3.12.44 \(\int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\) [1144]

Optimal. Leaf size=313 \[ \frac {16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+10439 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2224 A+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {10 a A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d} \]

[Out]

10/143*a*A*cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^(3/2)*sin(d*x+c)/d+2/13*A*cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^(5/2
)*sin(d*x+c)/d+2/15015*a^3*(8368*A+10439*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+2/9009*a^3*(2
224*A+2717*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+16/45045*a^3*(8368*A+10439*C)*sin(d*x+c)/d/
cos(d*x+c)^(1/2)/(a+a*sec(d*x+c))^(1/2)+8/45045*a^3*(8368*A+10439*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/d/(a+a*sec(d*
x+c))^(1/2)+2/1287*a^2*(136*A+143*C)*cos(d*x+c)^(7/2)*sin(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {4350, 4172, 4102, 4100, 3890, 3889} \begin {gather*} \frac {2 a^3 (2224 A+2717 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{9009 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (8368 A+10439 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{15015 d \sqrt {a \sec (c+d x)+a}}+\frac {8 a^3 (8368 A+10439 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{45045 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (136 A+143 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{1287 d}+\frac {2 A \sin (c+d x) \cos ^{\frac {11}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d}+\frac {10 a A \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(8368*
A + 10439*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*C
os[c + d*x]^(3/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2224*A + 2717*C)*Cos[c + d*x]^(5/
2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 143*C)*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[
c + d*x]]*Sin[c + d*x])/(1287*d) + (10*a*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(143*d)
+ (2*A*Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)

Rule 3889

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)], x_Symbol] :> Simp[-2*a*(Co
t[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[d*Csc[e + f*x]])), x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^
2, 0]

Rule 3890

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

Rule 4100

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(
B_.) + (A_)), x_Symbol] :> Simp[A*b^2*Cot[e + f*x]*((d*Csc[e + f*x])^n/(a*f*n*Sqrt[a + b*Csc[e + f*x]])), x] +
Dist[(A*b*(2*n + 1) + 2*a*B*n)/(2*a*d*n), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; Fr
eeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] &&
LtQ[n, 0]

Rule 4102

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x]
- Dist[b/(a*d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*(m - n - 1) - b*B*n - (a*
B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2
- b^2, 0] && GtQ[m, 1/2] && LtQ[n, -1]

Rule 4172

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Dis
t[1/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*(A*(m + n + 1) + C*n)*Csc[e +
f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -2
^(-1)] || EqQ[m + n + 1, 0])

Rule 4350

Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]

Rubi steps

\begin {align*} \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx\\ &=\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{5/2} \left (\frac {5 a A}{2}+\frac {1}{2} a (6 A+13 C) \sec (c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx}{13 a}\\ &=\frac {10 a A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {1}{4} a^2 (136 A+143 C)+\frac {1}{4} a^2 (96 A+143 C) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx}{143 a}\\ &=\frac {2 a^2 (136 A+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {10 a A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (2224 A+2717 C)+\frac {15}{8} a^3 (112 A+143 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{1287 a}\\ &=\frac {2 a^3 (2224 A+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {10 a A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (a^2 (8368 A+10439 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{3003}\\ &=\frac {2 a^3 (8368 A+10439 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2224 A+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {10 a A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (4 a^2 (8368 A+10439 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{15015}\\ &=\frac {8 a^3 (8368 A+10439 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2224 A+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {10 a A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}+\frac {\left (8 a^2 (8368 A+10439 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{45045}\\ &=\frac {16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {8 a^3 (8368 A+10439 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (8368 A+10439 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (2224 A+2717 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (136 A+143 C) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{1287 d}+\frac {10 a A \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{143 d}+\frac {2 A \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{13 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 6.60, size = 206, normalized size = 0.66 \begin {gather*} \frac {\cos ^{\frac {9}{2}}(c+d x) \sec ^5\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {1}{16} (21 A+26 C) \sin \left (\frac {1}{2} (c+d x)\right )+\frac {1}{192} (71 A+80 C) \sin \left (\frac {3}{2} (c+d x)\right )+\frac {3}{320} (17 A+16 C) \sin \left (\frac {5}{2} (c+d x)\right )+\frac {5}{224} (3 A+2 C) \sin \left (\frac {7}{2} (c+d x)\right )+\frac {1}{288} (7 A+2 C) \sin \left (\frac {9}{2} (c+d x)\right )+\frac {5}{704} A \sin \left (\frac {11}{2} (c+d x)\right )+\frac {1}{832} A \sin \left (\frac {13}{2} (c+d x)\right )\right )}{d (A+2 C+A \cos (2 c+2 d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[c + d*x]^(9/2)*Sec[(c + d*x)/2]^5*(a*(1 + Sec[c + d*x]))^(5/2)*(A + C*Sec[c + d*x]^2)*(((21*A + 26*C)*Sin
[(c + d*x)/2])/16 + ((71*A + 80*C)*Sin[(3*(c + d*x))/2])/192 + (3*(17*A + 16*C)*Sin[(5*(c + d*x))/2])/320 + (5
*(3*A + 2*C)*Sin[(7*(c + d*x))/2])/224 + ((7*A + 2*C)*Sin[(9*(c + d*x))/2])/288 + (5*A*Sin[(11*(c + d*x))/2])/
704 + (A*Sin[(13*(c + d*x))/2])/832))/(d*(A + 2*C + A*Cos[2*c + 2*d*x]))

