3.727 \(\int \sec (c+d x) (a+b \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=454 \[ \frac{2 (a-b) \sqrt{a+b} \left (15 a^2 b (21 A+11 C)+10 a^3 C-6 a b^2 (28 A+19 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{315 b^2 d}-\frac{2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}+\frac{4 a \left (-5 a^2 C+84 A b^2+57 b^2 C\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b d}+\frac{2 (a-b) \sqrt{a+b} \left (-3 a^2 b^2 (161 A+93 C)+10 a^4 C-21 b^4 (9 A+7 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^3 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}-\frac{4 a C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d} \]
[Out]
(2*(a - b)*Sqrt[a + b]*(10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cot[c + d*x]*EllipticE[ArcSi
n[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 +
Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*(a - b)*Sqrt[a + b]*(10*a^3*C + 21*b^3*(9*A + 7*C) + 15*a^2*b*(21*A
+ 11*C) - 6*a*b^2*(28*A + 19*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)
/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (4*a*(84
*A*b^2 - 5*a^2*C + 57*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) - (2*(10*a^2*C - 7*b^2*(9*A + 7*
C))*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) - (4*a*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b
*d) + (2*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)
________________________________________________________________________________________
Rubi [A] time = 1.03208, antiderivative size = 454, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used =
{4083, 4002, 4005, 3832, 4004} \[ -\frac{2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}+\frac{4 a \left (-5 a^2 C+84 A b^2+57 b^2 C\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b d}+\frac{2 (a-b) \sqrt{a+b} \left (15 a^2 b (21 A+11 C)+10 a^3 C-6 a b^2 (28 A+19 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \left (-3 a^2 b^2 (161 A+93 C)+10 a^4 C-21 b^4 (9 A+7 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{315 b^3 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}-\frac{4 a C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d} \]
Antiderivative was successfully verified.
[In]
Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]
[Out]
(2*(a - b)*Sqrt[a + b]*(10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cot[c + d*x]*EllipticE[ArcSi
n[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 +
Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*(a - b)*Sqrt[a + b]*(10*a^3*C + 21*b^3*(9*A + 7*C) + 15*a^2*b*(21*A
+ 11*C) - 6*a*b^2*(28*A + 19*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)
/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (4*a*(84
*A*b^2 - 5*a^2*C + 57*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) - (2*(10*a^2*C - 7*b^2*(9*A + 7*
C))*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) - (4*a*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b
*d) + (2*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)
Rule 4083
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(
m_), x_Symbol] :> -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*(m + 2)),
Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) - a*C*Csc[e + f*x], x], x], x] /; FreeQ
[{a, b, e, f, A, C, m}, x] && !LtQ[m, -1]
Rule 4002
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))
, x_Symbol] :> -Simp[(B*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[Csc[e + f*x
]*(a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1))*Csc[e + f*x], x], x], x] /; Fr
eeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
Rule 4005
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Dist[A - B, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[B, Int[(Csc[e + f*x]*(1 +
Csc[e + f*x]))/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && NeQ[A
^2 - B^2, 0]
Rule 3832
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*Rt[a + b, 2]*Sqr
t[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[e + f*x]))/(a - b))]*EllipticF[ArcSin[Sqrt[a + b*Csc[e +
f*x]]/Rt[a + b, 2]], (a + b)/(a - b)])/(b*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 4004
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[(-2*(A*b - a*B)*Rt[a + (b*B)/A, 2]*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Cs
c[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)]
)/(b^2*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Rubi steps
\begin{align*} \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac{2 \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac{1}{2} b (9 A+7 C)-a C \sec (c+d x)\right ) \, dx}{9 b}\\ &=-\frac{4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac{2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac{4 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac{3}{4} a b (21 A+13 C)-\frac{1}{4} \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) \sec (c+d x)\right ) \, dx}{63 b}\\ &=-\frac{2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac{4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac{2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac{8 \int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \left (\frac{3}{8} b \left (7 b^2 (9 A+7 C)+5 a^2 (21 A+11 C)\right )+\frac{3}{4} a \left (84 A b^2-5 a^2 C+57 b^2 C\right ) \sec (c+d x)\right ) \, dx}{315 b}\\ &=\frac{4 a \left (84 A b^2-5 a^2 C+57 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac{2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac{4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac{2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac{16 \int \frac{\sec (c+d x) \left (\frac{3}{16} a b \left (5 a^2 (63 A+31 C)+3 b^2 (119 A+87 C)\right )-\frac{3}{16} \left (10 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (161 A+93 C)\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{945 b}\\ &=\frac{4 a \left (84 A b^2-5 a^2 C+57 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac{2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac{4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac{2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac{\left ((a-b) \left (10 a^3 C+21 b^3 (9 A+7 C)+15 a^2 b (21 A+11 C)-6 a b^2 (28 A+19 C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b}-\frac{\left (10 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (161 A+93 C)\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{315 b}\\ &=\frac{2 (a-b) \sqrt{a+b} \left (10 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (161 A+93 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left (10 a^3 C+21 b^3 (9 A+7 C)+15 a^2 b (21 A+11 C)-6 a b^2 (28 A+19 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{315 b^2 d}+\frac{4 a \left (84 A b^2-5 a^2 C+57 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac{2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac{4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac{2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}\\ \end{align*}
Mathematica [A] time = 22.2411, size = 710, normalized size = 1.56 \[ \frac{4 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )} \left (\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \left ((a+b) \sec (c+d x) \left (\cos (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right )\right )^{3/2} \sqrt{\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{a+b}} \left (b \left (15 a^2 b (21 A+11 C)-10 a^3 C+6 a b^2 (28 A+19 C)+21 b^3 (9 A+7 C)\right ) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right )+\left (-3 a^2 b^2 (161 A+93 C)+10 a^4 C-21 b^4 (9 A+7 C)\right ) E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )\right )+\left (-3 a^2 b^2 (161 A+93 C)+10 a^4 C-21 b^4 (9 A+7 C)\right ) \cos (c+d x) \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)\right )}{315 b^2 d \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left (\frac{1}{2} (c+d x)\right )^{3/2} \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 C)}+\frac{\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \left (\frac{4 \left (483 a^2 A b^2+279 a^2 b^2 C-10 a^4 C+189 A b^4+147 b^4 C\right ) \sin (c+d x)}{315 b^2}+\frac{4}{315} \sec ^2(c+d x) \left (75 a^2 C \sin (c+d x)+63 A b^2 \sin (c+d x)+49 b^2 C \sin (c+d x)\right )+\frac{4 \sec (c+d x) \left (5 a^3 C \sin (c+d x)+231 a A b^2 \sin (c+d x)+163 a b^2 C \sin (c+d x)\right )}{315 b}+\frac{76}{63} a b C \tan (c+d x) \sec ^2(c+d x)+\frac{4}{9} b^2 C \tan (c+d x) \sec ^3(c+d x)\right )}{d (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]
[Out]
(4*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*S
ec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((a + b)*((10*a^4*C - 21*b^4*(9*A + 7*C)
- 3*a^2*b^2*(161*A + 93*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-10*a^3*C + 21*b^3*(9*A
+ 7*C) + 15*a^2*b*(21*A + 11*C) + 6*a*b^2*(28*A + 19*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])
*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]
+ (10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2
]^4*Tan[(c + d*x)/2]))/(315*b^2*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c +
2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(9/2)) + (Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec
[c + d*x]^2)*((4*(483*a^2*A*b^2 + 189*A*b^4 - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^2) +
(4*Sec[c + d*x]^2*(63*A*b^2*Sin[c + d*x] + 75*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/315 + (4*Sec[c + d*
x]*(231*a*A*b^2*Sin[c + d*x] + 5*a^3*C*Sin[c + d*x] + 163*a*b^2*C*Sin[c + d*x]))/(315*b) + (76*a*b*C*Sec[c + d
*x]^2*Tan[c + d*x])/63 + (4*b^2*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[
2*c + 2*d*x]))
________________________________________________________________________________________
Maple [B] time = 1.669, size = 4333, normalized size = 9.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(d*x+c)*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x)
[Out]
2/315/d/b^2*(cos(d*x+c)+1)^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))^2*(-315*A*sin(d*x+c)*cos(d*x+
c)^4*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c
))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2-155*C*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/
(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2
-315*A*sin(d*x+c)*cos(d*x+c)^5*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/
2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2-189*A*cos(d*x+c)^5*b^5-105*A*cos(d*x+c)^5
*a*b^4+10*C*cos(d*x+c)^5*a^4*b+199*C*cos(d*x+c)^5*a^3*b^2-279*C*cos(d*x+c)^5*a^2*b^3-65*C*cos(d*x+c)^5*a*b^4+7
14*A*cos(d*x+c)^4*a^2*b^3-5*C*cos(d*x+c)^4*a^4*b+272*C*cos(d*x+c)^4*a^2*b^3+294*A*cos(d*x+c)^3*a*b^4+80*C*cos(
d*x+c)^3*a^3*b^2+82*C*cos(d*x+c)^3*a*b^4-483*A*cos(d*x+c)^6*a^3*b^2-231*A*cos(d*x+c)^6*a^2*b^3-189*A*cos(d*x+c
)^6*a*b^4-5*C*cos(d*x+c)^6*a^4*b-279*C*cos(d*x+c)^6*a^3*b^2-163*C*cos(d*x+c)^6*a^2*b^3-147*C*cos(d*x+c)^6*a*b^
4+170*C*cos(d*x+c)^2*a^2*b^3+130*C*cos(d*x+c)*a*b^4+483*A*cos(d*x+c)^5*a^3*b^2-483*A*cos(d*x+c)^5*a^2*b^3-10*C
*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ell
ipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^5+147*C*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+
c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b)
