3.555 \(\int \frac {(a+a \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=271 \[ \frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (2 A+3 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]

[Out]

2/9*A*(a+a*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/21*(2*A+3*B)*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/a/d/se
c(d*x+c)^(5/2)+2/315*(73*A+99*B+63*C)*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+4/105*a^3*(32*A+41*B+
42*C)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+4/15*a^3*(17*A+21*B+27*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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+4/21*a^3*(11*A+13*B+21*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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/
d

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Rubi [A]  time = 0.65, antiderivative size = 271, 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 = {4086, 4017, 3996, 3787, 3771, 2639, 2641} \[ \frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {4 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {4 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 (2 A+3 B) \sin (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]

[Out]

(4*a^3*(17*A + 21*B + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(
11*A + 13*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(32*A + 4
1*B + 42*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c +
d*x]^(7/2)) + (2*(2*A + 3*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sec[c + d*x]^(5/2)) + (2*(73*A +
99*B + 63*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d*Sec[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 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3996

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.)
+ (A_)), x_Symbol] :> Simp[(A*a*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*n), x] + Dist[1/(d*n), Int[(d*Csc[e + f*x
])^(n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B},
x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]

Rule 4017

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(a*A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*n), x]
- Dist[b/(a*d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*(m - n - 1) - b*B*n - (a*
B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2
- b^2, 0] && GtQ[m, 1/2] && LtQ[n, -1]

Rule 4086

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*
Csc[e + f*x])^n)/(f*n), x] - Dist[1/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m -
b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 -
b^2, 0] &&  !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])

Rubi steps

\begin {align*} \int \frac {(a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \int \frac {(a+a \sec (c+d x))^3 \left (\frac {3}{2} a (2 A+3 B)+\frac {1}{2} a (A+9 C) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx}{9 a}\\ &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 \int \frac {(a+a \sec (c+d x))^2 \left (\frac {1}{4} a^2 (73 A+99 B+63 C)+\frac {1}{4} a^2 (13 A+9 B+63 C) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{63 a}\\ &=\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 \int \frac {(a+a \sec (c+d x)) \left (\frac {9}{4} a^3 (32 A+41 B+42 C)+\frac {3}{4} a^3 (23 A+24 B+63 C) \sec (c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{315 a}\\ &=\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {63}{8} a^4 (17 A+21 B+27 C)-\frac {45}{8} a^4 (11 A+13 B+21 C) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{945 a}\\ &=\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{21} \left (2 a^3 (11 A+13 B+21 C)\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (2 a^3 (17 A+21 B+27 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{21} \left (2 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (2 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {4 a^3 (17 A+21 B+27 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}\\ \end {align*}

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Mathematica [C]  time = 2.95, size = 214, normalized size = 0.79 \[ \frac {a^3 \sqrt {\sec (c+d x)} \left (-224 i (17 A+21 B+27 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+2 \cos (c+d x) (30 (97 A+107 B+84 C) \sin (c+d x)+14 (73 A+54 B+18 C) \sin (2 (c+d x))+270 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+5712 i A+90 B \sin (3 (c+d x))+7056 i B+9072 i C)+480 (11 A+13 B+21 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{2520 d} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]

[Out]

(a^3*Sqrt[Sec[c + d*x]]*(480*(11*A + 13*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (224*I)*(17*A
+ 21*B + 27*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c +
d*x))] + 2*Cos[c + d*x]*((5712*I)*A + (7056*I)*B + (9072*I)*C + 30*(97*A + 107*B + 84*C)*Sin[c + d*x] + 14*(73
*A + 54*B + 18*C)*Sin[2*(c + d*x)] + 270*A*Sin[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))
/(2520*d)

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fricas [F]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C a^{3} \sec \left (d x + c\right )^{5} + {\left (B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{4} + {\left (A + 3 \, B + 3 \, C\right )} a^{3} \sec \left (d x + c\right )^{3} + {\left (3 \, A + 3 \, B + C\right )} a^{3} \sec \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \sec \left (d x + c\right ) + A a^{3}}{\sec \left (d x + c\right )^{\frac {9}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^3/sec(d*x + c)^(9/2), x)

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maple [A]  time = 4.79, size = 514, normalized size = 1.90 \[ -\frac {4 \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 )}\, a^{3} \left (-560 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2200 A +360 B \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-3412 A -1296 B -252 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (2702 A +1806 B +882 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-738 A -624 B -378 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+165 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )-357 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )+195 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )-441 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )+315 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )-567 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )\right )}{315 \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} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x)

[Out]

-4/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-560*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*
c)^10+(2200*A+360*B)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-3412*A-1296*B-252*C)*sin(1/2*d*x+1/2*c)^6*cos(1
/2*d*x+1/2*c)+(2702*A+1806*B+882*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-738*A-624*B-378*C)*sin(1/2*d*x+1
/2*c)^2*cos(1/2*d*x+1/2*c)+165*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))-357*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))+195*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))-441*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)*EllipticE(cos(1/2*d*x+1/2*c
),2^(1/2))+315*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))-567*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
)))/(-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))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(9/2),x)

[Out]

int(((a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/cos(c + d*x))^(9/2), x)

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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))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(9/2),x)

[Out]

Timed out

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