________________________________________________________________________________________

Maple [A]
time = 0.26, size = 166, normalized size = 0.53

method	result	size
default	\(-\frac {2 a^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (3465 A \left (\cos ^{6}\left (d x +c \right )\right )+11970 A \left (\cos ^{5}\left (d x +c \right )\right )+18305 A \left (\cos ^{4}\left (d x +c \right )\right )+5005 C \left (\cos ^{4}\left (d x +c \right )\right )+20920 A \left (\cos ^{3}\left (d x +c \right )\right )+18590 C \left (\cos ^{3}\left (d x +c \right )\right )+25104 A \left (\cos ^{2}\left (d x +c \right )\right )+31317 C \left (\cos ^{2}\left (d x +c \right )\right )+33472 A \cos \left (d x +c \right )+41756 C \cos \left (d x +c \right )+66944 A +83512 C \right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{45045 d \sin \left (d x +c \right )}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45045/d*a^2*(-1+cos(d*x+c))*(3465*A*cos(d*x+c)^6+11970*A*cos(d*x+c)^5+18305*A*cos(d*x+c)^4+5005*C*cos(d*x+c
)^4+20920*A*cos(d*x+c)^3+18590*C*cos(d*x+c)^3+25104*A*cos(d*x+c)^2+31317*C*cos(d*x+c)^2+33472*A*cos(d*x+c)+417
56*C*cos(d*x+c)+66944*A+83512*C)*cos(d*x+c)^(1/2)*(a*(1+cos(d*x+c))/cos(d*x+c))^(1/2)/sin(d*x+c)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (271) = 542\).
time = 0.66, size = 852, normalized size = 2.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2882880*(sqrt(2)*(3783780*a^2*cos(12/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c)))*sin(13/2*d*
x + 13/2*c) + 1066065*a^2*cos(10/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c)))*sin(13/2*d*x + 13
/2*c) + 459459*a^2*cos(8/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c)))*sin(13/2*d*x + 13/2*c) +
193050*a^2*cos(6/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c)))*sin(13/2*d*x + 13/2*c) + 70070*a^
2*cos(4/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c)))*sin(13/2*d*x + 13/2*c) + 20475*a^2*cos(2/1
3*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c)))*sin(13/2*d*x + 13/2*c) - 3783780*a^2*cos(13/2*d*x +
13/2*c)*sin(12/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) - 1066065*a^2*cos(13/2*d*x + 13/2*
c)*sin(10/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) - 459459*a^2*cos(13/2*d*x + 13/2*c)*sin(
8/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) - 193050*a^2*cos(13/2*d*x + 13/2*c)*sin(6/13*arc
tan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) - 70070*a^2*cos(13/2*d*x + 13/2*c)*sin(4/13*arctan2(sin(
13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) - 20475*a^2*cos(13/2*d*x + 13/2*c)*sin(2/13*arctan2(sin(13/2*d*x
+ 13/2*c), cos(13/2*d*x + 13/2*c))) + 6930*a^2*sin(13/2*d*x + 13/2*c) + 20475*a^2*sin(11/13*arctan2(sin(13/2*d
*x + 13/2*c), cos(13/2*d*x + 13/2*c))) + 70070*a^2*sin(9/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/
2*c))) + 193050*a^2*sin(7/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) + 459459*a^2*sin(5/13*ar
ctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))) + 1066065*a^2*sin(3/13*arctan2(sin(13/2*d*x + 13/2*c),
cos(13/2*d*x + 13/2*c))) + 3783780*a^2*sin(1/13*arctan2(sin(13/2*d*x + 13/2*c), cos(13/2*d*x + 13/2*c))))*A*sq
rt(a) + 1144*sqrt(2)*(225*a^2*sin(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 378*a^2*sin(5/4*arctan2(s
in(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2100*a^2*sin(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 4095*a^2
*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 63*(65*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*
cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 7*(585*a^2*cos(4*d*x + 4*c) + 54*a^2*cos(2*d*x + 2*c) +
5*a^2)*sin(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*C*sqrt(a))/d

________________________________________________________________________________________

Fricas [A]
time = 2.97, size = 168, normalized size = 0.54 \begin {gather*} \frac {2 \, {\left (3465 \, A a^{2} \cos \left (d x + c\right )^{6} + 11970 \, A a^{2} \cos \left (d x + c\right )^{5} + 35 \, {\left (523 \, A + 143 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (2092 \, A + 1859 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 4 \, {\left (8368 \, A + 10439 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (8368 \, A + 10439 \, C\right )} a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*A*a^2*cos(d*x + c)^6 + 11970*A*a^2*cos(d*x + c)^5 + 35*(523*A + 143*C)*a^2*cos(d*x + c)^4 + 10*(
2092*A + 1859*C)*a^2*cos(d*x + c)^3 + 3*(8368*A + 10439*C)*a^2*cos(d*x + c)^2 + 4*(8368*A + 10439*C)*a^2*cos(d
*x + c) + 8*(8368*A + 10439*C)*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c)/(d
*cos(d*x + c) + d)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^{13/2}\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________