3.1216 \(\int \frac {(a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sqrt {\cos (c+d x)}} \, dx\)

Optimal. Leaf size=310 \[ \frac {8 a^4 (187 A+132 B+113 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {8 a^4 (24 A+19 B+16 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 (33 A+55 B+43 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]

[Out]

-8/15*a^4*(24*A+19*B+16*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
))/d+8/231*a^4*(187*A+132*B+113*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))/d+4/1155*a^4*(913*A+803*B+667*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/99*a*(11*B+8*C)*(a+a*cos(d*x+c))^3
*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/11*C*(a+a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(11/2)+2/231*(33*A+55*B+43*C)
*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(7/2)+4/3465*(891*A+946*B+769*C)*(a^4+a^4*cos(d*x+c))*sin(d*x+
c)/d/cos(d*x+c)^(5/2)+8/15*a^4*(24*A+19*B+16*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2)

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Rubi [A]  time = 0.92, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {4112, 3043, 2975, 2968, 3021, 2748, 2636, 2639, 2641} \[ \frac {8 a^4 (187 A+132 B+113 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {8 a^4 (24 A+19 B+16 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*a^4*(24*A + 19*B + 16*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(187*A + 132*B + 113*C)*EllipticF[(c +
d*x)/2, 2])/(231*d) + (4*a^4*(913*A + 803*B + 667*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(3/2)) + (8*a^4*(24*A
+ 19*B + 16*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*(11*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x
])/(99*d*Cos[c + d*x]^(9/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (2*(33*A
+ 55*B + 43*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (4*(891*A + 946*B + 769*
C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2))

Rule 2636

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

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]

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 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 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 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 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

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 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]

Rubi steps

\begin {align*} \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, 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 {13}{2}}(c+d x)} \, dx\\ &=\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \cos (c+d x))^4 \left (\frac {1}{2} a (11 B+8 C)+\frac {1}{2} a (11 A+C) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \cos (c+d x))^3 \left (\frac {3}{4} a^2 (33 A+55 B+43 C)+\frac {1}{4} a^2 (99 A+11 B+17 C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {8 \int \frac {(a+a \cos (c+d x))^2 \left (\frac {1}{4} a^3 (891 A+946 B+769 C)+\frac {1}{4} a^3 (396 A+121 B+124 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 \int \frac {(a+a \cos (c+d x)) \left (\frac {9}{8} a^4 (913 A+803 B+667 C)+\frac {3}{8} a^4 (957 A+517 B+463 C) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{3465 a}\\ &=\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 \int \frac {\frac {9}{8} a^5 (913 A+803 B+667 C)+\left (\frac {3}{8} a^5 (957 A+517 B+463 C)+\frac {9}{8} a^5 (913 A+803 B+667 C)\right ) \cos (c+d x)+\frac {3}{8} a^5 (957 A+517 B+463 C) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{3465 a}\\ &=\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {32 \int \frac {\frac {693}{8} a^5 (24 A+19 B+16 C)+\frac {45}{8} a^5 (187 A+132 B+113 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{10395 a}\\ &=\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{15} \left (4 a^4 (24 A+19 B+16 C)\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx+\frac {1}{231} \left (4 a^4 (187 A+132 B+113 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {8 a^4 (187 A+132 B+113 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {1}{15} \left (4 a^4 (24 A+19 B+16 C)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {8 a^4 (24 A+19 B+16 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (187 A+132 B+113 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)}\\ \end {align*}

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Mathematica [C]  time = 7.35, size = 1795, normalized size = 5.79 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],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)*(((24
*A + 19*B + 16*C)*Csc[c]*Sec[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^6*Sin[d*x])/(44*d) + (Sec[c]*Sec[c + d*x]^5*(
9*C*Sin[c] + 11*B*Sin[d*x] + 44*C*Sin[d*x]))/(396*d) + (Sec[c]*Sec[c + d*x]^4*(77*B*Sin[c] + 308*C*Sin[c] + 99
*A*Sin[d*x] + 396*B*Sin[d*x] + 675*C*Sin[d*x]))/(2772*d) + (Sec[c]*Sec[c + d*x]^3*(495*A*Sin[c] + 1980*B*Sin[c
] + 3375*C*Sin[c] + 2772*A*Sin[d*x] + 4697*B*Sin[d*x] + 4928*C*Sin[d*x]))/(13860*d) + (Sec[c]*Sec[c + d*x]*(25
85*A*Sin[c] + 2640*B*Sin[c] + 2260*C*Sin[c] + 7392*A*Sin[d*x] + 5852*B*Sin[d*x] + 4928*C*Sin[d*x]))/(4620*d) +
(Sec[c]*Sec[c + d*x]^2*(2772*A*Sin[c] + 4697*B*Sin[c] + 4928*C*Sin[c] + 7755*A*Sin[d*x] + 7920*B*Sin[d*x] + 6
780*C*Sin[d*x]))/(13860*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (17*A*Cos[c + d*x]^6*Csc[c]*H
ypergeometricPFQ[{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[-(Sq
rt[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]) - (4*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*Se
c[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]*S
in[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]) - (113*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - Arc
Tan[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]]]])/(231*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[
c]^2]) + (4*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)*((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])) + (19*B*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}, 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 + T
an[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]]]*Sqr
t[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d*(A +
2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (8*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}, Cos[d*x + ArcTan[Ta
n[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]]))/(15*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c
+ 2*d*x]))

