3.831 \(\int (a+b \sec (c+d x))^{5/2} (B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=384 \[ \frac{2 (a-b) \sqrt{a+b} \left (15 a^2 (7 B-C)-8 a b (7 B-15 C)+b^2 (63 B-25 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{105 b d}+\frac{2 \left (15 a^2 C+56 a b B+25 b^2 C\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 d}-\frac{2 (a-b) \sqrt{a+b} \left (161 a^2 b B+15 a^3 C+145 a b^2 C+63 b^3 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{105 b^2 d}+\frac{2 (5 a C+7 b B) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d} \]

[Out]

(-2*(a - b)*Sqrt[a + b]*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Cot[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))])/(105*b^2*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*B - 25*C) - 8*a*b*(7*B - 15*C) + 15*a^2*(7*B -
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))])/(105*b*d) + (2*(56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqr
t[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*
d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)

________________________________________________________________________________________

Rubi [A]  time = 0.677614, antiderivative size = 384, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {4056, 4058, 12, 3832, 4004} \[ \frac{2 \left (15 a^2 C+56 a b B+25 b^2 C\right ) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 d}+\frac{2 (a-b) \sqrt{a+b} \left (15 a^2 (7 B-C)-8 a b (7 B-15 C)+b^2 (63 B-25 C)\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{105 b d}-\frac{2 (a-b) \sqrt{a+b} \left (161 a^2 b B+15 a^3 C+145 a b^2 C+63 b^3 B\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{105 b^2 d}+\frac{2 (5 a C+7 b B) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(-2*(a - b)*Sqrt[a + b]*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Cot[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))])/(105*b^2*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*B - 25*C) - 8*a*b*(7*B - 15*C) + 15*a^2*(7*B -
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))])/(105*b*d) + (2*(56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqr
t[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*
d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)

Rule 4056

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

Rule 4058

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_
.) + (a_)], x_Symbol] :> Int[(A + (B - C)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Dist[C, 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, C}, x] && NeQ[a^2 - b^2, 0
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{2}{7} \int (a+b \sec (c+d x))^{3/2} \left (\frac{1}{2} (7 a B+5 b C) \sec (c+d x)+\frac{1}{2} (7 b B+5 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 (7 b B+5 a C) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{4}{35} \int \sqrt{a+b \sec (c+d x)} \left (\frac{1}{4} \left (35 a^2 B+21 b^2 B+40 a b C\right ) \sec (c+d x)+\frac{1}{4} \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{2 \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{105 d}+\frac{2 (7 b B+5 a C) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{8}{105} \int \frac{\frac{1}{8} \left (105 a^3 B+119 a b^2 B+135 a^2 b C+25 b^3 C\right ) \sec (c+d x)+\frac{1}{8} \left (161 a^2 b B+63 b^3 B+15 a^3 C+145 a b^2 C\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{2 \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{105 d}+\frac{2 (7 b B+5 a C) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{8}{105} \int \frac{\left (\frac{1}{8} \left (-161 a^2 b B-63 b^3 B-15 a^3 C-145 a b^2 C\right )+\frac{1}{8} \left (105 a^3 B+119 a b^2 B+135 a^2 b C+25 b^3 C\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx+\frac{1}{105} \left (161 a^2 b B+63 b^3 B+15 a^3 C+145 a b^2 C\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=-\frac{2 (a-b) \sqrt{a+b} \left (161 a^2 b B+63 b^3 B+15 a^3 C+145 a b^2 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}}}{105 b^2 d}+\frac{2 \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{105 d}+\frac{2 (7 b B+5 a C) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac{1}{105} \left ((a-b) \left (b^2 (63 B-25 C)-8 a b (7 B-15 C)+15 a^2 (7 B-C)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=-\frac{2 (a-b) \sqrt{a+b} \left (161 a^2 b B+63 b^3 B+15 a^3 C+145 a b^2 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}}}{105 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \left (b^2 (63 B-25 C)-8 a b (7 B-15 C)+15 a^2 (7 B-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}}}{105 b d}+\frac{2 \left (56 a b B+15 a^2 C+25 b^2 C\right ) \sqrt{a+b \sec (c+d x)} \tan (c+d x)}{105 d}+\frac{2 (7 b B+5 a C) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}\\ \end{align*}

