3.1213 \(\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=267 \[ \frac{8 a^4 (4 A+5 B+4 C) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 d}+\frac{4 a^4 (A-25 B-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (5 A+15 B+19 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d \sqrt{\cos (c+d x)}}-\frac{4 (6 A+25 B+34 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{15 d}+\frac{56 a^4 (A-C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a (5 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]

[Out]

(56*a^4*(A - C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^4*(4*A + 5*B + 4*C)*EllipticF[(c + d*x)/2, 2])/(3*d) +
(4*a^4*(A - 25*B - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(5*B + 8*C)*(a + a*Cos[c + d*x])^3*Si
n[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (
2*(5*A + 15*B + 19*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - (4*(6*A + 25*B + 34*
C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(15*d)

________________________________________________________________________________________

Rubi [A]  time = 0.874518, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.209, Rules used = {4112, 3043, 2975, 2976, 2968, 3023, 2748, 2641, 2639} \[ \frac{8 a^4 (4 A+5 B+4 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{4 a^4 (A-25 B-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (5 A+15 B+19 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d \sqrt{\cos (c+d x)}}-\frac{4 (6 A+25 B+34 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{15 d}+\frac{56 a^4 (A-C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a (5 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(56*a^4*(A - C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^4*(4*A + 5*B + 4*C)*EllipticF[(c + d*x)/2, 2])/(3*d) +
(4*a^4*(A - 25*B - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(5*B + 8*C)*(a + a*Cos[c + d*x])^3*Si
n[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (
2*(5*A + 15*B + 19*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - (4*(6*A + 25*B + 34*
C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(15*d)

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 3043

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[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(b*d*(n + 1)
*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C -
B*d)*(a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2,
0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])

Rule 2975

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*S
in[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c + a*d)), x] - Dist[b/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])
^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n +
1) - B*(b*c*m - a*d*(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] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n]
|| EqQ[c, 0])

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

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 \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \frac{(a+a \cos (c+d x))^4 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \cos (c+d x))^4 \left (\frac{1}{2} a (5 B+8 C)+\frac{5}{2} a (A-C) \cos (c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{5 a}\\ &=\frac{2 a (5 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 \int \frac{(a+a \cos (c+d x))^3 \left (\frac{3}{4} a^2 (5 A+15 B+19 C)+\frac{5}{4} a^2 (3 A-5 B-11 C) \cos (c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{15 a}\\ &=\frac{2 a (5 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (5 A+15 B+19 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 \int \frac{(a+a \cos (c+d x))^2 \left (\frac{5}{4} a^3 (9 A+20 B+23 C)-\frac{5}{4} a^3 (6 A+25 B+34 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{15 a}\\ &=\frac{2 a (5 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (5 A+15 B+19 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{4 (6 A+25 B+34 C) \sqrt{\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{16 \int \frac{(a+a \cos (c+d x)) \left (\frac{15}{8} a^4 (13 A+25 B+27 C)+\frac{15}{8} a^4 (A-25 B-41 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{75 a}\\ &=\frac{2 a (5 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (5 A+15 B+19 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{4 (6 A+25 B+34 C) \sqrt{\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{16 \int \frac{\frac{15}{8} a^5 (13 A+25 B+27 C)+\left (\frac{15}{8} a^5 (A-25 B-41 C)+\frac{15}{8} a^5 (13 A+25 B+27 C)\right ) \cos (c+d x)+\frac{15}{8} a^5 (A-25 B-41 C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{75 a}\\ &=\frac{4 a^4 (A-25 B-41 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a (5 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (5 A+15 B+19 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{4 (6 A+25 B+34 C) \sqrt{\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{32 \int \frac{\frac{75}{8} a^5 (4 A+5 B+4 C)+\frac{315}{8} a^5 (A-C) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{225 a}\\ &=\frac{4 a^4 (A-25 B-41 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a (5 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (5 A+15 B+19 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{4 (6 A+25 B+34 C) \sqrt{\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{5} \left (28 a^4 (A-C)\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (4 a^4 (4 A+5 B+4 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{56 a^4 (A-C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{8 a^4 (4 A+5 B+4 C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{4 a^4 (A-25 B-41 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 a (5 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (5 A+15 B+19 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{4 (6 A+25 B+34 C) \sqrt{\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}\\ \end{align*}

