3.1211 \(\int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=270 \[ \frac {8 a^4 (12 A+17 B+28 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {8 a^4 (19 A+24 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^4 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{105 d}+\frac {4 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{315 d}+\frac {2 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{21 d}+\frac {2 a (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}} \]

[Out]

8/15*a^4*(19*A+24*B+21*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+8/21*a^4*(12*A+17*B+28*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+2*C*(a+a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(1/2)+4/105*a^4*(73*A+83*B+7*C)*sin(d*x+c)*cos(d*x+c)^(
1/2)/d+2/9*a*(A-9*C)*(a+a*cos(d*x+c))^3*sin(d*x+c)*cos(d*x+c)^(1/2)/d+2/21*(5*A+3*B-21*C)*(a^2+a^2*cos(d*x+c))
^2*sin(d*x+c)*cos(d*x+c)^(1/2)/d+4/315*(86*A+81*B-126*C)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)*cos(d*x+c)^(1/2)/d

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Rubi [A]  time = 0.89, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4112, 3043, 2976, 2968, 3023, 2748, 2641, 2639} \[ \frac {8 a^4 (12 A+17 B+28 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {8 a^4 (19 A+24 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^4 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{105 d}+\frac {2 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{21 d}+\frac {4 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{315 d}+\frac {2 a (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(8*a^4*(19*A + 24*B + 21*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(12*A + 17*B + 28*C)*EllipticF[(c + d*x
)/2, 2])/(21*d) + (4*a^4*(73*A + 83*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(A - 9*C)*Sqrt[Co
s[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos
[c + d*x]]) + (2*(5*A + 3*B - 21*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (4*(8
6*A + 81*B - 126*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(315*d)

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 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 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 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 \cos ^{\frac {9}{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 {3}{2}}(c+d x)} \, dx\\ &=\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 \int \frac {(a+a \cos (c+d x))^4 \left (\frac {1}{2} a (B+8 C)+\frac {1}{2} a (A-9 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{a}\\ &=\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {4 \int \frac {(a+a \cos (c+d x))^3 \left (\frac {1}{4} a^2 (A+9 B+63 C)+\frac {3}{4} a^2 (5 A+3 B-21 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{9 a}\\ &=\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {8 \int \frac {(a+a \cos (c+d x))^2 \left (\frac {1}{4} a^3 (11 A+36 B+189 C)+\frac {1}{4} a^3 (86 A+81 B-126 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{63 a}\\ &=\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {16 \int \frac {(a+a \cos (c+d x)) \left (\frac {3}{8} a^4 (47 A+87 B+273 C)+\frac {9}{8} a^4 (73 A+83 B+7 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{315 a}\\ &=\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {16 \int \frac {\frac {3}{8} a^5 (47 A+87 B+273 C)+\left (\frac {9}{8} a^5 (73 A+83 B+7 C)+\frac {3}{8} a^5 (47 A+87 B+273 C)\right ) \cos (c+d x)+\frac {9}{8} a^5 (73 A+83 B+7 C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{315 a}\\ &=\frac {4 a^4 (73 A+83 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {32 \int \frac {\frac {45}{8} a^5 (12 A+17 B+28 C)+\frac {63}{8} a^5 (19 A+24 B+21 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{945 a}\\ &=\frac {4 a^4 (73 A+83 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {1}{15} \left (4 a^4 (19 A+24 B+21 C)\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (4 a^4 (12 A+17 B+28 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {8 a^4 (19 A+24 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (12 A+17 B+28 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^4 (73 A+83 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d}\\ \end {align*}

