\(\int \frac {\cos ^{\frac {3}{2}}(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^4} \, dx\) [1239]

Optimal result

Integrand size = 43, antiderivative size = 278 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=-\frac {(176 A-57 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}+\frac {(339 A-108 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}+\frac {(339 A-108 B+17 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{42 a^4 d}-\frac {(43 A-15 B+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{42 a^4 d (1+\cos (c+d x))^2}-\frac {(176 A-57 B+8 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 a^4 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(13 A-6 B-C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]

[Out]

-1/10*(176*A-57*B+8*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*(339*A-108*B+17*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-1/42*(43*A-15*B+C)*cos(d*x+c)^(5/2)*sin(d*x+c)/a^4/d/(1+cos(d*x+c))^2-1/30*(176*A-57*B+8*C)*cos(d*x
+c)^(3/2)*sin(d*x+c)/a^4/d/(1+cos(d*x+c))-1/7*(A-B+C)*cos(d*x+c)^(9/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^4-1/35*(1
3*A-6*B-C)*cos(d*x+c)^(7/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3+1/42*(339*A-108*B+17*C)*sin(d*x+c)*cos(d*x+c)^(1
/2)/a^4/d

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4197, 3120, 3056, 2827, 2719, 2715, 2720} \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\frac {(339 A-108 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}-\frac {(176 A-57 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac {(43 A-15 B+C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{42 a^4 d (\cos (c+d x)+1)^2}-\frac {(176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{30 a^4 d (\cos (c+d x)+1)}+\frac {(339 A-108 B+17 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{42 a^4 d}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac {(13 A-6 B-C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]

[In]

Int[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]

[Out]

-1/10*((176*A - 57*B + 8*C)*EllipticE[(c + d*x)/2, 2])/(a^4*d) + ((339*A - 108*B + 17*C)*EllipticF[(c + d*x)/2
, 2])/(42*a^4*d) + ((339*A - 108*B + 17*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(42*a^4*d) - ((43*A - 15*B + C)*Co
s[c + d*x]^(5/2)*Sin[c + d*x])/(42*a^4*d*(1 + Cos[c + d*x])^2) - ((176*A - 57*B + 8*C)*Cos[c + d*x]^(3/2)*Sin[
c + d*x])/(30*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*
x])^4) - ((13*A - 6*B - C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

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 {7}{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 {9}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (-\frac {1}{2} a (9 A-9 B-5 C)+\frac {1}{2} a (17 A-3 B+3 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(13 A-6 B-C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (-\frac {7}{2} a^2 (13 A-6 B-C)+\frac {1}{2} a^2 (124 A-33 B+12 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4} \\ & = -\frac {(43 A-15 B+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{42 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(13 A-6 B-C) \cos ^{\frac {7}{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 {25}{4} a^3 (43 A-15 B+C)+\frac {3}{4} a^3 (463 A-141 B+29 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6} \\ & = -\frac {(43 A-15 B+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{42 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(13 A-6 B-C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(176 A-57 B+8 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int \sqrt {\cos (c+d x)} \left (-\frac {21}{4} a^4 (176 A-57 B+8 C)+\frac {15}{4} a^4 (339 A-108 B+17 C) \cos (c+d x)\right ) \, dx}{105 a^8} \\ & = -\frac {(43 A-15 B+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{42 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(13 A-6 B-C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(176 A-57 B+8 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^4+a^4 \cos (c+d x)\right )}-\frac {(176 A-57 B+8 C) \int \sqrt {\cos (c+d x)} \, dx}{20 a^4}+\frac {(339 A-108 B+17 C) \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{28 a^4} \\ & = -\frac {(176 A-57 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}+\frac {(339 A-108 B+17 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{42 a^4 d}-\frac {(43 A-15 B+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{42 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(13 A-6 B-C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(176 A-57 B+8 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac {(339 A-108 B+17 C) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{84 a^4} \\ & = -\frac {(176 A-57 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}+\frac {(339 A-108 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}+\frac {(339 A-108 B+17 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{42 a^4 d}-\frac {(43 A-15 B+C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{42 a^4 d (1+\cos (c+d x))^2}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac {(13 A-6 B-C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(176 A-57 B+8 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 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 = 18.86 (sec) , antiderivative size = 1965, normalized size of antiderivative = 7.07 \[ \int \frac {\cos ^{\frac {3}{2}}(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 {904 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 {288 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)))}}{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 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 (96 A-37 B+8 C+80 A \cos (c)-20 B \cos (c)) \csc (c)}{5 d}+\frac {64 A \cos (d x) \sin (c)}{3 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 {16 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (96 A \sin \left (\frac {d x}{2}\right )-37 B \sin \left (\frac {d x}{2}\right )+8 C \sin \left (\frac {d x}{2}\right )\right )}{5 d}+\frac {8 \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (33 A \sin \left (\frac {d x}{2}\right )-26 B \sin \left (\frac {d x}{2}\right )+19 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 (629 A \sin \left (\frac {d x}{2}\right )-363 B \sin \left (\frac {d x}{2}\right )+167 C \sin \left (\frac {d x}{2}\right )\right )}{105 d}+\frac {64 A \cos (c) \sin (d x)}{3 d}-\frac {8 (629 A-363 B+167 C) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{105 d}+\frac {8 (33 A-26 B+19 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 {704 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 {228 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 {32 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[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]

[Out]

