (c+d x))+1859 C \cos (3 (c+d x))+224 A \cos (4 (c+d x))+286 C \cos (4 (c+d x))+224 A \cos (5 (c+d x))+286 C \cos (5 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{2310 d \cos ^{\frac {11}{2}}(c+d x)} \]

[In]

Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]

[Out]

(a*Sqrt[a*(1 + Cos[c + d*x])]*(1652*A + 1188*C + (4228*A + 4147*C)*Cos[c + d*x] + 2*(728*A + 737*C)*Cos[2*(c +
d*x)] + 1456*A*Cos[3*(c + d*x)] + 1859*C*Cos[3*(c + d*x)] + 224*A*Cos[4*(c + d*x)] + 286*C*Cos[4*(c + d*x)] +
224*A*Cos[5*(c + d*x)] + 286*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(2310*d*Cos[c + d*x]^(11/2))

Maple [A] (verified)

Time = 13.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.54

method	result	size
default	\(\frac {2 a \sin \left (d x +c \right ) \left (896 A \left (\cos ^{5}\left (d x +c \right )\right )+1144 C \left (\cos ^{5}\left (d x +c \right )\right )+448 A \left (\cos ^{4}\left (d x +c \right )\right )+572 C \left (\cos ^{4}\left (d x +c \right )\right )+336 A \left (\cos ^{3}\left (d x +c \right )\right )+429 C \left (\cos ^{3}\left (d x +c \right )\right )+280 A \left (\cos ^{2}\left (d x +c \right )\right )+165 C \left (\cos ^{2}\left (d x +c \right )\right )+245 A \cos \left (d x +c \right )+105 A \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{1155 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {11}{2}}}\)	\(144\)
parts	\(\frac {2 A \sin \left (d x +c \right ) \left (128 \left (\cos ^{5}\left (d x +c \right )\right )+64 \left (\cos ^{4}\left (d x +c \right )\right )+48 \left (\cos ^{3}\left (d x +c \right )\right )+40 \left (\cos ^{2}\left (d x +c \right )\right )+35 \cos \left (d x +c \right )+15\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a}{165 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {11}{2}}}+\frac {2 C \sin \left (d x +c \right ) \left (104 \left (\cos ^{3}\left (d x +c \right )\right )+52 \left (\cos ^{2}\left (d x +c \right )\right )+39 \cos \left (d x +c \right )+15\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a}{105 d \left (1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )^{\frac {7}{2}}}\)	\(168\)

[In]

int((a+cos(d*x+c)*a)^(3/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x,method=_RETURNVERBOSE)

[Out]

2/1155*a/d*sin(d*x+c)*(896*A*cos(d*x+c)^5+1144*C*cos(d*x+c)^5+448*A*cos(d*x+c)^4+572*C*cos(d*x+c)^4+336*A*cos(
d*x+c)^3+429*C*cos(d*x+c)^3+280*A*cos(d*x+c)^2+165*C*cos(d*x+c)^2+245*A*cos(d*x+c)+105*A)*(a*(1+cos(d*x+c)))^(
1/2)/(1+cos(d*x+c))/cos(d*x+c)^(11/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.52 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 \, {\left (8 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{5} + 4 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \, {\left (112 \, A + 143 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (56 \, A + 33 \, C\right )} a \cos \left (d x + c\right )^{2} + 245 \, A a \cos \left (d x + c\right ) + 105 \, A a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{1155 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}} \]

[In]

integrate((a+a*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="fricas")

[Out]

2/1155*(8*(112*A + 143*C)*a*cos(d*x + c)^5 + 4*(112*A + 143*C)*a*cos(d*x + c)^4 + 3*(112*A + 143*C)*a*cos(d*x
+ c)^3 + 5*(56*A + 33*C)*a*cos(d*x + c)^2 + 245*A*a*cos(d*x + c) + 105*A*a)*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)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((a+a*cos(d*x+c))**(3/2)*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(13/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (230) = 460\).