Mathematica [C]  time = 7.07729, size = 1449, normalized size = 5.43 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-((2
3*A - 20*B - 61*C + 33*A*Cos[2*c] + 20*B*Cos[2*c] + 5*C*Cos[2*c])*Csc[c]*Sec[c])/(40*d) + ((4*A + B)*Cos[d*x]*
Sin[c])/(12*d) + (A*Cos[2*d*x]*Sin[2*c])/(40*d) + ((4*A + B)*Cos[c]*Sin[d*x])/(12*d) + (C*Sec[c]*Sec[c + d*x]^
3*Sin[d*x])/(20*d) + (Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] + 5*B*Sin[d*x] + 20*C*Sin[d*x]))/(60*d) + (Sec[c]*Sec[
c + d*x]*(5*B*Sin[c] + 20*C*Sin[c] + 15*A*Sin[d*x] + 60*B*Sin[d*x] + 99*C*Sin[d*x]))/(60*d) + (A*Cos[2*c]*Sin[
2*d*x])/(40*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^6*Csc[c]*Hypergeometric
PFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(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]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] +
A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin
[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(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 - ArcTa
n[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[
1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*
Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(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]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (7*A*Cos[
c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Hyperg
eometricPFQ[{-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 + Ar
cTan[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]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (7*C*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2
]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Co
s[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]]))/(10*d*(A + 2*C + 2*B*Cos[c +
d*x] + A*Cos[2*c + 2*d*x]))

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Maple [B]  time = 9.332, size = 1214, normalized size = 4.6 \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)^(5/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

8/15*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*
c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^3*(-24*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+128*A*sin(
1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+20*B*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+140*B*cos(1/2*d*x+1/2*c)*sin(
1/2*d*x+1/2*c)^4-186*A*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+25*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*
x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+20*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2
*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+21*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2
-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+20*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(
1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-21*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*
EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-35*B*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-80*A*(sin(1/2*d*x+1/2*c)^2)
^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*c)^2-61*sin(1/2*
d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)*C+84*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*
(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2-100*B*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x
+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2-84*C*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellip
ticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2-80*C*(2*sin(1/2*d*x+1/2*c)^
2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+80*C*(2*sin
(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*sin(1/2*d*x+1/2*
c)^4+80*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*
sin(1/2*d*x+1/2*c)^4-84*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+
1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4+100*B*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/
2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4+84*C*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*
d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4+102*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2
*c)^4-19*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)*A+218*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4-150*B*cos(1/2
*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-198*C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1
/2*d*x+1/2*c)^2)^(1/2)/(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)^(5/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{6} +{\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{5} +{\left (A + 4 \, B + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{4} + 2 \,{\left (2 \, A + 3 \, B + 2 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{3} +{\left (6 \, A + 4 \, B + C\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} +{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + A a^{4} \cos \left (d x + c\right )^{2}\right )} \sqrt{\cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*a^4*cos(d*x + c)^2*sec(d*x + c)^6 + (B + 4*C)*a^4*cos(d*x + c)^2*sec(d*x + c)^5 + (A + 4*B + 6*C)*
a^4*cos(d*x + c)^2*sec(d*x + c)^4 + 2*(2*A + 3*B + 2*C)*a^4*cos(d*x + c)^2*sec(d*x + c)^3 + (6*A + 4*B + C)*a^
4*cos(d*x + c)^2*sec(d*x + c)^2 + (4*A + B)*a^4*cos(d*x + c)^2*sec(d*x + c) + A*a^4*cos(d*x + c)^2)*sqrt(cos(d
*x + c)), 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)**(5/2)*(a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^4*cos(d*x + c)^(5/2), x)