3.370 \(\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=617 \[ \frac{\sqrt{a+b} \left (4 a^2 b (71 A+52 B)+8 a^3 (9 A+16 B)+2 a b^2 (59 A+132 B)+15 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{192 a d}+\frac{\left (284 a^2 A b+128 a^3 B+264 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\left (36 a^2 A+104 a b B+59 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{(a-b) \sqrt{a+b} \left (284 a^2 A b+128 a^3 B+264 a b^2 B+15 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a b d}-\frac{\sqrt{a+b} \left (120 a^2 A b^2+48 a^4 A+160 a^3 b B+40 a b^3 B-5 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^2 d}+\frac{a (8 a B+11 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d} \]

[Out]

((a - b)*Sqrt[a + b]*(284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*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))])/(192*a*b*d) + (Sqrt[a + b]*(15*A*b^3 + 8*a^3*(9*A + 16*B) + 4*a^2*b*(71*A + 52*B) + 2*a*b^2*(5
9*A + 132*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))])/(192*a*d) - (Sqrt[a + b]*(48*a^4*A + 120*a
^2*A*b^2 - 5*A*b^4 + 160*a^3*b*B + 40*a*b^3*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))
])/(64*a^2*d) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(19
2*a*d) + ((36*a^2*A + 59*A*b^2 + 104*a*b*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*(1
1*A*b + 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*A*Cos[c + d*x]^3*(a + b*Sec[c
+ d*x])^(3/2)*Sin[c + d*x])/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 1.83221, antiderivative size = 617, 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 = {4025, 4094, 4104, 4058, 3921, 3784, 3832, 4004} \[ \frac{\left (284 a^2 A b+128 a^3 B+264 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\left (36 a^2 A+104 a b B+59 A b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{\sqrt{a+b} \left (4 a^2 b (71 A+52 B)+8 a^3 (9 A+16 B)+2 a b^2 (59 A+132 B)+15 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a d}+\frac{(a-b) \sqrt{a+b} \left (284 a^2 A b+128 a^3 B+264 a b^2 B+15 A b^3\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{192 a b d}-\frac{\sqrt{a+b} \left (120 a^2 A b^2+48 a^4 A+160 a^3 b B+40 a b^3 B-5 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{64 a^2 d}+\frac{a (8 a B+11 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]

[Out]

((a - b)*Sqrt[a + b]*(284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*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))])/(192*a*b*d) + (Sqrt[a + b]*(15*A*b^3 + 8*a^3*(9*A + 16*B) + 4*a^2*b*(71*A + 52*B) + 2*a*b^2*(5
9*A + 132*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))])/(192*a*d) - (Sqrt[a + b]*(48*a^4*A + 120*a
^2*A*b^2 - 5*A*b^4 + 160*a^3*b*B + 40*a*b^3*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))
])/(64*a^2*d) + ((284*a^2*A*b + 15*A*b^3 + 128*a^3*B + 264*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(19
2*a*d) + ((36*a^2*A + 59*A*b^2 + 104*a*b*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*(1
1*A*b + 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*A*Cos[c + d*x]^3*(a + b*Sec[c
+ d*x])^(3/2)*Sin[c + d*x])/(4*d)

Rule 4025

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

Rule 4094

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]