)^(1/2))*b^5-189*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*
x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^5+189*A*cos(d*x+c)^5*sin(d*x+c)*(co
s(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d
*x+c),((a-b)/(a+b))^(1/2))*b^5-147*C*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*c
os(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^5-10*C*cos(d*x+c)
^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+
cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^5+147*C*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2
)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^
5-189*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1
/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^5+189*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(c
os(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b
)/(a+b))^(1/2))*b^5-147*C*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/
(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b^5-10*C*cos(d*x+c)^5*a^5-147*
C*cos(d*x+c)^5*b^5+126*A*cos(d*x+c)^4*b^5+98*C*cos(d*x+c)^4*b^5+63*A*cos(d*x+c)^2*b^5+14*C*cos(d*x+c)^2*b^5+10
*C*cos(d*x+c)^6*a^5-279*C*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/
(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3-261*C*cos(d*x+c)^5*sin
(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*
x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4-10*C*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/
(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b+2
79*C*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)
*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2+279*C*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(
cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-
b)/(a+b))^(1/2))*a^2*b^3+147*C*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x
+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4-483*A*cos(d*x+c)^4*
sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos
(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3-357*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/
2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a
*b^4+483*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))
^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2+483*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*
x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c
),((a-b)/(a+b))^(1/2))*a^2*b^3+189*A*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*c
os(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4+10*C*cos(d*x+
c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-
1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b-155*C*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^
(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2)
)*a^3*b^2-279*C*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c
)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3-261*C*cos(d*x+c)^4*sin(d*x+c)*(c
os(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(
d*x+c),((a-b)/(a+b))^(1/2))*a*b^4-10*C*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a
*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b+279*C*cos(d
*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE
((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2+279*C*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)
+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^
(1/2))*a^2*b^3+147*C*cos(d*x+c)^4*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(
d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4-483*A*cos(d*x+c)^5*sin(d*x+c)
*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/s
in(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^3-357*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b
)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4+483*A
*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*Ell
ipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b^2+483*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(
d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(
a+b))^(1/2))*a^2*b^3+189*A*cos(d*x+c)^5*sin(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))
/(cos(d*x+c)+1))^(1/2)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^4+10*C*cos(d*x+c)^5*sin(d
*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF((-1+cos(d*x+
c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4*b+35*C*b^5)/(b+a*cos(d*x+c))/cos(d*x+c)^4/sin(d*x+c)^5
________________________________________________________________________________________
Maxima [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(sec(d*x+c)*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="maxima")
[Out]
Timed out
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C b^{2} \sec \left (d x + c\right )^{5} + 2 \, C a b \sec \left (d x + c\right )^{4} + 2 \, A a b \sec \left (d x + c\right )^{2} + A a^{2} \sec \left (d x + c\right ) +{\left (C a^{2} + A b^{2}\right )} \sec \left (d x + c\right )^{3}\right )} \sqrt{b \sec \left (d x + c\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)*(a+b*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
integral((C*b^2*sec(d*x + c)^5 + 2*C*a*b*sec(d*x + c)^4 + 2*A*a*b*sec(d*x + c)^2 + A*a^2*sec(d*x + c) + (C*a^2
+ A*b^2)*sec(d*x + c)^3)*sqrt(b*sec(d*x + c) + a), x)
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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(sec(d*x+c)*(a+b*sec(d*x+c))**(5/2)*(A+C*sec(d*x+c)**2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)*(a+b*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)*(b*sec(d*x + c) + a)^(5/2)*sec(d*x + c), x)