3.594 \(\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx\)
Optimal. Leaf size=284 \[ \frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]
[Out]
(2*a^2*(84*A + 110*B + 99*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(336*A
+ 374*B + 429*C)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(336*A + 374*B +
429*C)*Sin[c + d*x])/(3465*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(336*A + 374*B + 429*C)*S
qrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Sec[c + d*x]]
*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9
/2))
________________________________________________________________________________________
Rubi [A] time = 0.739475, antiderivative size = 284, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used =
{4086, 4017, 4015, 3805, 3804} \[ \frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]
[Out]
(2*a^2*(84*A + 110*B + 99*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(336*A
+ 374*B + 429*C)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(336*A + 374*B +
429*C)*Sin[c + d*x])/(3465*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(336*A + 374*B + 429*C)*S
qrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Sec[c + d*x]]
*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9
/2))
Rule 4086
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + 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] - Dist[1/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m -
b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 -
b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Rule 4017
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 4015
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 3805
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 3804
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]
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{1}{2} a (3 A+11 B)+\frac{1}{2} a (6 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac{2 a (3 A+11 B) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{4 \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{1}{4} a^2 (84 A+110 B+99 C)+\frac{3}{4} a^2 (24 A+22 B+33 C) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 a (3 A+11 B) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{1}{231} (a (336 A+374 B+429 C)) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 a (3 A+11 B) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{(4 a (336 A+374 B+429 C)) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{1155}\\ &=\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a (3 A+11 B) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{(8 a (336 A+374 B+429 C)) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx}{3465}\\ &=\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{16 a^2 (336 A+374 B+429 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{3465 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (3 A+11 B) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 2.33363, size = 158, normalized size = 0.56 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} ((34734 A+44 (799 B+759 C)) \cos (c+d x)+8 (1743 A+1507 B+1287 C) \cos (2 (c+d x))+4935 A \cos (3 (c+d x))+1470 A \cos (4 (c+d x))+315 A \cos (5 (c+d x))+55482 A+3740 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+59158 B+1980 C \cos (3 (c+d x))+65208 C)}{27720 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]
[Out]
(a*(55482*A + 59158*B + 65208*C + (34734*A + 44*(799*B + 759*C))*Cos[c + d*x] + 8*(1743*A + 1507*B + 1287*C)*C
os[2*(c + d*x)] + 4935*A*Cos[3*(c + d*x)] + 3740*B*Cos[3*(c + d*x)] + 1980*C*Cos[3*(c + d*x)] + 1470*A*Cos[4*(
c + d*x)] + 770*B*Cos[4*(c + d*x)] + 315*A*Cos[5*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(277
20*d*Sqrt[Sec[c + d*x]])
________________________________________________________________________________________
Maple [A] time = 0.403, size = 197, normalized size = 0.7 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 315\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+735\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+385\,B \left ( \cos \left ( dx+c \right ) \right ) ^{4}+840\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+935\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+495\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1008\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+1122\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+1287\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+1344\,A\cos \left ( dx+c \right ) +1496\,B\cos \left ( dx+c \right ) +1716\,C\cos \left ( dx+c \right ) +2688\,A+2992\,B+3432\,C \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{3465\,d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x)
[Out]
-2/3465/d*a*(-1+cos(d*x+c))*(315*A*cos(d*x+c)^5+735*A*cos(d*x+c)^4+385*B*cos(d*x+c)^4+840*A*cos(d*x+c)^3+935*B
*cos(d*x+c)^3+495*C*cos(d*x+c)^3+1008*A*cos(d*x+c)^2+1122*B*cos(d*x+c)^2+1287*C*cos(d*x+c)^2+1344*A*cos(d*x+c)
+1496*B*cos(d*x+c)+1716*C*cos(d*x+c)+2688*A+2992*B+3432*C)*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*cos(d*x+c)^6*(1
