3.1244 \(\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx\)
Optimal. Leaf size=276 \[ \frac{(4 A+17 B-108 C) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{42 a^4 d}+\frac{(A+8 B-57 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac{(A+8 B-57 C) \sin (c+d x)}{10 a^4 d \sqrt{\cos (c+d x)}}+\frac{(4 A+17 B-108 C) \sin (c+d x)}{42 a^4 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}+\frac{(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)^2}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^4} \]
[Out]
((A + 8*B - 57*C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((4*A + 17*B - 108*C)*EllipticF[(c + d*x)/2, 2])/(42
*a^4*d) - ((A + 8*B - 57*C)*Sin[c + d*x])/(10*a^4*d*Sqrt[Cos[c + d*x]]) + ((13*A + 29*B - 141*C)*Sin[c + d*x])
/(210*a^4*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^2) + ((4*A + 17*B - 108*C)*Sin[c + d*x])/(42*a^4*d*Sqrt[Cos[
c + d*x]]*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(7*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4) + (
(3*A + 4*B - 11*C)*Sin[c + d*x])/(35*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3)
________________________________________________________________________________________
Rubi [A] time = 0.826389, antiderivative size = 276, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used =
{4112, 3041, 2978, 2748, 2636, 2639, 2641} \[ \frac{(4 A+17 B-108 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{42 a^4 d}+\frac{(A+8 B-57 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac{(A+8 B-57 C) \sin (c+d x)}{10 a^4 d \sqrt{\cos (c+d x)}}+\frac{(4 A+17 B-108 C) \sin (c+d x)}{42 a^4 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}+\frac{(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)^2}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^4),x]
[Out]
((A + 8*B - 57*C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((4*A + 17*B - 108*C)*EllipticF[(c + d*x)/2, 2])/(42
*a^4*d) - ((A + 8*B - 57*C)*Sin[c + d*x])/(10*a^4*d*Sqrt[Cos[c + d*x]]) + ((13*A + 29*B - 141*C)*Sin[c + d*x])
/(210*a^4*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^2) + ((4*A + 17*B - 108*C)*Sin[c + d*x])/(42*a^4*d*Sqrt[Cos[
c + d*x]]*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(7*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4) + (
(3*A + 4*B - 11*C)*Sin[c + d*x])/(35*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3)
Rule 4112
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sec[(e_.)
+ (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[d^(m + 2), Int[(b + a*Cos[e + f*x])^m*(d*
Cos[e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}
, x] && !IntegerQ[n] && IntegerQ[m]
Rule 3041
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((a*A - b*B + a*C)*Cos[e + f*x]*(
a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(f*(b*c - a*d)*(2*m + 1)), x] + Dist[1/(b*(b*c - a*d)*(2*m
+ 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(
b*c*m + a*d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c*(2*m + 1) - a*d*(m - n -
1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^
2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)]
Rule 2978
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*
x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,
0])
Rule 2748
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Rule 2636
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]
Rule 2639
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]
Rule 2641
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx &=\int \frac{C+B \cos (c+d x)+A \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^4} \, dx\\ &=-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac{\int \frac{\frac{1}{2} a (A-B+15 C)+\frac{7}{2} a (A+B-C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac{\int \frac{\frac{1}{2} a^2 (2 A-9 B+86 C)+\frac{5}{2} a^2 (3 A+4 B-11 C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=\frac{(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt{\cos (c+d x)} (1+\cos (c+d x))^2}-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac{\int \frac{-\frac{1}{4} a^3 (A+83 B-657 C)+\frac{3}{4} a^3 (13 A+29 B-141 C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx}{105 a^6}\\ &=\frac{(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt{\cos (c+d x)} (1+\cos (c+d x))^2}-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac{(4 A+17 B-108 C) \sin (c+d x)}{42 d \sqrt{\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right )}+\frac{\int \frac{-\frac{21}{4} a^4 (A+8 B-57 C)+\frac{5}{4} a^4 (4 A+17 B-108 C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{105 a^8}\\ &=\frac{(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt{\cos (c+d x)} (1+\cos (c+d x))^2}-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac{(4 A+17 B-108 C) \sin (c+d x)}{42 d \sqrt{\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right )}+\frac{(4 A+17 B-108 C) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{84 a^4}-\frac{(A+8 B-57 C) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{20 a^4}\\ &=\frac{(4 A+17 B-108 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{42 a^4 d}-\frac{(A+8 B-57 C) \sin (c+d x)}{10 a^4 d \sqrt{\cos (c+d x)}}+\frac{(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt{\cos (c+d x)} (1+\cos (c+d x))^2}-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac{(4 A+17 B-108 C) \sin (c+d x)}{42 d \sqrt{\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right )}+\frac{(A+8 B-57 C) \int \sqrt{\cos (c+d x)} \, dx}{20 a^4}\\ &=\frac{(A+8 B-57 C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^4 d}+\frac{(4 A+17 B-108 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{42 a^4 d}-\frac{(A+8 B-57 C) \sin (c+d x)}{10 a^4 d \sqrt{\cos (c+d x)}}+\frac{(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt{\cos (c+d x)} (1+\cos (c+d x))^2}-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac{(4 A+17 B-108 C) \sin (c+d x)}{42 d \sqrt{\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 7.44565, size = 2316, normalized size = 8.39 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^4),x]
[Out]
(((2*I)/5)*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2
*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*
d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)
