3.1215 \(\int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=274 \[ \frac {8 a^4 (28 A+17 B+12 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {8 a^4 (21 A+24 B+19 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 (21 A+24 B+19 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{45 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^4 (287 A+253 B+193 C) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}}+\frac {2 (63 A+117 B+97 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]

[Out]

-8/15*a^4*(21*A+24*B+19*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*(28*A+17*B+12*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+2/63*a*(9*B+8*C)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/9*C*(a+a*cos(d*x+c))^4*sin(d*x+c)
/d/cos(d*x+c)^(9/2)+2/315*(63*A+117*B+97*C)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(5/2)+4/45*(21*A+24
*B+19*C)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d/cos(d*x+c)^(3/2)+4/105*a^4*(287*A+253*B+193*C)*sin(d*x+c)/d/cos(d*x
+c)^(1/2)

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Rubi [A]  time = 0.89, antiderivative size = 274, 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, 2975, 2968, 3021, 2748, 2641, 2639} \[ \frac {8 a^4 (28 A+17 B+12 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {8 a^4 (21 A+24 B+19 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (63 A+117 B+97 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 (21 A+24 B+19 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{45 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^4 (287 A+253 B+193 C) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}}+\frac {2 a (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*a^4*(21*A + 24*B + 19*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(28*A + 17*B + 12*C)*EllipticF[(c + d*
x)/2, 2])/(21*d) + (4*a^4*(287*A + 253*B + 193*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*a*(9*B + 8*C)*
(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(9*
d*Cos[c + d*x]^(9/2)) + (2*(63*A + 117*B + 97*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^
(5/2)) + (4*(21*A + 24*B + 19*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(45*d*Cos[c + d*x]^(3/2))

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 \sqrt {\cos (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 {11}{2}}(c+d x)} \, dx\\ &=\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \cos (c+d x))^4 \left (\frac {1}{2} a (9 B+8 C)+\frac {1}{2} a (9 A-C) \cos (c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac {2 a (9 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \cos (c+d x))^3 \left (\frac {1}{4} a^2 (63 A+117 B+97 C)+\frac {3}{4} a^2 (21 A-3 B-5 C) \cos (c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac {2 a (9 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (63 A+117 B+97 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {8 \int \frac {(a+a \cos (c+d x))^2 \left (\frac {21}{4} a^3 (21 A+24 B+19 C)+\frac {1}{4} a^3 (126 A-81 B-86 C) \cos (c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac {2 a (9 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (63 A+117 B+97 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 (21 A+24 B+19 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{45 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {16 \int \frac {(a+a \cos (c+d x)) \left (\frac {9}{8} a^4 (287 A+253 B+193 C)-\frac {9}{8} a^4 (7 A+83 B+73 C) \cos (c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{945 a}\\ &=\frac {2 a (9 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (63 A+117 B+97 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 (21 A+24 B+19 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{45 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {16 \int \frac {\frac {9}{8} a^5 (287 A+253 B+193 C)+\left (-\frac {9}{8} a^5 (7 A+83 B+73 C)+\frac {9}{8} a^5 (287 A+253 B+193 C)\right ) \cos (c+d x)-\frac {9}{8} a^5 (7 A+83 B+73 C) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{945 a}\\ &=\frac {4 a^4 (287 A+253 B+193 C) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}}+\frac {2 a (9 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (63 A+117 B+97 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 (21 A+24 B+19 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{45 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {32 \int \frac {\frac {45}{8} a^5 (28 A+17 B+12 C)-\frac {63}{8} a^5 (21 A+24 B+19 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{945 a}\\ &=\frac {4 a^4 (287 A+253 B+193 C) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}}+\frac {2 a (9 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (63 A+117 B+97 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 (21 A+24 B+19 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{45 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{21} \left (4 a^4 (28 A+17 B+12 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx-\frac {1}{15} \left (4 a^4 (21 A+24 B+19 C)\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {8 a^4 (21 A+24 B+19 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (28 A+17 B+12 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^4 (287 A+253 B+193 C) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}}+\frac {2 a (9 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (63 A+117 B+97 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 (21 A+24 B+19 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{45 d \cos ^{\frac {3}{2}}(c+d x)}\\ \end {align*}

