3.384 \(\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=630 \[ \frac{\left (-6 a^2 b (A+5 B)+4 a^3 (4 A+3 B)+5 a b^2 (7 A-18 B)+105 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{24 a^4 d \sqrt{a+b}}+\frac{\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{24 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (41 a^2 A b^2+16 a^4 A-42 a^3 b B+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{24 a^4 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (41 a^2 A b^2+16 a^4 A-42 a^3 b B+90 a b^3 B-105 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{24 a^4 b d \sqrt{a+b}}+\frac{\sqrt{a+b} \left (12 a^2 A b-8 a^3 B-30 a b^2 B+35 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{8 a^5 d}-\frac{(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{12 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}} \]

[Out]

((16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*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))])/(24*a^4*b*Sqrt[a + b]*d) + ((105*A*b^3 + 5*a*b^2*(7*A - 18*B) + 4*a^3*(4*A + 3*B) - 6*a^2*b*(A + 5*B)
)*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))])/(24*a^4*Sqrt[a + b]*d) + (Sqrt[a + b]*(12*a^2*A*b + 35
*A*b^3 - 8*a^3*B - 30*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, 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))])/(8*a^5*d) + (
(16*a^2*A + 35*A*b^2 - 30*a*b*B)*Sin[c + d*x])/(24*a^3*d*Sqrt[a + b*Sec[c + d*x]]) - ((7*A*b - 6*a*B)*Cos[c +
d*x]*Sin[c + d*x])/(12*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*Sqrt[a + b*Sec
[c + d*x]]) + (b*(16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*Tan[c + d*x])/(24*a^4*(a^2 -
b^2)*d*Sqrt[a + b*Sec[c + d*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.66949, antiderivative size = 630, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {4034, 4104, 4060, 4058, 3921, 3784, 3832, 4004} \[ \frac{\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{24 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (41 a^2 A b^2+16 a^4 A-42 a^3 b B+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{24 a^4 d \left (a^2-b^2\right ) \sqrt{a+b \sec (c+d x)}}+\frac{\left (-6 a^2 b (A+5 B)+4 a^3 (4 A+3 B)+5 a b^2 (7 A-18 B)+105 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{24 a^4 d \sqrt{a+b}}+\frac{\left (41 a^2 A b^2+16 a^4 A-42 a^3 b B+90 a b^3 B-105 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{24 a^4 b d \sqrt{a+b}}+\frac{\sqrt{a+b} \left (12 a^2 A b-8 a^3 B-30 a b^2 B+35 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{8 a^5 d}-\frac{(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{12 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*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))])/(24*a^4*b*Sqrt[a + b]*d) + ((105*A*b^3 + 5*a*b^2*(7*A - 18*B) + 4*a^3*(4*A + 3*B) - 6*a^2*b*(A + 5*B)
)*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))])/(24*a^4*Sqrt[a + b]*d) + (Sqrt[a + b]*(12*a^2*A*b + 35
*A*b^3 - 8*a^3*B - 30*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, 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))])/(8*a^5*d) + (
(16*a^2*A + 35*A*b^2 - 30*a*b*B)*Sin[c + d*x])/(24*a^3*d*Sqrt[a + b*Sec[c + d*x]]) - ((7*A*b - 6*a*B)*Cos[c +
d*x]*Sin[c + d*x])/(12*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*Sqrt[a + b*Sec
[c + d*x]]) + (b*(16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*Tan[c + d*x])/(24*a^4*(a^2 -
b^2)*d*Sqrt[a + b*Sec[c + d*x]])

Rule 4034

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

Rule 4104

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

Rule 4060

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

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 3921

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c, In
t[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 3784

