3.6.71 \(\int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^3} \, dx\) [571]

Optimal. Leaf size=308 \[ \frac {(9 A-49 B+119 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(9 A-49 B+119 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )} \]

[Out]

1/6*(3*A-13*B+33*C)*sec(d*x+c)^(3/2)*sin(d*x+c)/a^3/d-1/5*(A-B+C)*sec(d*x+c)^(9/2)*sin(d*x+c)/d/(a+a*sec(d*x+c
))^3+1/3*(B-2*C)*sec(d*x+c)^(7/2)*sin(d*x+c)/a/d/(a+a*sec(d*x+c))^2-1/30*(9*A-49*B+119*C)*sec(d*x+c)^(5/2)*sin
(d*x+c)/d/(a^3+a^3*sec(d*x+c))-1/10*(9*A-49*B+119*C)*sin(d*x+c)*sec(d*x+c)^(1/2)/a^3/d+1/10*(9*A-49*B+119*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)/a^3/d+1/6*(3*A-13*B+33*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)/a^3/d

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Rubi [A]
time = 0.42, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4169, 4104, 3872, 3853, 3856, 2719, 2720} \begin {gather*} -\frac {(9 A-49 B+119 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{30 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac {(3 A-13 B+33 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{6 a^3 d}-\frac {(9 A-49 B+119 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac {(9 A-49 B+119 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {(A-B+C) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac {(B-2 C) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]

[Out]

((9*A - 49*B + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13
*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((9*A - 49*B + 119*C)*
Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((3*A - 13*B + 33*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d)
- ((A - B + C)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((B - 2*C)*Sec[c + d*x]^(7/2)*S
in[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - ((9*A - 49*B + 119*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^
3 + a^3*Sec[c + d*x]))

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3853

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

Rule 3856

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 3872

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 4104

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

Rule 4169

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*A - b*B + a*C))*Cot[e + f*x]*(a + b*C
sc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Csc[e + f*x])^(m
+ 1)*(d*Csc[e + f*x])^n*Simp[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(m + n + 1) - a*(A*(m + n + 1) - C*(m -
n)))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]

Rubi steps

\begin {align*} \int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx &=-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {\int \frac {\sec ^{\frac {7}{2}}(c+d x) \left (\frac {1}{2} a (3 A+7 B-7 C)+\frac {1}{2} a (3 A-3 B+13 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (\frac {25}{2} a^2 (B-2 C)+\frac {3}{2} a^2 (3 A-8 B+23 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \sec ^{\frac {3}{2}}(c+d x) \left (-\frac {3}{4} a^3 (9 A-49 B+119 C)+\frac {15}{4} a^3 (3 A-13 B+33 C) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(3 A-13 B+33 C) \int \sec ^{\frac {5}{2}}(c+d x) \, dx}{4 a^3}-\frac {(9 A-49 B+119 C) \int \sec ^{\frac {3}{2}}(c+d x) \, dx}{20 a^3}\\ &=-\frac {(9 A-49 B+119 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {(3 A-13 B+33 C) \int \sqrt {\sec (c+d x)} \, dx}{12 a^3}+\frac {(9 A-49 B+119 C) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}\\ &=-\frac {(9 A-49 B+119 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\left ((3 A-13 B+33 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{12 a^3}+\frac {\left ((9 A-49 B+119 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}\\ &=\frac {(9 A-49 B+119 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}-\frac {(9 A-49 B+119 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac {(3 A-13 B+33 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {(A-B+C) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(B-2 C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}-\frac {(9 A-49 B+119 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 8.86, size = 1462, normalized size = 4.75 \begin {gather*} -\frac {6 \sqrt {2} A e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )\right ) \sec \left (\frac {c}{2}\right ) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {98 \sqrt {2} B e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )\right ) \sec \left (\frac {c}{2}\right ) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{15 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}-\frac {238 \sqrt {2} C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )\right ) \sec \left (\frac {c}{2}\right ) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{15 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {4 A \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sec \left (\frac {c}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (c)}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}-\frac {52 B \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sec \left (\frac {c}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (c)}{3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {44 C \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \csc \left (\frac {c}{2}\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sec \left (\frac {c}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sin (c)}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3}+\frac {\cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {4 (9 A-49 B+119 C) \cos (d x) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right )}{5 d}+\frac {4 \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )-B \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{5 d}+\frac {8 \sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3 A \sin \left (\frac {d x}{2}\right )-8 B \sin \left (\frac {d x}{2}\right )+13 C \sin \left (\frac {d x}{2}\right )\right )}{15 d}+\frac {8 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3 A \sin \left (\frac {d x}{2}\right )-13 B \sin \left (\frac {d x}{2}\right )+29 C \sin \left (\frac {d x}{2}\right )\right )}{3 d}+\frac {32 C \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac {8 (4 C+3 A \cos (c)-13 B \cos (c)+33 C \cos (c)) \sec (c) \tan \left (\frac {c}{2}\right )}{3 d}+\frac {8 (3 A-8 B+13 C) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{15 d}+\frac {4 (A-B+C) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{5 d}\right )}{(A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]

[Out]