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Mathematica [C]  time = 7.26, size = 1748, normalized size = 6.38 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[Cos[c + d*x]]*(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*((-183*A - 192*B - 152*C + 15*A*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(36*d) + (S
ec[c]*Sec[c + d*x]^4*(7*C*Sin[c] + 9*B*Sin[d*x] + 36*C*Sin[d*x]))/(252*d) + (Sec[c]*Sec[c + d*x]^3*(45*B*Sin[c
] + 180*C*Sin[c] + 63*A*Sin[d*x] + 252*B*Sin[d*x] + 427*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]*(140*A*Si
n[c] + 235*B*Sin[c] + 240*C*Sin[c] + 693*A*Sin[d*x] + 672*B*Sin[d*x] + 532*C*Sin[d*x]))/(420*d) + (Sec[c]*Sec[
c + d*x]^2*(63*A*Sin[c] + 252*B*Sin[c] + 427*C*Sin[c] + 420*A*Sin[d*x] + 705*B*Sin[d*x] + 720*C*Sin[d*x]))/(12
60*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]]]])/(3*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 - A
rcTan[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 - A
rcTan[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]) + (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)*((Hypergeometri
cPFQ[{-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 + A
rcTan[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[Ta
n[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])) + (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 + A
rcTan[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*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 + ArcTa
n[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]]))/(30*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))

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fricas [F]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C a^{4} \sec \left (d x + c\right )^{6} + {\left (B + 4 \, C\right )} a^{4} \sec \left (d x + c\right )^{5} + {\left (A + 4 \, B + 6 \, C\right )} a^{4} \sec \left (d x + c\right )^{4} + 2 \, {\left (2 \, A + 3 \, B + 2 \, C\right )} a^{4} \sec \left (d x + c\right )^{3} + {\left (6 \, A + 4 \, B + C\right )} a^{4} \sec \left (d x + c\right )^{2} + {\left (4 \, A + B\right )} a^{4} \sec \left (d x + c\right ) + A a^{4}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(1/2)*(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*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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(1/2)*(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*sqrt(cos(d*x + c)), x)

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maple [B]  time = 23.18, size = 1427, normalized size = 5.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(1/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^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))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))+3/16*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*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)/(-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)*(-1/56*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)))+(1/4*A+3/
8*B+1/4*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+(3/8*A+1/4*B+1/16*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)^(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*(-1/144*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)*(El
lipticF(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*(1/16*A+1/4*B+3/8*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*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(s
in(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(cos(d*x+c)^(1/2)*(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 = 10.77, size = 724, normalized size = 2.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(A*a^4*ellipticE(c/2 + (d*x)/2, 2) + 4*A*a^4*ellipticF(c/2 + (d*x)/2, 2)))/d + (2*((34*A*a^4*sin(c + d*x))/
(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (A*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2))
)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(5*d) + (8*((11*C*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c
+ d*x)^2)^(1/2)) + (3*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],
1/4, cos(c + d*x)^2))/(21*d) - (8*((61*C*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (5*C*
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))/(1
35*d) + (2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((289*C*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c +
d*x)^2)^(1/2)) + (66*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))))/(45*d) + (2*B*a^4*ellipticF(c/2 + (d*x)/2, 2))/d + (8*A*a^4*sin(
c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(3*d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) - (8*A*a
^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2))/(15*d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2))
+ (8*B*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*B*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*B*a^4*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/(5*d*cos(c + d*x)^(5/2)*(sin(c
+ d*x)^2)^(1/2)) + (2*B*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*C*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))

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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)**(1/2)*(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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