/cos(d*x+c))^(11/2)/sin(d*x+c)
________________________________________________________________________________________
Maxima [B] time = 2.51371, size = 1604, normalized size = 5.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="maxima")
[Out]
1/110880*(21*sqrt(2)*(3630*a*cos(10/11*arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c)))*sin(11/2*d*x +
11/2*c) + 990*a*cos(8/11*arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c)))*sin(11/2*d*x + 11/2*c) + 42
9*a*cos(6/11*arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c)))*sin(11/2*d*x + 11/2*c) + 165*a*cos(4/11*
arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c)))*sin(11/2*d*x + 11/2*c) + 55*a*cos(2/11*arctan2(sin(11
/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c)))*sin(11/2*d*x + 11/2*c) - 3630*a*cos(11/2*d*x + 11/2*c)*sin(10/11*ar
ctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c))) - 990*a*cos(11/2*d*x + 11/2*c)*sin(8/11*arctan2(sin(11/
2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c))) - 429*a*cos(11/2*d*x + 11/2*c)*sin(6/11*arctan2(sin(11/2*d*x + 11/2*
c), cos(11/2*d*x + 11/2*c))) - 165*a*cos(11/2*d*x + 11/2*c)*sin(4/11*arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*
d*x + 11/2*c))) - 55*a*cos(11/2*d*x + 11/2*c)*sin(2/11*arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c))
) + 30*a*sin(11/2*d*x + 11/2*c) + 55*a*sin(9/11*arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c))) + 165
*a*sin(7/11*arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c))) + 429*a*sin(5/11*arctan2(sin(11/2*d*x + 1
1/2*c), cos(11/2*d*x + 11/2*c))) + 990*a*sin(3/11*arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c))) + 3
630*a*sin(1/11*arctan2(sin(11/2*d*x + 11/2*c), cos(11/2*d*x + 11/2*c))))*A*sqrt(a) + 22*sqrt(2)*(3780*a*cos(8/
9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c)))*sin(9/2*d*x + 9/2*c) + 1050*a*cos(2/3*arctan2(sin(9/2*d
*x + 9/2*c), cos(9/2*d*x + 9/2*c)))*sin(9/2*d*x + 9/2*c) + 378*a*cos(4/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2
*d*x + 9/2*c)))*sin(9/2*d*x + 9/2*c) + 135*a*cos(2/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c)))*sin(
9/2*d*x + 9/2*c) - 3780*a*cos(9/2*d*x + 9/2*c)*sin(8/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) -
1050*a*cos(9/2*d*x + 9/2*c)*sin(2/3*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) - 378*a*cos(9/2*d*x +
9/2*c)*sin(4/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) - 135*a*cos(9/2*d*x + 9/2*c)*sin(2/9*arct
an2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) + 70*a*sin(9/2*d*x + 9/2*c) + 135*a*sin(7/9*arctan2(sin(9/2*d
*x + 9/2*c), cos(9/2*d*x + 9/2*c))) + 378*a*sin(5/9*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) + 105
0*a*sin(1/3*arctan2(sin(9/2*d*x + 9/2*c), cos(9/2*d*x + 9/2*c))) + 3780*a*sin(1/9*arctan2(sin(9/2*d*x + 9/2*c)
, cos(9/2*d*x + 9/2*c))))*B*sqrt(a) + 132*sqrt(2)*(735*a*cos(6/7*arctan2(sin(7/2*d*x + 7/2*c), cos(7/2*d*x + 7
/2*c)))*sin(7/2*d*x + 7/2*c) + 175*a*cos(4/7*arctan2(sin(7/2*d*x + 7/2*c), cos(7/2*d*x + 7/2*c)))*sin(7/2*d*x
+ 7/2*c) + 63*a*cos(2/7*arctan2(sin(7/2*d*x + 7/2*c), cos(7/2*d*x + 7/2*c)))*sin(7/2*d*x + 7/2*c) - 735*a*cos(
7/2*d*x + 7/2*c)*sin(6/7*arctan2(sin(7/2*d*x + 7/2*c), cos(7/2*d*x + 7/2*c))) - 175*a*cos(7/2*d*x + 7/2*c)*sin
(4/7*arctan2(sin(7/2*d*x + 7/2*c), cos(7/2*d*x + 7/2*c))) - 63*a*cos(7/2*d*x + 7/2*c)*sin(2/7*arctan2(sin(7/2*
d*x + 7/2*c), cos(7/2*d*x + 7/2*c))) + 30*a*sin(7/2*d*x + 7/2*c) + 63*a*sin(5/7*arctan2(sin(7/2*d*x + 7/2*c),
cos(7/2*d*x + 7/2*c))) + 175*a*sin(3/7*arctan2(sin(7/2*d*x + 7/2*c), cos(7/2*d*x + 7/2*c))) + 735*a*sin(1/7*ar
ctan2(sin(7/2*d*x + 7/2*c), cos(7/2*d*x + 7/2*c))))*C*sqrt(a))/d
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Fricas [A] time = 0.506846, size = 460, normalized size = 1.62 \begin{align*} \frac{2 \,{\left (315 \, A a \cos \left (d x + c\right )^{6} + 35 \,{\left (21 \, A + 11 \, B\right )} a \cos \left (d x + c\right )^{5} + 5 \,{\left (168 \, A + 187 \, B + 99 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \,{\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{3} + 4 \,{\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )^{2} + 8 \,{\left (336 \, A + 374 \, B + 429 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3465 \,{\left (d \cos \left (d x + c\right ) + d\right )} \sqrt{\cos \left (d x + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="fricas")
[Out]
2/3465*(315*A*a*cos(d*x + c)^6 + 35*(21*A + 11*B)*a*cos(d*x + c)^5 + 5*(168*A + 187*B + 99*C)*a*cos(d*x + c)^4
+ 3*(336*A + 374*B + 429*C)*a*cos(d*x + c)^3 + 4*(336*A + 374*B + 429*C)*a*cos(d*x + c)^2 + 8*(336*A + 374*B
+ 429*C)*a*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/((d*cos(d*x + c) + d)*sqrt(cos(d
*x + c)))
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(11/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\sec \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x, algorithm="giac")
[Out]
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^(3/2)/sec(d*x + c)^(11/2), x)