(-6*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]
^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7
/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A +
2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (98*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 +
E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c +
d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec
[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2
*d*x])*(a + a*Sec[c + d*x])^3) - (238*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2
*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)
*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec
[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*A
*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B
*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c +
d*x])^3) - (52*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d
*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x
])*(a + a*Sec[c + d*x])^3) + (44*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*
Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A
*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*
Sec[c + d*x]^2)*((-4*(9*A - 49*B + 119*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5
*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(3*A*Sin[(d*x)/2
] - 8*B*Sin[(d*x)/2] + 13*C*Sin[(d*x)/2]))/(15*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] - 13*B*Si
n[(d*x)/2] + 29*C*Sin[(d*x)/2]))/(3*d) + (32*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (8*(4*C + 3*A*Cos[c] - 13
*B*Cos[c] + 33*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d) + (8*(3*A - 8*B + 13*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d)
+ (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a
+ a*Sec[c + d*x])^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1012\) vs. \(2(328)=656\).
time = 0.17, size = 1013, normalized size = 3.29

method	result	size
default	\(\text {Expression too large to display}\)	\(1013\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/4*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/a^3*((-4*B+12*C)*(cos(1/2*d*x+1/2*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))-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)/cos(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^4+s
in(1/2*d*x+1/2*c)^2)^(1/2)+(A-B+C)*(1/5*(-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
)^5+4/5*(-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)^3+18/5*(-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)-8/5*(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))+18/5*(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)*(E
llipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))))+8*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^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)))+1/3*(4*C-2*B)*(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)*(2*EllipticF(c
os(1/2*d*x+1/2*c),2^(1/2))-3*EllipticE(cos(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*
(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*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3*Ellip
ticE(cos(1/2*d*x+1/2*c),2^(1/2)))*cos(1/2*d*x+1/2*c)-12*sin(1/2*d*x+1/2*c)^6+20*sin(1/2*d*x+1/2*c)^4-7*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)^2-1
)+(8*B-24*C)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1
/2)*(2*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)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*E
llipticE(cos(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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.07, size = 593, normalized size = 1.93 \begin {gather*} -\frac {5 \, {\left (\sqrt {2} {\left (3 i \, A - 13 i \, B + 33 i \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (3 i \, A - 13 i \, B + 33 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (3 i \, A - 13 i \, B + 33 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (3 i \, A - 13 i \, B + 33 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (-3 i \, A + 13 i \, B - 33 i \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-3 i \, A + 13 i \, B - 33 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-3 i \, A + 13 i \, B - 33 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-3 i \, A + 13 i \, B - 33 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 \, {\left (\sqrt {2} {\left (-9 i \, A + 49 i \, B - 119 i \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-9 i \, A + 49 i \, B - 119 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-9 i \, A + 49 i \, B - 119 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-9 i \, A + 49 i \, B - 119 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (\sqrt {2} {\left (9 i \, A - 49 i \, B + 119 i \, C\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (9 i \, A - 49 i \, B + 119 i \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (9 i \, A - 49 i \, B + 119 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (9 i \, A - 49 i \, B + 119 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + \frac {2 \, {\left (3 \, {\left (9 \, A - 49 \, B + 119 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (33 \, A - 188 \, B + 453 \, C\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (9 \, A - 59 \, B + 139 \, C\right )} \cos \left (d x + c\right )^{2} - 60 \, {\left (B - 2 \, C\right )} \cos \left (d x + c\right ) - 20 \, C\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/60*(5*(sqrt(2)*(3*I*A - 13*I*B + 33*I*C)*cos(d*x + c)^4 + 3*sqrt(2)*(3*I*A - 13*I*B + 33*I*C)*cos(d*x + c)^
3 + 3*sqrt(2)*(3*I*A - 13*I*B + 33*I*C)*cos(d*x + c)^2 + sqrt(2)*(3*I*A - 13*I*B + 33*I*C)*cos(d*x + c))*weier
strassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*(sqrt(2)*(-3*I*A + 13*I*B - 33*I*C)*cos(d*x + c)^4 +
3*sqrt(2)*(-3*I*A + 13*I*B - 33*I*C)*cos(d*x + c)^3 + 3*sqrt(2)*(-3*I*A + 13*I*B - 33*I*C)*cos(d*x + c)^2 + sq
rt(2)*(-3*I*A + 13*I*B - 33*I*C)*cos(d*x + c))*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 3*(
sqrt(2)*(-9*I*A + 49*I*B - 119*I*C)*cos(d*x + c)^4 + 3*sqrt(2)*(-9*I*A + 49*I*B - 119*I*C)*cos(d*x + c)^3 + 3*
sqrt(2)*(-9*I*A + 49*I*B - 119*I*C)*cos(d*x + c)^2 + sqrt(2)*(-9*I*A + 49*I*B - 119*I*C)*cos(d*x + c))*weierst
rassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*(sqrt(2)*(9*I*A - 49*I*B + 119*
I*C)*cos(d*x + c)^4 + 3*sqrt(2)*(9*I*A - 49*I*B + 119*I*C)*cos(d*x + c)^3 + 3*sqrt(2)*(9*I*A - 49*I*B + 119*I*
C)*cos(d*x + c)^2 + sqrt(2)*(9*I*A - 49*I*B + 119*I*C)*cos(d*x + c))*weierstrassZeta(-4, 0, weierstrassPInvers
e(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(3*(9*A - 49*B + 119*C)*cos(d*x + c)^4 + 2*(33*A - 188*B + 453*C)
*cos(d*x + c)^3 + 5*(9*A - 59*B + 139*C)*cos(d*x + c)^2 - 60*(B - 2*C)*cos(d*x + c) - 20*C)*sin(d*x + c)/sqrt(
cos(d*x + c)))/(a^3*d*cos(d*x + c)^4 + 3*a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + a^3*d*cos(d*x + c))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 7318 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sec(d*x + c)^(7/2)/(a*sec(d*x + c) + a)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1/cos(c + d*x))^(7/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^3,x)

[Out]

int(((1/cos(c + d*x))^(7/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + a/cos(c + d*x))^3, x)

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