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(2*Rt[a + b, 2]*Sqrt[(b*(1 - Csc[c + d*x])
)/(a + b)]*Sqrt[-((b*(1 + Csc[c + d*x]))/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[c + d*x]]/Rt[a
+ b, 2]], (a + b)/(a - b)])/(a*d*Cot[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && 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 \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx &=\frac{A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}}-\frac{\int \frac{\cos ^2(c+d x) \left (\frac{1}{2} (7 A b-6 a B)-2 a A \sec (c+d x)-\frac{5}{2} A b \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx}{3 a}\\ &=-\frac{(7 A b-6 a B) \cos (c+d x) \sin (c+d x)}{12 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}}+\frac{\int \frac{\cos (c+d x) \left (\frac{1}{4} \left (16 a^2 A+5 b (7 A b-6 a B)\right )+\frac{3}{2} a (A b+2 a B) \sec (c+d x)-\frac{3}{4} b (7 A b-6 a B) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx}{6 a^2}\\ &=\frac{\left (16 a^2 A+35 A b^2-30 a b B\right ) \sin (c+d x)}{24 a^3 d \sqrt{a+b \sec (c+d x)}}-\frac{(7 A b-6 a B) \cos (c+d x) \sin (c+d x)}{12 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}}-\frac{\int \frac{\frac{3}{8} \left (12 a^2 A b+35 A b^3-8 a^3 B-30 a b^2 B\right )+\frac{3}{4} a b (7 A b-6 a B) \sec (c+d x)-\frac{1}{8} b \left (16 a^2 A+35 A b^2-30 a b B\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx}{6 a^3}\\ &=\frac{\left (16 a^2 A+35 A b^2-30 a b B\right ) \sin (c+d x)}{24 a^3 d \sqrt{a+b \sec (c+d x)}}-\frac{(7 A b-6 a B) \cos (c+d x) \sin (c+d x)}{12 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\right ) \tan (c+d x)}{24 a^4 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{\int \frac{-\frac{3}{16} \left (a^2-b^2\right ) \left (12 a^2 A b+35 A b^3-8 a^3 B-30 a b^2 B\right )-\frac{1}{8} a b \left (11 a^2 A b-35 A b^3-6 a^3 B+30 a b^2 B\right ) \sec (c+d x)-\frac{1}{16} b \left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )}\\ &=\frac{\left (16 a^2 A+35 A b^2-30 a b B\right ) \sin (c+d x)}{24 a^3 d \sqrt{a+b \sec (c+d x)}}-\frac{(7 A b-6 a B) \cos (c+d x) \sin (c+d x)}{12 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\right ) \tan (c+d x)}{24 a^4 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}+\frac{\int \frac{-\frac{3}{16} \left (a^2-b^2\right ) \left (12 a^2 A b+35 A b^3-8 a^3 B-30 a b^2 B\right )+\left (-\frac{1}{8} a b \left (11 a^2 A b-35 A b^3-6 a^3 B+30 a b^2 B\right )+\frac{1}{16} b \left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )}-\frac{\left (b \left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\right )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{48 a^4 \left (a^2-b^2\right )}\\ &=\frac{\left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\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}}}{24 a^4 b \sqrt{a+b} d}+\frac{\left (16 a^2 A+35 A b^2-30 a b B\right ) \sin (c+d x)}{24 a^3 d \sqrt{a+b \sec (c+d x)}}-\frac{(7 A b-6 a B) \cos (c+d x) \sin (c+d x)}{12 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\right ) \tan (c+d x)}{24 a^4 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}-\frac{\left (12 a^2 A b+35 A b^3-8 a^3 B-30 a b^2 B\right ) \int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx}{16 a^4}+\frac{\left (b \left (105 A b^3+5 a b^2 (7 A-18 B)+4 a^3 (4 A+3 B)-6 a^2 b (A+5 B)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{48 a^4 (a+b)}\\ &=\frac{\left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\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}}}{24 a^4 b \sqrt{a+b} d}+\frac{\left (105 A b^3+5 a b^2 (7 A-18 B)+4 a^3 (4 A+3 B)-6 a^2 b (A+5 B)\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}}}{24 a^4 \sqrt{a+b} d}+\frac{\sqrt{a+b} \left (12 a^2 A b+35 A b^3-8 a^3 B-30 a b^2 B\right ) \cot (c+d x) \Pi \left (\frac{a+b}{a};\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}}}{8 a^5 d}+\frac{\left (16 a^2 A+35 A b^2-30 a b B\right ) \sin (c+d x)}{24 a^3 d \sqrt{a+b \sec (c+d x)}}-\frac{(7 A b-6 a B) \cos (c+d x) \sin (c+d x)}{12 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}}+\frac{b \left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\right ) \tan (c+d x)}{24 a^4 \left (a^2-b^2\right ) d \sqrt{a+b \sec (c+d x)}}\\ \end{align*}

