3.1240 \(\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^4} \, dx\)
Optimal. Leaf size=244 \[ -\frac{(108 A-17 B-4 C) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{42 a^4 d}+\frac{(57 A-8 B-C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac{(141 A-29 B-13 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{210 a^4 d (\cos (c+d x)+1)^2}-\frac{(108 A-17 B-4 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{42 a^4 d (\cos (c+d x)+1)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(11 A-4 B-3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
[Out]
((57*A - 8*B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) - ((108*A - 17*B - 4*C)*EllipticF[(c + d*x)/2, 2])/(42
*a^4*d) - ((141*A - 29*B - 13*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) - ((108*A -
17*B - 4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(42*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(7/2)*
Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((11*A - 4*B - 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d*(a +
a*Cos[c + d*x])^3)
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Rubi [A] time = 0.790196, antiderivative size = 244, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used =
{4112, 3041, 2977, 2748, 2641, 2639} \[ -\frac{(108 A-17 B-4 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{42 a^4 d}+\frac{(57 A-8 B-C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac{(141 A-29 B-13 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{210 a^4 d (\cos (c+d x)+1)^2}-\frac{(108 A-17 B-4 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{42 a^4 d (\cos (c+d x)+1)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(11 A-4 B-3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]
[Out]
((57*A - 8*B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) - ((108*A - 17*B - 4*C)*EllipticF[(c + d*x)/2, 2])/(42
*a^4*d) - ((141*A - 29*B - 13*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) - ((108*A -
17*B - 4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(42*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(7/2)*
Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((11*A - 4*B - 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d*(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 2977
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, 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 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]
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]
Rubi steps
\begin{align*} \int \frac{\sqrt{\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx &=\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx\\ &=-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (-\frac{7}{2} a (A-B-C)+\frac{1}{2} a (15 A-B+C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(11 A-4 B-3 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (-\frac{5}{2} a^2 (11 A-4 B-3 C)+\frac{1}{2} a^2 (86 A-9 B+2 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(141 A-29 B-13 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(11 A-4 B-3 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\sqrt{\cos (c+d x)} \left (-\frac{3}{4} a^3 (141 A-29 B-13 C)+\frac{1}{4} a^3 (657 A-83 B-C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(141 A-29 B-13 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(11 A-4 B-3 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(108 A-17 B-4 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{42 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{\int \frac{-\frac{5}{4} a^4 (108 A-17 B-4 C)+\frac{21}{4} a^4 (57 A-8 B-C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{105 a^8}\\ &=-\frac{(141 A-29 B-13 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(11 A-4 B-3 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(108 A-17 B-4 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{42 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac{(108 A-17 B-4 C) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{84 a^4}+\frac{(57 A-8 B-C) \int \sqrt{\cos (c+d x)} \, dx}{20 a^4}\\ &=\frac{(57 A-8 B-C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac{(108 A-17 B-4 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{42 a^4 d}-\frac{(141 A-29 B-13 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac{(A-B+C) \cos ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{(11 A-4 B-3 C) \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{(108 A-17 B-4 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{42 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 7.39016, size = 2286, normalized size = 9.37 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]
[Out]
(((114*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*Hyperg
eometric2F1[-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) - (((2*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) + (288*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 - A
rcTan[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]]]])/(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) - (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) - (32*C*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 - Si
n[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTa
n[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
) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-16*(37*A - 8*B - C + 20*A*Cos[c])*Csc[c])
/(5*d) - (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(37*A*Sin[(d*x)/2] - 8*B*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) + (4*S
ec[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*(26*A*Sin[(d*x)/2] - 19*B*Sin[(d*x)/2] + 12*C*Sin[(d*x)/2]))/(35*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/
2]^3*(363*A*Sin[(d*x)/2] - 167*B*Sin[(d*x)/2] + 41*C*Sin[(d*x)/2]))/(105*d) + (8*(363*A - 167*B + 41*C)*Sec[c/
2 + (d*x)/2]^2*Tan[c/2])/(105*d) - (8*(26*A - 19*B + 12*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 = 3.206, size = 666, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^4,x)
[Out]
1/840*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(6216*A*cos(1/2*d*x+1/2*c)^10+2160*A*(sin(1/2*d*
x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^7
+4788*A*cos(1/2*d*x+1/2*c)^7*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticE(cos(1/2*
d*x+1/2*c),2^(1/2))-1344*B*cos(1/2*d*x+1/2*c)^10-340*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1
)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^7-672*B*cos(1/2*d*x+1/2*c)^7*(sin(1/2*d*x+1/2
*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-168*C*cos(1/2*d*x+1/2*c)^
10-80*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*c
os(1/2*d*x+1/2*c)^7-84*C*cos(1/2*d*x+1/2*c)^7*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*E
llipticE(cos(1/2*d*x+1/2*c),2^(1/2))-10776*A*cos(1/2*d*x+1/2*c)^8+2684*B*cos(1/2*d*x+1/2*c)^8+88*C*cos(1/2*d*x
+1/2*c)^8+5598*A*cos(1/2*d*x+1/2*c)^6-1902*B*cos(1/2*d*x+1/2*c)^6+306*C*cos(1/2*d*x+1/2*c)^6-1224*A*cos(1/2*d*
x+1/2*c)^4+706*B*cos(1/2*d*x+1/2*c)^4-328*C*cos(1/2*d*x+1/2*c)^4+201*A*cos(1/2*d*x+1/2*c)^2-159*B*cos(1/2*d*x+
1/2*c)^2+117*C*cos(1/2*d*x+1/2*c)^2-15*A+15*B-15*C)/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(cos(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^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} \sec \left (d x + c\right )^{4} + 4 \, a^{4} \sec \left (d x + c\right )^{3} + 6 \, a^{4} \sec \left (d x + c\right )^{2} + 4 \, a^{4} \sec \left (d x + c\right ) + a^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^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*sec(d*x + c)^4 + 4*a^4*sec(d*x + c)^3
+ 6*a^4*sec(d*x + c)^2 + 4*a^4*sec(d*x + c) + a^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(cos(d*x+c)**(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**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{{\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 )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^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)*sqrt(cos(d*x + c))/(a*sec(d*x + c) + a)^4, x)