(-904*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[
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) + (288*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]]]]*Sqr
t[-(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*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*S
ec[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) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c +
d*x] + C*Sec[c + d*x]^2)*((16*(96*A - 37*B + 8*C + 80*A*Cos[c] - 20*B*Cos[c])*Csc[c])/(5*d) + (64*A*Cos[d*x]*S
in[c])/(3*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) + (1
6*Sec[c/2]*Sec[c/2 + (d*x)/2]*(96*A*Sin[(d*x)/2] - 37*B*Sin[(d*x)/2] + 8*C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*
Sec[c/2 + (d*x)/2]^5*(33*A*Sin[(d*x)/2] - 26*B*Sin[(d*x)/2] + 19*C*Sin[(d*x)/2]))/(35*d) - (8*Sec[c/2]*Sec[c/2
+ (d*x)/2]^3*(629*A*Sin[(d*x)/2] - 363*B*Sin[(d*x)/2] + 167*C*Sin[(d*x)/2]))/(105*d) + (64*A*Cos[c]*Sin[d*x])
/(3*d) - (8*(629*A - 363*B + 167*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(105*d) + (8*(33*A - 26*B + 19*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) + (704*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Se
c[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) - (228*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]*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) + (32*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*S
ec[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[T
an[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. \(679\) vs. \(2(306)=612\).

Time = 4.29 (sec) , antiderivative size = 680, normalized size of antiderivative = 2.45

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 (-15 C -15 A +15 B -5598 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+12768 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-25588 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+1224 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+12234 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+1902 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-706 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-6216 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+10776 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+6780 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}+14784 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 )-2160 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}-4788 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 )+340 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}+672 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 )-1882 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+243 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-201 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1344 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+159 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} C +2240 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-2684 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}\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}\)	\(680\)

[In]

int(cos(d*x+c)^(3/2)*(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)*(-15*C-15*A+15*B-5598*B*cos(1/2*d*x+1/2*c)^6+12
768*A*cos(1/2*d*x+1/2*c)^10-25588*A*cos(1/2*d*x+1/2*c)^8+1224*B*cos(1/2*d*x+1/2*c)^4+12234*A*cos(1/2*d*x+1/2*c
)^6+1902*C*cos(1/2*d*x+1/2*c)^6-706*C*cos(1/2*d*x+1/2*c)^4-6216*B*cos(1/2*d*x+1/2*c)^10+10776*B*cos(1/2*d*x+1/
2*c)^8-1882*A*cos(1/2*d*x+1/2*c)^4+243*A*cos(1/2*d*x+1/2*c)^2-201*B*cos(1/2*d*x+1/2*c)^2+1344*C*cos(1/2*d*x+1/
2*c)^10+159*cos(1/2*d*x+1/2*c)^2*C+2240*A*cos(1/2*d*x+1/2*c)^12-2684*C*cos(1/2*d*x+1/2*c)^8+6780*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
+14784*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))-2160*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)^7-4788*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))+340*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))*cos(1/2*d*x+1/2*c)^7+672*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))/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

Fricas [C] (verification not implemented)

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

Time = 0.16 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.37 \[ \int \frac {\cos ^{\frac {3}{2}}(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 (140 \, A \cos \left (d x + c\right )^{4} + 7 \, {\left (368 \, A - 111 \, B + 24 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (6259 \, A - 1968 \, B + 337 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (5548 \, A - 1761 \, B + 284 \, C\right )} \cos \left (d x + c\right ) + 1695 \, A - 540 \, B + 85 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, {\left (\sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (339 i \, A - 108 i \, B + 17 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 (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-339 i \, A + 108 i \, B - 17 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 (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (176 i \, A - 57 i \, B + 8 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 (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-176 i \, A + 57 i \, B - 8 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)^(3/2)*(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*(140*A*cos(d*x + c)^4 + 7*(368*A - 111*B + 24*C)*cos(d*x + c)^3 + (6259*A - 1968*B + 337*C)*cos(d*x +
c)^2 + (5548*A - 1761*B + 284*C)*cos(d*x + c) + 1695*A - 540*B + 85*C)*sqrt(cos(d*x + c))*sin(d*x + c) - 5*(s
qrt(2)*(339*I*A - 108*I*B + 17*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(339*I*A - 108*I*B + 17*I*C)*cos(d*x + c)^3 + 6
*sqrt(2)*(339*I*A - 108*I*B + 17*I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(339*I*A - 108*I*B + 17*I*C)*cos(d*x + c) + s
qrt(2)*(339*I*A - 108*I*B + 17*I*C))*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 5*(sqrt(2)*(-
339*I*A + 108*I*B - 17*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(-339*I*A + 108*I*B - 17*I*C)*cos(d*x + c)^3 + 6*sqrt(2
)*(-339*I*A + 108*I*B - 17*I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(-339*I*A + 108*I*B - 17*I*C)*cos(d*x + c) + sqrt(2
)*(-339*I*A + 108*I*B - 17*I*C))*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*(sqrt(2)*(176*
I*A - 57*I*B + 8*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(176*I*A - 57*I*B + 8*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(176*I*
A - 57*I*B + 8*I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(176*I*A - 57*I*B + 8*I*C)*cos(d*x + c) + sqrt(2)*(176*I*A - 57
*I*B + 8*I*C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*(sqrt(2)
*(-176*I*A + 57*I*B - 8*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(-176*I*A + 57*I*B - 8*I*C)*cos(d*x + c)^3 + 6*sqrt(2)
*(-176*I*A + 57*I*B - 8*I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(-176*I*A + 57*I*B - 8*I*C)*cos(d*x + c) + sqrt(2)*(-1
76*I*A + 57*I*B - 8*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. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**(3/2)*(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. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)^(3/2)*(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]

\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate(cos(d*x+c)^(3/2)*(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)*cos(d*x + c)^(3/2)/(a*sec(d*x + c) + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^4} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^4} \,d x \]

[In]

int((cos(c + d*x)^(3/2)*(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)^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^4, x)