3.821 \(\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} (B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=573 \[ -\frac{2 (a-b) \sqrt{a+b} \left (6 a^2 b^2 (11 B-24 C)+4 a^3 b (22 B-9 C)-48 a^4 C+3 a b^3 (143 B-471 C)-3 b^4 (539 B-225 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{3465 b^4 d}-\frac{2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^3 d}+\frac{2 \left (88 a^2 b B-48 a^3 C-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{3465 b^3 d}+\frac{2 \left (-144 a^2 b^2 C+88 a^3 b B-48 a^4 C+429 a b^3 B+675 b^4 C\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3465 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left (363 a^2 b^3 B-108 a^3 b^2 C+88 a^4 b B-48 a^5 C+2088 a b^4 C+1617 b^5 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3465 b^5 d}+\frac{2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{99 b^2 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d} \]

[Out]

(-2*(a - b)*Sqrt[a + b]*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 108*a^3*b^2*C + 2088*a*b^4*C)*Co
t[c + d*x]*EllipticE[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))])/(3465*b^5*d) - (2*(a - b)*Sqrt[a + b]*(3*a*b^3*(143*B - 47
1*C) - 3*b^4*(539*B - 225*C) + 6*a^2*b^2*(11*B - 24*C) + 4*a^3*b*(22*B - 9*C) - 48*a^4*C)*Cot[c + d*x]*Ellipti
cF[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))])/(3465*b^4*d) + (2*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 144*a^2*b^2*C + 675
*b^4*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^3*d) + (2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 204*a*b^
2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^3*d) - (2*(44*a*b*B - 24*a^2*C - 81*b^2*C)*(a + b*Sec[c
+ d*x])^(5/2)*Tan[c + d*x])/(693*b^3*d) + (2*(11*b*B - 6*a*C)*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c +
d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(11*b*d)

________________________________________________________________________________________

Rubi [A]  time = 1.98108, antiderivative size = 573, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {4072, 4033, 4092, 4082, 4002, 4005, 3832, 4004} \[ -\frac{2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^3 d}+\frac{2 \left (88 a^2 b B-48 a^3 C-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{3465 b^3 d}+\frac{2 \left (-144 a^2 b^2 C+88 a^3 b B-48 a^4 C+429 a b^3 B+675 b^4 C\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3465 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left (6 a^2 b^2 (11 B-24 C)+4 a^3 b (22 B-9 C)-48 a^4 C+3 a b^3 (143 B-471 C)-3 b^4 (539 B-225 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3465 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left (363 a^2 b^3 B-108 a^3 b^2 C+88 a^4 b B-48 a^5 C+2088 a b^4 C+1617 b^5 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3465 b^5 d}+\frac{2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{99 b^2 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a - b)*Sqrt[a + b]*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 108*a^3*b^2*C + 2088*a*b^4*C)*Co
t[c + d*x]*EllipticE[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))])/(3465*b^5*d) - (2*(a - b)*Sqrt[a + b]*(3*a*b^3*(143*B - 47
1*C) - 3*b^4*(539*B - 225*C) + 6*a^2*b^2*(11*B - 24*C) + 4*a^3*b*(22*B - 9*C) - 48*a^4*C)*Cot[c + d*x]*Ellipti
cF[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))])/(3465*b^4*d) + (2*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 144*a^2*b^2*C + 675
*b^4*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^3*d) + (2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 204*a*b^
2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^3*d) - (2*(44*a*b*B - 24*a^2*C - 81*b^2*C)*(a + b*Sec[c
+ d*x])^(5/2)*Tan[c + d*x])/(693*b^3*d) + (2*(11*b*B - 6*a*C)*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c +
d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(11*b*d)

Rule 4072

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

Rule 4033

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

Rule 4092

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(
e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(C*Csc[e + f*x]*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m
+ 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2)
+ A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m
}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 4082

Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + 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) + (b*B*
(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, 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 ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^4(c+d x) (a+b \sec (c+d x))^{3/2} (B+C \sec (c+d x)) \, dx\\ &=\frac{2 C \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{11 b d}+\frac{2 \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (2 a C+\frac{9}{2} b C \sec (c+d x)+\frac{1}{2} (11 b B-6 a C) \sec ^2(c+d x)\right ) \, dx}{11 b}\\ &=\frac{2 (11 b B-6 a C) \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{99 b^2 d}+\frac{2 C \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{11 b d}+\frac{4 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac{1}{2} a (11 b B-6 a C)+\frac{1}{4} b (77 b B-6 a C) \sec (c+d x)-\frac{1}{4} \left (44 a b B-24 a^2 C-81 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx}{99 b^2}\\ &=-\frac{2 \left (44 a b B-24 a^2 C-81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{99 b^2 d}+\frac{2 C \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{11 b d}+\frac{8 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (-\frac{3}{8} b \left (22 a b B-12 a^2 C-135 b^2 C\right )+\frac{1}{8} \left (88 a^2 b B+539 b^3 B-48 a^3 C-204 a b^2 C\right ) \sec (c+d x)\right ) \, dx}{693 b^3}\\ &=\frac{2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-204 a b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^3 d}-\frac{2 \left (44 a b B-24 a^2 C-81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{99 b^2 d}+\frac{2 C \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{11 b d}+\frac{16 \int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \left (-\frac{3}{16} b \left (22 a^2 b B-539 b^3 B-12 a^3 C-471 a b^2 C\right )+\frac{3}{16} \left (88 a^3 b B+429 a b^3 B-48 a^4 C-144 a^2 b^2 C+675 b^4 C\right ) \sec (c+d x)\right ) \, dx}{3465 b^3}\\ &=\frac{2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-144 a^2 b^2 C+675 b^4 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^3 d}+\frac{2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-204 a b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^3 d}-\frac{2 \left (44 a b B-24 a^2 C-81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{99 b^2 d}+\frac{2 C \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{11 b d}+\frac{32 \int \frac{\sec (c+d x) \left (\frac{3}{32} b \left (22 a^3 b B+2046 a b^3 B-12 a^4 C+1269 a^2 b^2 C+675 b^4 C\right )+\frac{3}{32} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-108 a^3 b^2 C+2088 a b^4 C\right ) \sec (c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{10395 b^3}\\ &=\frac{2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-144 a^2 b^2 C+675 b^4 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^3 d}+\frac{2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-204 a b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^3 d}-\frac{2 \left (44 a b B-24 a^2 C-81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{99 b^2 d}+\frac{2 C \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{11 b d}+\frac{\left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-108 a^3 b^2 C+2088 a b^4 C\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{3465 b^3}+\frac{\left (32 \left (\frac{3}{32} b \left (22 a^3 b B+2046 a b^3 B-12 a^4 C+1269 a^2 b^2 C+675 b^4 C\right )-\frac{3}{32} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-108 a^3 b^2 C+2088 a b^4 C\right )\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{10395 b^3}\\ &=-\frac{2 (a-b) \sqrt{a+b} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-108 a^3 b^2 C+2088 a b^4 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}}}{3465 b^5 d}-\frac{2 (a-b) \sqrt{a+b} \left (3 a b^3 (143 B-471 C)-3 b^4 (539 B-225 C)+a^3 b (88 B-36 C)+6 a^2 b^2 (11 B-24 C)-48 a^4 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}}}{3465 b^4 d}+\frac{2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-144 a^2 b^2 C+675 b^4 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{3465 b^3 d}+\frac{2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-204 a b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^3 d}-\frac{2 \left (44 a b B-24 a^2 C-81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^3 d}+\frac{2 (11 b B-6 a C) \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{99 b^2 d}+\frac{2 C \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{11 b d}\\ \end{align*}