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C a^{4} \sec \left (d x + c\right )^{6} + {\left (B + 4 \, C\right )} a^{4} \sec \left (d x + c\right )^{5} + {\left (A + 4 \, B + 6 \, C\right )} a^{4} \sec \left (d x + c\right )^{4} + 2 \, {\left (2 \, A + 3 \, B + 2 \, C\right )} a^{4} \sec \left (d x + c\right )^{3} + {\left (6 \, A + 4 \, B + C\right )} a^{4} \sec \left (d x + c\right )^{2} + {\left (4 \, A + B\right )} a^{4} \sec \left (d x + c\right ) + A a^{4}}{\sqrt {\cos \left (d x + c\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 24.42, size = 1505, normalized size = 4.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*(1/16*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)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2
^(1/2))+(1/16*A+1/4*B+3/8*C)*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-
1/2+cos(1/2*d*x+1/2*c)^2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2
+cos(1/2*d*x+1/2*c)^2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1
/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+(3/8*A+1/4*B+1/16*C)*(-1/6*cos(1/2*
d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+1/3*(sin(1/2*d*x
+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipt
icF(cos(1/2*d*x+1/2*c),2^(1/2)))+(1/4*A+1/16*B)*(-(-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)^(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))+2*(-2*sin(1/2*d*x
+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2
*d*x+1/2*c)^2-1)+1/16*C*(-1/352*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+
cos(1/2*d*x+1/2*c)^2)^6-9/616*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+co
s(1/2*d*x+1/2*c)^2)^4-15/154*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos
(1/2*d*x+1/2*c)^2)^2+15/77*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*
c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+(1/16*B+1/4*C)*(-1/144*cos(1/2*d*x+1/2
*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^5-7/180*cos(1/2*d*x+1/2*c
)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^3-14/15*sin(1/2*d*x+1/2*c)^
2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/
2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*
x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4
+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))))-1/
5*(1/4*A+3/8*B+1/4*C)/(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)^2*(12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)
*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*s
in(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+24*sin(1/2*d*x+1/2*c)^4*cos(1/2
*d*x+1/2*c)+3*(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))-8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2))/sin(1/2*d
*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 10.13, size = 830, normalized size = 2.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*((11*B*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (3*B*a^4*sin(c + d*x))/(cos(c + d*x)
^(7/2)*(sin(c + d*x)^2)^(1/2)))*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(21*d) - (8*((61*B*a^4*sin(c + d*
x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (5*B*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^
(1/2)))*hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2))/(135*d) + (8*((13*C*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)
*(sin(c + d*x)^2)^(1/2)) + (5*C*a^4*sin(c + d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2)))*hypergeom([-5/4
, 1/2], -1/4, cos(c + d*x)^2))/(45*d) + (8*((75*C*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)
) + (7*C*a^4*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)))*hypergeom([-3/4, 1/2], 5/4, cos(c + d*
x)^2))/(231*d) + (2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((289*B*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*
(sin(c + d*x)^2)^(1/2)) + (66*B*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (5*B*a^4*sin(c
+ d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2))))/(45*d) + (2*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)
*((377*C*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (218*C*a^4*sin(c + d*x))/(cos(c + d*x
)^(7/2)*(sin(c + d*x)^2)^(1/2)) + (21*C*a^4*sin(c + d*x))/(cos(c + d*x)^(11/2)*(sin(c + d*x)^2)^(1/2))))/(231*
d) + (2*A*a^4*ellipticF(c/2 + (d*x)/2, 2))/d + (8*A*a^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^
2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (4*A*a^4*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c +
d*x)^2))/(d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (8*A*a^4*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, co
s(c + d*x)^2))/(5*d*cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (2*A*a^4*sin(c + d*x)*hypergeom([-7/4, 1/2],
-3/4, cos(c + d*x)^2))/(7*d*cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)) + (32*B*a^4*sin(c + d*x)*hypergeom([-3/
4, 1/2], 5/4, cos(c + d*x)^2))/(21*d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) - (32*C*a^4*sin(c + d*x)*hyper
geom([-5/4, 1/2], 3/4, cos(c + d*x)^2))/(15*d*cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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