Mathematica [B]  time = 22.9373, size = 2913, normalized size = 7.59 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Sin[c + d*x])
/(105*b) + (2*Sec[c + d*x]^2*(7*b^2*B*Sin[c + d*x] + 15*a*b*C*Sin[c + d*x]))/35 + (2*Sec[c + d*x]*(77*a*b*B*Si
n[c + d*x] + 45*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/105 + (2*b^2*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(
d*(b + a*Cos[c + d*x])^2) + (2*((-23*a^2*b*B)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*b^3*B)/(5*
Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^3*C)/(7*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (29*a
*b^2*C)/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*B*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c +
d*x]]) + (8*a*b^2*B*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (a^4*C*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[
b + a*Cos[c + d*x]]) - (2*a^2*b*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (5*b^3*C*Sqrt[Sec[c + d*
x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (23*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x
]]) - (3*a*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (a^4*C*Cos[2*(c + d*x)]*S
qrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (29*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[
b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*((-2*(Cos[c + d*x]/(1 +
Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (
a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)
/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*b*B +
63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)))/(105*b*d*(b + a*Cos[c + d*x])
^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*(-(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((-2*(C
os[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[
(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcS
in[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]
+ (161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)))/(105*b*Sqrt[b
+ a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]
*Tan[(c + d*x)/2]*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2
*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*
B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a +
b)*(1 + Cos[c + d*x]))] + (161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d
*x)/2]^2)))/(105*b*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[b + a*Cos[c + d*x]]*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x])
)^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)
] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/
(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*b*B + 63*b^3*B + 15
*a^3*C + 145*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^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]))/(105*b*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^
2*Sec[c + d*x]]) + (2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-3*Sqrt[Cos[c + d*x]/(1
+ Cos[c + d*x])]*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a -
b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]]
, (a - b)/(a + b)])*Sec[c + d*x]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c +
d*x])))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + ((Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((
161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*
a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*
Sec[c + d*x]*(-((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)))/((b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x])))^(3/2) + (161*a^2*b*B + 63*b^3*B +
15*a^3*C + 145*a*b^2*C)*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + ((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^
2*C)*Sec[(c + d*x)/2]^2*(-1 + Tan[(c + d*x)/2]^2))/2 - (2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*Sec[c + d*x]
*(-(b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x
)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*S
ec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2])))/Sqrt[(b +
a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63
*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) +
8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x]*Tan
[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]))/(105*b*Sqrt[Sec[(c + d*x)/2]^2])))

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Maple [B]  time = 1.117, size = 3637, 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((a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

-2/105/d/b*(cos(d*x+c)+1)^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))^2*(63*B*cos(d*x+c)^4*b^4+105*B
*cos(d*x+c)^3*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
ipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^3*b+105*B*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-42*B*cos(d*x+c)^3*b^4-21*B*cos(d*x+c)*b^4-161*B*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)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*si
n(d*x+c)*a^3*b+15*C*cos(d*x+c)^5*a^4-15*C*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)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^4+25*C*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))*b^4-15*C*sin(d*x+c)*cos(d*x+c)^3*(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+25*C*sin(d*x+c)*cos(d*x+c)^3*(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^4-238*B*cos(d*x+c)^3*a^2*b^2-98*B*cos(d*x+c
)^2*a*b^3-63*B*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)*
EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*b^4+63*B*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))*sin(d*x+c)*b^4-63*B*cos(d*x+c)^3*(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))*sin(d*x+c)*b^4+63*B*cos(d*x+c)^3*(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)*EllipticF((-1+cos(d*x+c))/sin(d
*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*b^4-161*B*cos(d*x+c)^4*(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))*sin(d*x+c)*a^2*b^2-
63*B*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)*EllipticE(
(-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a*b^3+161*B*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))*sin(d*x+c)*a^2*b^2+119*B*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))*sin(d*x+c)*a*b^3-161*B*cos(d*x+c)^3*(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)*EllipticE((-1+cos(d*x+c))/sin(
d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a^3*b-161*B*cos(d*x+c)^3*(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))*sin(d*x+c)*a^2*b
^2-63*B*cos(d*x+c)^3*(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)*Ellipti
cE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*sin(d*x+c)*a*b^3+161*B*cos(d*x+c)^3*(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))*sin(d*x+c)*a^2*b^2+119*B*cos(d*x+c)^3*(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))*sin(d*x+c)*a*b^3-161*B*cos(d*x+c)^4
*a^3*b+161*B*cos(d*x+c)^4*a^2*b^2+35*B*cos(d*x+c)^4*a*b^3+161*B*cos(d*x+c)^5*a^3*b+77*B*cos(d*x+c)^5*a^2*b^2+6
3*B*cos(d*x+c)^5*a*b^3+15*C*cos(d*x+c)^4*a^3*b-55*C*cos(d*x+c)^4*a^2*b^2+145*C*cos(d*x+c)^4*a*b^3-60*C*cos(d*x
+c)^3*a^3*b-110*C*cos(d*x+c)^3*a*b^3-90*C*cos(d*x+c)^2*a^2*b^2-60*C*cos(d*x+c)*a*b^3+45*C*cos(d*x+c)^5*a^3*b+1
45*C*cos(d*x+c)^5*a^2*b^2+25*C*cos(d*x+c)^5*a*b^3-15*C*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)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a
^3*b-145*C*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)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a^2*b^2-145*C*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)*EllipticE((-1+cos(d*x+c))/sin(d*x+c
),((a-b)/(a+b))^(1/2))*a*b^3+15*C*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+135*C*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^2*b^2+145*C*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*b^3-15*C*sin(d*x+c)*cos(d*x+c)^3*(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-145*C*sin(d*x+c)*cos(d*x+c)^3*(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^2-145*C*sin(d*x+c)*cos(d*x+c)^3*(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^3+15*C*sin(d*x+
c)*cos(d*x+c)^3*(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+135*C*sin(d*x+c)*cos(d*x+c)^3*(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^2+145*C*sin(d*x+c)*cos(d*x+c)^3*(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^3-15*C*cos(d*x+c)^4*a^4-10*C*cos(d*x
+c)^2*b^4+25*C*cos(d*x+c)^4*b^4-15*C*b^4)/(b+a*cos(d*x+c))/cos(d*x+c)^3/sin(d*x+c)^5

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(5/2)*(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)^(5/2), x)