\(\int \frac {\sqrt {\cos (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^4} \, dx\) [1240]

Optimal result

Integrand size = 43, antiderivative size = 244 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {(57 A-8 B-C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac {(108 A-17 B-4 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}-\frac {(141 A-29 B-13 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac {(108 A-17 B-4 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{42 a^4 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(11 A-4 B-3 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]

[Out]

1/10*(57*A-8*B-C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^4/d-
1/42*(108*A-17*B-4*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/a^
4/d-1/210*(141*A-29*B-13*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/a^4/d/(1+cos(d*x+c))^2-1/7*(A-B+C)*cos(d*x+c)^(7/2)*si
n(d*x+c)/d/(a+a*cos(d*x+c))^4-1/35*(11*A-4*B-3*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3-1/42*(108
*A-17*B-4*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/a^4/d/(1+cos(d*x+c))

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4197, 3120, 3056, 2827, 2720, 2719} \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {(108 A-17 B-4 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}+\frac {(57 A-8 B-C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac {(141 A-29 B-13 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{210 a^4 d (\cos (c+d x)+1)^2}-\frac {(108 A-17 B-4 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{42 a^4 d (\cos (c+d x)+1)}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac {(11 A-4 B-3 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

[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 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 2827

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 3056

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 3120

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 4197

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]

Rubi steps \begin{align*} \text {integral}& = \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) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 15.79 (sec) , antiderivative size = 1932, normalized size of antiderivative = 7.92 \[ \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=\frac {288 A \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \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 ^2(c)} (a+a \sec (c+d x))^4}-\frac {136 B \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \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 ^2(c)} (a+a \sec (c+d x))^4}-\frac {32 C \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \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 ^2(c)} (a+a \sec (c+d x))^4}+\frac {\cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {16 (37 A-8 B-C+20 A \cos (c)) \csc (c)}{5 d}-\frac {16 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (37 A \sin \left (\frac {d x}{2}\right )-8 B \sin \left (\frac {d x}{2}\right )-C \sin \left (\frac {d x}{2}\right )\right )}{5 d}+\frac {4 \sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )-B \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{7 d}-\frac {8 \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (26 A \sin \left (\frac {d x}{2}\right )-19 B \sin \left (\frac {d x}{2}\right )+12 C \sin \left (\frac {d x}{2}\right )\right )}{35 d}+\frac {8 \sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (363 A \sin \left (\frac {d x}{2}\right )-167 B \sin \left (\frac {d x}{2}\right )+41 C \sin \left (\frac {d x}{2}\right )\right )}{105 d}+\frac {8 (363 A-167 B+41 C) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{105 d}-\frac {8 (26 A-19 B+12 C) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{35 d}+\frac {4 (A-B+C) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{7 d}\right )}{\cos ^{\frac {3}{2}}(c+d x) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {228 A \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {32 B \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {4 C \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4} \]

[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]

(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 - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[C
ot[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 + 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) - (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*Se
c[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 + C
ot[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) + (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*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*(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) - (228*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)*((HypergeometricPFQ[{-
1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[T
an[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1
+ Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*
Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A
+ 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (32*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)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d
*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*
x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x
+ ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos
[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x
] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (4*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)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*S
in[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt
[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c]
)/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[C
os[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(665\) vs. \(2(276)=552\).

Time = 4.07 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.73

method	result	size
default	\(\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (6216 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+2160 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+4788 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1344 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-340 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-672 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-168 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-80 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-84 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-10776 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+2684 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+88 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+5598 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-1902 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+306 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-1224 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+706 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-328 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+201 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-159 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+117 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} C -15 A +15 B -15 C \right )}{840 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(666\)

[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,method=_RETURNVERBOSE)

[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*cos(1/2*d*x+1/2*c)^2*C-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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.15 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.66 \[ \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=-\frac {2 \, {\left (21 \, {\left (37 \, A - 8 \, B - C\right )} \cos \left (d x + c\right )^{3} + {\left (1968 \, A - 337 \, B - 104 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (1761 \, A - 284 \, B - 73 \, C\right )} \cos \left (d x + c\right ) + 540 \, A - 85 \, B - 20 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 5 \, {\left (\sqrt {2} {\left (-108 i \, A + 17 i \, B + 4 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-108 i \, A + 17 i \, B + 4 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-108 i \, A + 17 i \, B + 4 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-108 i \, A + 17 i \, B + 4 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-108 i \, A + 17 i \, B + 4 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (108 i \, A - 17 i \, B - 4 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (108 i \, A - 17 i \, B - 4 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (108 i \, A - 17 i \, B - 4 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (108 i \, A - 17 i \, B - 4 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (108 i \, A - 17 i \, B - 4 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (-57 i \, A + 8 i \, B + i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-57 i \, A + 8 i \, B + i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-57 i \, A + 8 i \, B + i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-57 i \, A + 8 i \, B + i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-57 i \, A + 8 i \, B + i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (\sqrt {2} {\left (57 i \, A - 8 i \, B - i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (57 i \, A - 8 i \, B - i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (57 i \, A - 8 i \, B - i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (57 i \, A - 8 i \, B - i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (57 i \, A - 8 i \, B - i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

[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]

-1/420*(2*(21*(37*A - 8*B - C)*cos(d*x + c)^3 + (1968*A - 337*B - 104*C)*cos(d*x + c)^2 + (1761*A - 284*B - 73
*C)*cos(d*x + c) + 540*A - 85*B - 20*C)*sqrt(cos(d*x + c))*sin(d*x + c) + 5*(sqrt(2)*(-108*I*A + 17*I*B + 4*I*
C)*cos(d*x + c)^4 + 4*sqrt(2)*(-108*I*A + 17*I*B + 4*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(-108*I*A + 17*I*B + 4*I*
C)*cos(d*x + c)^2 + 4*sqrt(2)*(-108*I*A + 17*I*B + 4*I*C)*cos(d*x + c) + sqrt(2)*(-108*I*A + 17*I*B + 4*I*C))*
weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*(sqrt(2)*(108*I*A - 17*I*B - 4*I*C)*cos(d*x + c)
^4 + 4*sqrt(2)*(108*I*A - 17*I*B - 4*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(108*I*A - 17*I*B - 4*I*C)*cos(d*x + c)^2
+ 4*sqrt(2)*(108*I*A - 17*I*B - 4*I*C)*cos(d*x + c) + sqrt(2)*(108*I*A - 17*I*B - 4*I*C))*weierstrassPInverse
(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*(sqrt(2)*(-57*I*A + 8*I*B + I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(-57*I
*A + 8*I*B + I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(-57*I*A + 8*I*B + I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(-57*I*A + 8*I
*B + I*C)*cos(d*x + c) + sqrt(2)*(-57*I*A + 8*I*B + I*C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, co
s(d*x + c) + I*sin(d*x + c))) + 21*(sqrt(2)*(57*I*A - 8*I*B - I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(57*I*A - 8*I*B
- I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(57*I*A - 8*I*B - I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(57*I*A - 8*I*B - I*C)*cos
(d*x + c) + sqrt(2)*(57*I*A - 8*I*B - I*C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I
*sin(d*x + c))))/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c
) + a^4*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]

[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

Maxima [F(-1)]

Timed out. \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]

[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

Giac [F]

\[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {\cos \left (d x + c\right )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\int \frac {\sqrt {\cos \left (c+d\,x\right )}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^4} \,d x \]

[In]

int((cos(c + d*x)^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^4,x)

[Out]

int((cos(c + d*x)^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^4, x)