Mathematica [B]  time = 26.5058, size = 4220, normalized size = 7.36 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((-2*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b
^2*C - 2088*a*b^4*C)*Sin[c + d*x])/(3465*b^4) + (2*Sec[c + d*x]^4*(11*b*B*Sin[c + d*x] + 12*a*C*Sin[c + d*x]))
/99 + (2*Sec[c + d*x]^3*(110*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 81*b^2*C*Sin[c + d*x]))/(693*b) + (2*
Sec[c + d*x]^2*(33*a^2*b*B*Sin[c + d*x] + 539*b^3*B*Sin[c + d*x] - 18*a^3*C*Sin[c + d*x] + 606*a*b^2*C*Sin[c +
d*x]))/(3465*b^2) + (2*Sec[c + d*x]*(-44*a^3*b*B*Sin[c + d*x] + 968*a*b^3*B*Sin[c + d*x] + 24*a^4*C*Sin[c + d
*x] + 57*a^2*b^2*C*Sin[c + d*x] + 675*b^4*C*Sin[c + d*x]))/(3465*b^3) + (2*b*C*Sec[c + d*x]^4*Tan[c + d*x])/11
))/(d*(b + a*Cos[c + d*x])) + (2*((-11*a^2*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*B)/(3
15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b^2*B)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]
]) + (16*a^5*C)/(1155*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (12*a^3*C)/(385*b*Sqrt[b + a*Cos[c +
d*x]]*Sqrt[Sec[c + d*x]]) - (232*a*b*C)/(385*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^5*B*Sqrt[Sec[
c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (31*a^3*B*Sqrt[Sec[c + d*x]])/(315*b*Sqrt[b + a*Cos[c + d*x]])
+ (13*a*b*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (13*a^2*C*Sqrt[Sec[c + d*x]])/(55*Sqrt[b + a
*Cos[c + d*x]]) + (16*a^6*C*Sqrt[Sec[c + d*x]])/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]) + (32*a^4*C*Sqrt[Sec[c + d
*x]])/(1155*b^2*Sqrt[b + a*Cos[c + d*x]]) + (15*b^2*C*Sqrt[Sec[c + d*x]])/(77*Sqrt[b + a*Cos[c + d*x]]) - (8*a
^5*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (11*a^3*B*Cos[2*(c + d*x)]*Sqrt
[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) - (7*a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b +
a*Cos[c + d*x]]) - (232*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(385*Sqrt[b + a*Cos[c + d*x]]) + (16*a^6*C*
Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]) + (12*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec
[c + d*x]])/(385*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/
2)*(2*(a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Sqrt[Cos[c
+ d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d
*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B + 9*C) - 6*a^2*b^2*(11*B + 24*C) + 3*b^4*(5
39*B + 225*C) + 3*a*b^3*(143*B + 471*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a +
b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-88*a^4*b*B - 363*a^2*b^3*B -
1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Ta
n[(c + d*x)/2]))/(3465*b^4*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*((a*Sqrt[Cos[(
c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108
*a^3*b^2*C - 2088*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c
+ d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B + 9*C
) - 6*a^2*b^2*(11*B + 24*C) + 3*b^4*(539*B + 225*C) + 3*a*b^3*(143*B + 471*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c +
d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a
+ b)] + (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Cos[c + d*x]*(b +
a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*b^4*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)
/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 16
17*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c +
d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-48*a^
4*C + 4*a^3*b*(22*B + 9*C) - 6*a^2*b^2*(11*B + 24*C) + 3*b^4*(539*B + 225*C) + 3*a*b^3*(143*B + 471*C))*Sqrt[C
os[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(
c + d*x)/2]], (a - b)/(a + b)] + (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a
*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*b^4*Sqrt[b + a*Cos[c + d
*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-88*a^4*b*B - 363*a^2*b^3*B - 161
7*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (
(a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Sqrt[(b + a*Cos[c
+ d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin
[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b
*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B + 9*C) - 6*a^2*b^2*(11*B + 24*C) + 3*b^4*(539*B + 225*C) + 3*a*b^3*(143*B
+ 471*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/
(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*
x]/(1 + Cos[c + d*x])] + ((a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*
a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[
c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))
/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B + 9*C) - 6*a^
2*b^2*(11*B + 24*C) + 3*b^4*(539*B + 225*C) + 3*a*b^3*(143*B + 471*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*E
llipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b +
a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c +
d*x]))] - a*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Cos[c + d*x]
*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108
*a^3*b^2*C - 2088*a*b^4*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-88*a^4*b*
B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Se
c[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B + 9*C) - 6*a^2*b^2*(11*B + 24*C) +
3*b^4*(539*B + 225*C) + 3*a*b^3*(143*B + 471*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*
x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c +
d*x)/2]^2)/(a + b)]) + ((a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a
*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c +
d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3465*b^4*Sqrt[b + a
*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C +
108*a^3*b^2*C - 2088*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Co
s[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B +
9*C) - 6*a^2*b^2*(11*B + 24*C) + 3*b^4*(539*B + 225*C) + 3*a*b^3*(143*B + 471*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c
+ d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/
(a + b)] + (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Cos[c + d*x]*(
b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) +
Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3465*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqr
t[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))

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Maple [B]  time = 2.543, size = 5368, normalized size = 9.4 \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)^3*(a+b*sec(d*x+c))^(3/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

result too large to display

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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)^3*(a+b*sec(d*x+c))^(3/2)*(B*sec(d*x+c)+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 \sec \left (d x + c\right )^{6} + B a \sec \left (d x + c\right )^{4} +{\left (C a + B b\right )} \sec \left (d x + c\right )^{5}\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)^3*(a+b*sec(d*x+c))^(3/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*b*sec(d*x + c)^6 + B*a*sec(d*x + c)^4 + (C*a + B*b)*sec(d*x + c)^5)*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)**3*(a+b*sec(d*x+c))**(3/2)*(B*sec(d*x+c)+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} + B \sec \left (d x + c\right )\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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