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Mathematica [C]  time = 6.80, size = 1742, normalized size = 6.45 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^(9/2)*(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)*(-1/1
20*((76*A + 96*B + 69*C + 76*A*Cos[2*c] + 96*B*Cos[2*c] + 99*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((204*A + 191*B +
112*C)*Cos[d*x]*Sin[c])/(336*d) + ((127*A + 72*B + 18*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((4*A + B)*Cos[3*d*x]*
Sin[3*c])/(112*d) + (A*Cos[4*d*x]*Sin[4*c])/(288*d) + ((204*A + 191*B + 112*C)*Cos[c]*Sin[d*x])/(336*d) + (C*S
ec[c]*Sec[c + d*x]*Sin[d*x])/(4*d) + ((127*A + 72*B + 18*C)*Cos[2*c]*Sin[2*d*x])/(720*d) + ((4*A + B)*Cos[3*c]
*Sin[3*d*x])/(112*d) + (A*Cos[4*c]*Sin[4*d*x])/(288*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]*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
]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (17*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 - 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*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]) - (19*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])/Sq
rt[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]]))/(30*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (
4*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 + 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])) - (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}, 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]]))/(10*d*(A + 2*C + 2*B*
Cos[c + d*x] + A*Cos[2*c + 2*d*x]))

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{6} + {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{5} + {\left (A + 4 \, B + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{4} + 2 \, {\left (2 \, A + 3 \, B + 2 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{3} + {\left (6 \, A + 4 \, B + C\right )} a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right )^{2} + {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{4} \sec \left (d x + c\right ) + A a^{4} \cos \left (d x + c\right )^{4}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(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)^4*sec(d*x + c)^6 + (B + 4*C)*a^4*cos(d*x + c)^4*sec(d*x + c)^5 + (A + 4*B + 6*C)*
a^4*cos(d*x + c)^4*sec(d*x + c)^4 + 2*(2*A + 3*B + 2*C)*a^4*cos(d*x + c)^4*sec(d*x + c)^3 + (6*A + 4*B + C)*a^
4*cos(d*x + c)^4*sec(d*x + c)^2 + (4*A + B)*a^4*cos(d*x + c)^4*sec(d*x + c) + A*a^4*cos(d*x + c)^4)*sqrt(cos(d
*x + c)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(9/2)*(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)^(9/2), x)

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maple [B]  time = 6.37, size = 786, normalized size = 2.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(9/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

-4/315*a^4*(-560*A*(-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)^
10+40*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(64*A+9*B)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)-
4*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(1177*A+387*B+63*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/
2*c)+28*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(161*A+96*B+39*C)*sin(1/2*d*x+1/2*c)^4*cos(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)*(227*A+167*B+133*C)*sin(1/2*d*x+1/2*c)^2*cos(1
/2*d*x+1/2*c)+360*A*(-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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-798*A*(-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)
)+510*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1008*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))+840*C*(si
n(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))*(-2*sin(1/2*d
*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)-882*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)*(-
2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))/(-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(cos(d*x+c)^(9/2)*(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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mupad [B]  time = 5.97, size = 548, normalized size = 2.03 \[ \frac {2\,\left (3\,A\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,A\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,A\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{3\,d}+\frac {2\,\left (4\,B\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+3\,B\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,B\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}-\frac {2\,\left (\frac {66\,A\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {17\,A\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{77\,d}+\frac {4\,C\,a^4\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {12\,C\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,C\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {8\,A\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {208\,A\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{385\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,B\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^(9/2)*(a + a/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

(2*(3*A*a^4*ellipticE(c/2 + (d*x)/2, 2) + 4*A*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*A*a^4*cos(c + d*x)^(1/2)*sin
(c + d*x)))/(3*d) + (2*(4*B*a^4*ellipticE(c/2 + (d*x)/2, 2) + 3*B*a^4*ellipticF(c/2 + (d*x)/2, 2) + 2*B*a^4*co
s(c + d*x)^(1/2)*sin(c + d*x)))/d - (2*((66*A*a^4*cos(c + d*x)^(7/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (1
7*A*a^4*cos(c + d*x)^(11/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2))*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2)
)/(77*d) + (4*C*a^4*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (12*C*a^4
*ellipticE(c/2 + (d*x)/2, 2))/d + (8*C*a^4*ellipticF(c/2 + (d*x)/2, 2))/d - (8*A*a^4*cos(c + d*x)^(9/2)*sin(c
+ d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (208*A*a^4*cos(c + d*x)^(11
/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 19/4, cos(c + d*x)^2))/(385*d*(sin(c + d*x)^2)^(1/2)) - (8*B*a^4*cos(c
+ d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*B*a^
4*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) +
(2*C*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
)) - (2*C*a^4*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(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(cos(d*x+c)**(9/2)*(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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