*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4
, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*
d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*
x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c
+ d*x])^4) + (((16*I)/5)*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c
+ d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*
(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I
*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeo
metric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(
-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1
+ E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])
*(a + a*Sec[c + d*x])^4) - (((114*I)/5)*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c +
d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c
])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*
x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]
) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*C
os[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*
c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Co
s[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (32*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5
/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - Ar
cTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]
*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^
2]*(a + a*Sec[c + d*x])^4) - (136*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x
- ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]
]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Si
n[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Se
c[c + d*x])^4) + (288*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Co
t[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - S
in[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcT
an[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4
) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(20*C - A*Cos[c] - 8*B*Cos[c] + 37*C*Cos
[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + 83*B*Sin[(d*x)/2] -
237*C*Sin[(d*x)/2]))/(105*d) - (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 8*B*Sin[(d*x)/2] - 37*C*Sin[(
d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d) +
(8*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(2*A*Sin[(d*x)/2] - 9*B*Sin[(d*x)/2] + 16*C*Sin[(d*x)/2]))/(35*d) + (64*C*Se
c[c]*Sec[c + d*x]*Sin[d*x])/d - (8*(A + 83*B - 237*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(105*d) + (8*(2*A - 9*B +
16*C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(35*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^6*Tan[c/2])/(7*d)))/(Cos[c +
d*x]^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4)
________________________________________________________________________________________
Maple [B] time = 4.102, size = 1017, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+a*sec(d*x+c))^4,x)
[Out]
1/840*(-4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1
/2*c)^2)^(1/2)*(21*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-20*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+168*B*El
lipticE(cos(1/2*d*x+1/2*c),2^(1/2))-85*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1197*C*EllipticE(cos(1/2*d*x+1/
2*c),2^(1/2))+540*C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+12*(sin(1/2
*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(21
*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-20*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+168*B*EllipticE(cos(1/2*d*
x+1/2*c),2^(1/2))-85*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1197*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+540*
C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-12*(sin(1/2*d*x+1/2*c)^2)^(1/
2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(21*A*EllipticE(cos(1
/2*d*x+1/2*c),2^(1/2))-20*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+168*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-
85*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1197*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+540*C*EllipticF(cos(1/
2*d*x+1/2*c),2^(1/2)))*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1
/2*c)^2-1)^(1/2)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(21*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/
2))-20*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+168*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-85*B*EllipticF(cos(
1/2*d*x+1/2*c),2^(1/2))-1197*C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+540*C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2
)))*cos(1/2*d*x+1/2*c)-168*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(A+8*B-57*C)*sin(1/2*d*x+1/2*c
)^10+4*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(148*A+1259*B-9036*C)*sin(1/2*d*x+1/2*c)^8-14*(-2*
sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(53*A+499*B-3621*C)*sin(1/2*d*x+1/2*c)^6+2*(-2*sin(1/2*d*x+1/
2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(181*A+2108*B-15597*C)*sin(1/2*d*x+1/2*c)^4-(-2*sin(1/2*d*x+1/2*c)^4+sin(1/
2*d*x+1/2*c)^2)^(1/2)*(59*A+907*B-7053*C)*sin(1/2*d*x+1/2*c)^2)/a^4/cos(1/2*d*x+1/2*c)^7/(-2*sin(1/2*d*x+1/2*c
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
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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)+C*sec(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+a*sec(d*x+c))^4,x, algorithm="maxima")
[Out]
Timed out
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt{\cos \left (d x + c\right )}}{a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right ) + a^{4} \cos \left (d x + c\right )^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+a*sec(d*x+c))^4,x, algorithm="fricas")
[Out]
integral((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(cos(d*x + c))/(a^4*cos(d*x + c)^4*sec(d*x + c)^4 + 4*a^4
*cos(d*x + c)^4*sec(d*x + c)^3 + 6*a^4*cos(d*x + c)^4*sec(d*x + c)^2 + 4*a^4*cos(d*x + c)^4*sec(d*x + c) + a^4
*cos(d*x + c)^4), 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)+C*sec(d*x+c)**2)/cos(d*x+c)**(7/2)/(a+a*sec(d*x+c))**4,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(7/2)/(a+a*sec(d*x+c))^4,x, algorithm="giac")
[Out]
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)/((a*sec(d*x + c) + a)^4*cos(d*x + c)^(7/2)), x)