3.11.2 \(\int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) [1002]

Optimal. Leaf size=336 \[ \frac {2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {2 (2 A b+3 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]

[Out]

2/315*a*(24*A*b^2+99*a*b*B+7*a^2*(7*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/21*(2*A*b+3*B*a)*(a+b*sec(d*x+c))^
2*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/9*A*(a+b*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/63*(8*A*b^3+15*a^3*B+
54*a*b^2*B+9*a^2*b*(5*A+7*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/15*(27*a^2*b*B+15*b^3*B+9*a*b^2*(3*A+5*C)+a^3*(7
*A+9*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2
)*sec(d*x+c)^(1/2)/d+2/21*(5*a^3*B+21*a*b^2*B+7*b^3*(A+3*C)+3*a^2*b*(5*A+7*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/co
s(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d

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Rubi [A]
time = 0.57, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4179, 4159, 4132, 3856, 2719, 4130, 2720} \begin {gather*} \frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{63 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right )}{21 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{15 d}+\frac {2 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]

[Out]

(2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2
]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*Sqrt[Cos[c +
d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Si
n[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sin[c + d*
x])/(63*d*Sqrt[Sec[c + d*x]]) + (2*(2*A*b + 3*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/
2)) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))

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 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4130

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e
+ f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rule 4132

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 4159

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_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x]
+ Dist[1/(d*n), Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rule 4179

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/(d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*
m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]

Rubi steps

\begin {align*} \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2}{9} \int \frac {(a+b \sec (c+d x))^2 \left (\frac {3}{2} (2 A b+3 a B)+\frac {1}{2} (7 a A+9 b B+9 a C) \sec (c+d x)+\frac {1}{2} b (A+9 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 (2 A b+3 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {4}{63} \int \frac {(a+b \sec (c+d x)) \left (\frac {1}{4} \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right )+\frac {1}{4} \left (86 a A b+45 a^2 B+63 b^2 B+126 a b C\right ) \sec (c+d x)+\frac {1}{4} b (13 A b+9 a B+63 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (2 A b+3 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {8}{315} \int \frac {-\frac {15}{8} \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right )-\frac {21}{8} \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sec (c+d x)-\frac {5}{8} b^2 (13 A b+9 a B+63 b C) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (2 A b+3 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {8}{315} \int \frac {-\frac {15}{8} \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right )-\frac {5}{8} b^2 (13 A b+9 a B+63 b C) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx-\frac {1}{15} \left (-27 a^2 b B-15 b^3 B-9 a b^2 (3 A+5 C)-a^3 (7 A+9 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {2 (2 A b+3 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {1}{21} \left (-5 a^3 B-21 a b^2 B-7 b^3 (A+3 C)-3 a^2 b (5 A+7 C)\right ) \int \sqrt {\sec (c+d x)} \, dx-\frac {1}{15} \left (\left (-27 a^2 b B-15 b^3 B-9 a b^2 (3 A+5 C)-a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {2 (2 A b+3 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {1}{21} \left (\left (-5 a^3 B-21 a b^2 B-7 b^3 (A+3 C)-3 a^2 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {2 (2 A b+3 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\\ \end {align*}

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Mathematica [A]
time = 6.31, size = 323, normalized size = 0.96 \begin {gather*} \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (336 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+240 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \left (7 a \left (108 A b^2+108 a b B+a^2 (43 A+36 C)\right ) \cos (c+d x)+5 \left (84 A b^3+78 a^3 B+252 a b^2 B+6 a^2 (39 A b+42 b C)+18 a^2 (3 A b+a B) \cos (2 (c+d x))+7 a^3 A \cos (3 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{1260 d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {9}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]

[Out]

((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(336*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5
*C) + a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3
*C) + 3*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(7*a*(108*A*b^2 + 108*a*b*B + a^2*
(43*A + 36*C))*Cos[c + d*x] + 5*(84*A*b^3 + 78*a^3*B + 252*a*b^2*B + 6*a^2*(39*A*b + 42*b*C) + 18*a^2*(3*A*b +
a*B)*Cos[2*(c + d*x)] + 7*a^3*A*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d*(b + a*Cos[c + d*x])^3*(A + 2*C
+ 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(974\) vs. \(2(360)=720\).
time = 0.11, size = 975, normalized size = 2.90

method	result	size
default	\(\text {Expression too large to display}\)	\(975\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),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)*(-1120*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^
10*a^3+(2240*A*a^3+2160*A*a^2*b+720*B*a^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-2072*A*a^3-3240*A*a^2*b-1
512*A*a*b^2-1080*B*a^3-1512*B*a^2*b-504*C*a^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(952*A*a^3+2520*A*a^2*b
+1512*A*a*b^2+420*A*b^3+840*B*a^3+1512*B*a^2*b+1260*B*a*b^2+504*C*a^3+1260*C*a^2*b)*sin(1/2*d*x+1/2*c)^4*cos(1
/2*d*x+1/2*c)+(-168*A*a^3-720*A*a^2*b-378*A*a*b^2-210*A*b^3-240*B*a^3-378*B*a^2*b-630*B*a*b^2-126*C*a^3-630*C*
a^2*b)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+225*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)
^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a^2*b+105*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*b^3-147*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a^3-567*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*
c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a*b^2+75*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a^3+315*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a*b^2-567*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1
/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a^2*b-315*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x
+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*b^3+315*C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x
+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a^2*b+315*C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*b^3-189*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a^3-945*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d
*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a*b^2)/(-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

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Maxima [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((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.27, size = 387, normalized size = 1.15 \begin {gather*} -\frac {15 \, \sqrt {2} {\left (5 i \, B a^{3} + 3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 21 i \, B a b^{2} + 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{3} - 3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b - 21 i \, B a b^{2} - 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{3} - 27 i \, B a^{2} b - 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2} - 15 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (i \, {\left (7 \, A + 9 \, C\right )} a^{3} + 27 i \, B a^{2} b + 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2} + 15 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, A a^{3} \cos \left (d x + c\right )^{4} + 45 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{3} + 27 \, B a^{2} b + 27 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, B a^{3} + 3 \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 21 \, B a b^{2} + 7 \, A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

-1/315*(15*sqrt(2)*(5*I*B*a^3 + 3*I*(5*A + 7*C)*a^2*b + 21*I*B*a*b^2 + 7*I*(A + 3*C)*b^3)*weierstrassPInverse(
-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-5*I*B*a^3 - 3*I*(5*A + 7*C)*a^2*b - 21*I*B*a*b^2 - 7*I*(A
+ 3*C)*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(-I*(7*A + 9*C)*a^3 - 27*I
*B*a^2*b - 9*I*(3*A + 5*C)*a*b^2 - 15*I*B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c)
+ I*sin(d*x + c))) + 21*sqrt(2)*(I*(7*A + 9*C)*a^3 + 27*I*B*a^2*b + 9*I*(3*A + 5*C)*a*b^2 + 15*I*B*b^3)*weiers
trassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(35*A*a^3*cos(d*x + c)^4 + 45*
(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^3 + 7*((7*A + 9*C)*a^3 + 27*B*a^2*b + 27*A*a*b^2)*cos(d*x + c)^2 + 15*(5*B*a^
3 + 3*(5*A + 7*C)*a^2*b + 21*B*a*b^2 + 7*A*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3879 deep

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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((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3/sec(d*x + c)^(9/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(9/2),x)

[Out]

int(((a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(9/2), x)

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