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 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 \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx &=\frac{a A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac{1}{4} \int \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left (-\frac{1}{2} a (11 A b+8 a B)-\left (3 a^2 A+4 A b^2+8 a b B\right ) \sec (c+d x)-\frac{1}{2} b (3 a A+8 b B) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a (11 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{a A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac{1}{12} \int \frac{\cos ^2(c+d x) \left (-\frac{1}{4} a \left (36 a^2 A+59 A b^2+104 a b B\right )-\frac{1}{2} \left (49 a^2 A b+24 A b^3+16 a^3 B+72 a b^2 B\right ) \sec (c+d x)-\frac{3}{4} b \left (17 a A b+8 a^2 B+16 b^2 B\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\left (36 a^2 A+59 A b^2+104 a b B\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{a (11 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{a A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\int \frac{\cos (c+d x) \left (\frac{1}{8} a \left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right )+\frac{1}{4} a \left (36 a^3 A+161 a A b^2+152 a^2 b B+96 b^3 B\right ) \sec (c+d x)+\frac{1}{8} a b \left (36 a^2 A+59 A b^2+104 a b B\right ) \sec ^2(c+d x)\right )}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a}\\ &=\frac{\left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac{\left (36 a^2 A+59 A b^2+104 a b B\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{a (11 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{a A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac{\int \frac{-\frac{3}{16} a \left (48 a^4 A+120 a^2 A b^2-5 A b^4+160 a^3 b B+40 a b^3 B\right )-\frac{1}{8} a^2 b \left (36 a^2 A+59 A b^2+104 a b B\right ) \sec (c+d x)+\frac{1}{16} a b \left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a^2}\\ &=\frac{\left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac{\left (36 a^2 A+59 A b^2+104 a b B\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{a (11 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{a A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}-\frac{\int \frac{-\frac{3}{16} a \left (48 a^4 A+120 a^2 A b^2-5 A b^4+160 a^3 b B+40 a b^3 B\right )+\left (-\frac{1}{8} a^2 b \left (36 a^2 A+59 A b^2+104 a b B\right )-\frac{1}{16} a b \left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{24 a^2}-\frac{\left (b \left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a}\\ &=\frac{(a-b) \sqrt{a+b} \left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 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}}}{192 a b d}+\frac{\left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac{\left (36 a^2 A+59 A b^2+104 a b B\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{a (11 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{a A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}+\frac{\left (48 a^4 A+120 a^2 A b^2-5 A b^4+160 a^3 b B+40 a b^3 B\right ) \int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx}{128 a}+\frac{\left (b \left (15 A b^3+8 a^3 (9 A+16 B)+4 a^2 b (71 A+52 B)+2 a b^2 (59 A+132 B)\right )\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{384 a}\\ &=\frac{(a-b) \sqrt{a+b} \left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 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}}}{192 a b d}+\frac{\sqrt{a+b} \left (15 A b^3+8 a^3 (9 A+16 B)+4 a^2 b (71 A+52 B)+2 a b^2 (59 A+132 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}}}{192 a d}-\frac{\sqrt{a+b} \left (48 a^4 A+120 a^2 A b^2-5 A b^4+160 a^3 b B+40 a b^3 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}}}{64 a^2 d}+\frac{\left (284 a^2 A b+15 A b^3+128 a^3 B+264 a b^2 B\right ) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac{\left (36 a^2 A+59 A b^2+104 a b B\right ) \cos (c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{96 d}+\frac{a (11 A b+8 a B) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac{a A \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{4 d}\\ \end{align*}

Mathematica [B]  time = 24.5393, size = 5186, normalized size = 8.41 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]

[Out]

Result too large to show

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Maple [B]  time = 0.612, size = 4231, normalized size = 6.9 \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)^4*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x)

[Out]

