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

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

[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

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Rubi [A]
time = 0.57, antiderivative size = 278, normalized size of antiderivative = 1.00, 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 = {4197, 3120, 3056, 2827, 2719, 2715, 2720} \begin {gather*} \frac {(339 A-108 B+17 C) F\left (\left .\frac {1}{2} (c+d x)\right |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} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 ^{\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) F\left (\left .\frac {1}{2} (c+d x)\right |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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 7.72, size = 2319, normalized size = 8.34 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[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]

(((-352*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) + (((114*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*S
ec[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*Hype
rgeometric2F1[-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)*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*Si
n[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) - (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]]]]*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*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 -
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 - Ar
cTan[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] - 2
0*B*Cos[c])*Csc[c])/(5*d) + (64*A*Cos[d*x]*Sin[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) + (16*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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(679\) vs. \(2(306)=612\).
time = 0.23, size = 680, normalized size = 2.45

method	result	size
default	\(-\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-1882 A \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-706 C \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+243 A \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+159 C \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 C -15 A +15 B -2684 C \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-25588 A \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6216 B \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1224 B \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-201 B \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2240 A \left (\cos ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1902 C \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12234 A \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10776 B \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5598 B \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12768 A \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1344 C \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+672 C \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+6780 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+14784 A \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \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 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4788 B \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \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 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{840 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(680\)

Verification of antiderivative is not currently implemented for this CAS.

[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+1344*C*cos(1/2*d*x+1/2*c)^10+2
240*A*cos(1/2*d*x+1/2*c)^12-6216*B*cos(1/2*d*x+1/2*c)^10+12768*A*cos(1/2*d*x+1/2*c)^10-25588*A*cos(1/2*d*x+1/2
*c)^8+10776*B*cos(1/2*d*x+1/2*c)^8-2684*C*cos(1/2*d*x+1/2*c)^8+12234*A*cos(1/2*d*x+1/2*c)^6+1902*C*cos(1/2*d*x
+1/2*c)^6-1882*A*cos(1/2*d*x+1/2*c)^4+1224*B*cos(1/2*d*x+1/2*c)^4-706*C*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+159*C*cos(1/2*d*x+1/2*c)^2-5598*B*cos(1/2*d*x+1/2*c)^6-2160*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))+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)))/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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.14, size = 659, normalized size = 2.37 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: SystemError >> excessive stack use: stack is 3006 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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