Mathematica [B]  time = 22.3852, size = 2343, normalized size = 3.72 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(-((a^4*A - a^2*A*b^2 + 24*A*b^4 - 24*a*b^3*B)*Sin[c + d*x])/(12*a^4*(-
a^2 + b^2)) - (2*(A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x]))/(a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])) + ((-11*A
*b + 6*a*B)*Sin[2*(c + d*x)])/(24*a^3) + (A*Sin[3*(c + d*x)])/(12*a^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) - ((b
+ a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2
]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(16*a^5*A*Tan[(c + d*x)/2] + 16*a^4*A*b*Tan[(c + d*x)/2]
+ 41*a^3*A*b^2*Tan[(c + d*x)/2] + 41*a^2*A*b^3*Tan[(c + d*x)/2] - 105*a*A*b^4*Tan[(c + d*x)/2] - 105*A*b^5*Ta
n[(c + d*x)/2] - 42*a^4*b*B*Tan[(c + d*x)/2] - 42*a^3*b^2*B*Tan[(c + d*x)/2] + 90*a^2*b^3*B*Tan[(c + d*x)/2] +
90*a*b^4*B*Tan[(c + d*x)/2] - 32*a^5*A*Tan[(c + d*x)/2]^3 - 82*a^3*A*b^2*Tan[(c + d*x)/2]^3 + 210*a*A*b^4*Tan
[(c + d*x)/2]^3 + 84*a^4*b*B*Tan[(c + d*x)/2]^3 - 180*a^2*b^3*B*Tan[(c + d*x)/2]^3 + 16*a^5*A*Tan[(c + d*x)/2]
^5 - 16*a^4*A*b*Tan[(c + d*x)/2]^5 + 41*a^3*A*b^2*Tan[(c + d*x)/2]^5 - 41*a^2*A*b^3*Tan[(c + d*x)/2]^5 - 105*a
*A*b^4*Tan[(c + d*x)/2]^5 + 105*A*b^5*Tan[(c + d*x)/2]^5 - 42*a^4*b*B*Tan[(c + d*x)/2]^5 + 42*a^3*b^2*B*Tan[(c
+ d*x)/2]^5 + 90*a^2*b^3*B*Tan[(c + d*x)/2]^5 - 90*a*b^4*B*Tan[(c + d*x)/2]^5 + 72*a^4*A*b*EllipticPi[-1, -Ar
cSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*T
an[(c + d*x)/2]^2)/(a + b)] + 138*a^2*A*b^3*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1
- Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 210*A*b^5*Elliptic
Pi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/
2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*a^5*B*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sq
rt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 132*a^3*b^2*B
*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(
c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 180*a*b^4*B*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)
/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 7
2*a^4*A*b*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)
/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 138*a^2*A*b^3*EllipticPi[-1, -Arc
Sin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c
+ d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 210*A*b^5*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a
+ b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]
^2)/(a + b)] - 48*a^5*B*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 -
Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 132*a^3*b^2*B*Ellip
ticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a
+ b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 180*a*b^4*B*EllipticPi[-1, -ArcSin[Tan[(c + d*x)
/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*
Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*Ellipti
cE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a +
b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(a + b)*(-35*A*b^3 + 12*a^3*B - 2*a^2*b*(5*A +
9*B) + 3*a*b^2*(7*A + 10*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2
]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*a^4*(a^2
- b^2)*d*(a + b*Sec[c + d*x])^(3/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c
+ d*x)/2]^2)))

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Maple [B]  time = 0.717, size = 5086, normalized size = 8.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B \cos \left (d x + c\right )^{3} \sec \left (d x + c\right ) + A \cos \left (d x + c\right )^{3}\right )} \sqrt{b \sec \left (d x + c\right ) + a}}{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*cos(d*x + c)^3*sec(d*x + c) + A*cos(d*x + c)^3)*sqrt(b*sec(d*x + c) + a)/(b^2*sec(d*x + c)^2 + 2*a
*b*sec(d*x + c) + a^2), 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(cos(d*x+c)**3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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