-1/192/d/a*(-1+cos(d*x+c))^2*(48*A*a^4*cos(d*x+c)^6+288*A*(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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^4*sin(d*x+c)+24*
A*a^4*cos(d*x+c)^4-72*A*a^4*cos(d*x+c)^2+64*B*cos(d*x+c)^3*a^4+15*A*cos(d*x+c)^2*b^4-128*B*cos(d*x+c)^2*a^4+64
*B*cos(d*x+c)^5*a^4-30*A*(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
ipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*b^4*sin(d*x+c)+284*A*(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*sin(d*x+c)+15*A*(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)*Elli
pticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a*b^3*sin(d*x+c)+72*A*(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*sin(d*x+c)-644*A*(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*sin(d*x+c)+118*A*(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*
sin(d*x+c)+960*B*(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)*EllipticPi(
(-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^3*b*sin(d*x+c)+240*B*(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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a*
b^3*sin(d*x+c)+128*B*(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))*a^3*b*sin(d*x+c)+264*B*(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*sin(d*x+c)+264*B*b^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)*sin(d
*x+c)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a-608*B*a^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)*sin(d*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^
(1/2))*b+208*B*a^2*b^2*(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)*sin(d
*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))-384*B*(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*sin(
d*x+c)+172*A*cos(d*x+c)^3*a^3*b+133*A*cos(d*x+c)^3*a*b^3+472*B*cos(d*x+c)^3*a^2*b^2-284*A*cos(d*x+c)^2*a^3*b+3
0*A*cos(d*x+c)^2*a^2*b^2-144*B*cos(d*x+c)^2*a^3*b+264*B*cos(d*x+c)^2*a*b^3-72*A*cos(d*x+c)*a^3*b-284*A*cos(d*x
+c)*a^2*b^2-118*A*cos(d*x+c)*a*b^3-128*B*cos(d*x+c)*a^3*b-208*B*cos(d*x+c)*a^2*b^2+272*B*cos(d*x+c)^4*a^3*b-15
*A*cos(d*x+c)^2*a*b^3-264*B*cos(d*x+c)^2*a^2*b^2-264*B*cos(d*x+c)*a*b^3+184*A*cos(d*x+c)^5*a^3*b+254*A*cos(d*x
+c)^4*a^2*b^2+15*A*(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^4*sin(d*x+c)-144*A*(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*sin(d*
x+c)+128*B*(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*sin(d*x+c)+720*A*cos(d*x+c)*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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))
^(1/2))*a^2*b^2+284*A*cos(d*x+c)*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+284*A*cos(d*x+c)*a^2*b^2*(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)*sin(d*x+c)*EllipticE((-1+cos(d*x+
c))/sin(d*x+c),((a-b)/(a+b))^(1/2))+15*A*cos(d*x+c)*b^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)*sin(d*x+c)*EllipticE((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a+72*A*cos(d*
x+c)*a^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)*sin(d*x+c)*Elliptic
F((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b-644*A*cos(d*x+c)*a^2*b^2*(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)*sin(d*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))
^(1/2))+118*A*cos(d*x+c)*b^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)
*sin(d*x+c)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*a+960*B*cos(d*x+c)*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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),
-1,((a-b)/(a+b))^(1/2))*a^3*b+240*B*cos(d*x+c)*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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a*b^3+128*B*cos(d*
x+c)*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+264*B*cos(d*x+c)*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^2+264*B*cos(d*x+c)*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^3-608*B*cos(d*x+c)*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+208*B*cos(d*x+c)*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^2-384*B*cos(d*x+c)*s
in(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^3+288*A*cos(d*x+c)*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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^
4-30*A*cos(d*x+c)*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)
*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*b^4+15*A*cos(d*x+c)*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^4-144*A*cos(d*x+c)*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+128*B*cos(d*x+c)*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)*EllipticE((-1+cos(d*x+c))/sin(
d*x+c),((a-b)/(a+b))^(1/2))*a^4+720*A*(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)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*a^2*b^2*sin(d*x+c)+284*A*(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*sin(d*x+c)-15*A*cos(d*x+c)*b^4)*(cos(d*x+c)+1)^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)/(b
+a*cos(d*x+c))/sin(d*x+c)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*cos(d*x + c)^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{2} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} + A a^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )\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(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="fricas")

[Out]

integral((B*b^2*cos(d*x + c)^4*sec(d*x + c)^3 + A*a^2*cos(d*x + c)^4 + (2*B*a*b + A*b^2)*cos(d*x + c)^4*sec(d*
x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c)^4*sec(d*x + c))*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(cos(d*x+c)**4*(a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*cos(d*x + c)^4, x)