Time = 0.39 (sec) , antiderivative size = 620, normalized size of antiderivative = 2.33 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {4 \, {\left (\frac {11 \, {\left (\frac {105 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {245 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {273 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {171 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {38 \, \sqrt {2} a^{\frac {3}{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 )}^{3}}{{\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 {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}} + \frac {7 \, {\left (\frac {165 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {495 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1056 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1254 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {781 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {299 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {46 \, \sqrt {2} a^{\frac {3}{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 )}^{5}}{{\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 {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {\sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 1\right )}}\right )}}{1155 \, d} \]

[In]

integrate((a+a*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="maxima")

[Out]

4/1155*(11*(105*sqrt(2)*a^(3/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 245*sqrt(2)*a^(3/2)*sin(d*x + c)^3/(cos(d*x
+ c) + 1)^3 + 273*sqrt(2)*a^(3/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 171*sqrt(2)*a^(3/2)*sin(d*x + c)^7/(co
s(d*x + c) + 1)^7 + 38*sqrt(2)*a^(3/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)*C*(sin(d*x + c)^2/(cos(d*x + c) +
1)^2 + 1)^3/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(9/2)*(3*sin(d
*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + sin(d*x + c)^6/(cos(d*x + c) + 1)^6 +
1)) + 7*(165*sqrt(2)*a^(3/2)*sin(d*x + c)/(cos(d*x + c) + 1) - 495*sqrt(2)*a^(3/2)*sin(d*x + c)^3/(cos(d*x +
c) + 1)^3 + 1056*sqrt(2)*a^(3/2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1254*sqrt(2)*a^(3/2)*sin(d*x + c)^7/(co
s(d*x + c) + 1)^7 + 781*sqrt(2)*a^(3/2)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 299*sqrt(2)*a^(3/2)*sin(d*x + c)
^11/(cos(d*x + c) + 1)^11 + 46*sqrt(2)*a^(3/2)*sin(d*x + c)^13/(cos(d*x + c) + 1)^13)*A*(sin(d*x + c)^2/(cos(d
*x + c) + 1)^2 + 1)^5/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(13
/2)*(5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*sin(d*x + c)^6/(cos(d
*x + c) + 1)^6 + 5*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1)))/d

Giac [F(-1)]

Timed out. \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((a+a*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 8.18 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.34 \[ \int \frac {(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\frac {\sqrt {a+a\,\cos \left (c+d\,x\right )}\,\left (-\frac {16\,C\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{3\,d}-\frac {16\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,A+23\,C\right )}{15\,d}+\frac {48\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\left (28\,A+27\,C\right )}{35\,d}+\frac {16\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\left (112\,A+143\,C\right )}{105\,d}+\frac {32\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,\left (112\,A+143\,C\right )}{1155\,d}\right )}{20\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+20\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )+10\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+10\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )+2\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )+2\,\sqrt {\cos \left (c+d\,x\right )}\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )} \]

[In]

int(((A + C*cos(c + d*x)^2)*(a + a*cos(c + d*x))^(3/2))/cos(c + d*x)^(13/2),x)

[Out]

((a + a*cos(c + d*x))^(1/2)*((48*a*exp((c*11i)/2 + (d*x*11i)/2)*sin((3*c)/2 + (3*d*x)/2)*(28*A + 27*C))/(35*d)
- (16*a*exp((c*11i)/2 + (d*x*11i)/2)*sin(c/2 + (d*x)/2)*(12*A + 23*C))/(15*d) - (16*C*a*exp((c*11i)/2 + (d*x*
11i)/2)*sin((5*c)/2 + (5*d*x)/2))/(3*d) + (16*a*exp((c*11i)/2 + (d*x*11i)/2)*sin((7*c)/2 + (7*d*x)/2)*(112*A +
143*C))/(105*d) + (32*a*exp((c*11i)/2 + (d*x*11i)/2)*sin((11*c)/2 + (11*d*x)/2)*(112*A + 143*C))/(1155*d)))/(
20*cos(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos(c/2 + (d*x)/2) + 20*cos(c + d*x)^(1/2)*exp((c*11i)/2 +
(d*x*11i)/2)*cos((3*c)/2 + (3*d*x)/2) + 10*cos(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos((5*c)/2 + (5*d*
x)/2) + 10*cos(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos((7*c)/2 + (7*d*x)/2) + 2*cos(c + d*x)^(1/2)*exp
((c*11i)/2 + (d*x*11i)/2)*cos((9*c)/2 + (9*d*x)/2) + 2*cos(c + d*x)^(1/2)*exp((c*11i)/2 + (d*x*11i)/2)*cos((11
*c)/2 + (11*d*x)/2))