Integrand size = 43, antiderivative size = 234 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\frac {(B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}+\frac {(3 A+4 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}+\frac {(15 A-B-83 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{210 a^4 d (1+\cos (c+d x))^2}-\frac {(B+8 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{10 a^4 d (1+\cos (c+d x))}-\frac {(A-B+C) \sqrt {\cos (c+d x)} \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(5 A+2 B-9 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3} \]
1/10*(B+8*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))/a^4/d+1/42*(3*A+4*B+17*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))/a^4/d+1/210 *(15*A-B-83*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/a^4/d/(1+cos(d*x+c))^2-1/10*(B+ 8*C)*sin(d*x+c)*cos(d*x+c)^(1/2)/a^4/d/(1+cos(d*x+c))-1/7*(A-B+C)*sin(d*x+ c)*cos(d*x+c)^(1/2)/d/(a+a*cos(d*x+c))^4+1/35*(5*A+2*B-9*C)*sin(d*x+c)*cos (d*x+c)^(1/2)/a/d/(a+a*cos(d*x+c))^3
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 19.73 (sec) , antiderivative size = 1626, normalized size of antiderivative = 6.95 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx =\text {Too large to display} \]
Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^4),x]
(-8*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, S in[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(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]*(a + a*Sec[c + d*x])^4) - (32*B*Cos[c/2 + (d *x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Co t[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(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]*(a + a*Sec[c + d*x])^4) - (136*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2 ]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/ 2]*Sec[c + d*x]^2*(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]*Si n[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]*(a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-16*(B + 8*C)*Csc[c])/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^ 3*(15*A*Sin[(d*x)/2] - B*Sin[(d*x)/2] - 83*C*Sin[(d*x)/2]))/(105*d) + (...
Time = 1.61 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4600, 3042, 3520, 27, 3042, 3457, 25, 3042, 3457, 27, 3042, 3457, 3042, 3227, 3042, 3119, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sec (c+d x)+C \sec (c+d x)^2}{\cos (c+d x)^{5/2} (a \sec (c+d x)+a)^4}dx\) |
\(\Big \downarrow \) 4600 |
\(\displaystyle \int \frac {A \cos ^2(c+d x)+B \cos (c+d x)+C}{\sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
\(\Big \downarrow \) 3520 |
\(\displaystyle \frac {\int -\frac {a (A-B-13 C)-a (9 A+5 B-5 C) \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3}dx}{7 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {a (A-B-13 C)-a (9 A+5 B-5 C) \cos (c+d x)}{\sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3}dx}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {a (A-B-13 C)-a (9 A+5 B-5 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3457 |
\(\displaystyle -\frac {\frac {\int -\frac {7 (B+8 C) a^2+3 (5 A+2 B-9 C) \cos (c+d x) a^2}{\sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {-\frac {\int \frac {7 (B+8 C) a^2+3 (5 A+2 B-9 C) \cos (c+d x) a^2}{\sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2}dx}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {\int \frac {7 (B+8 C) a^2+3 (5 A+2 B-9 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3457 |
\(\displaystyle -\frac {-\frac {\frac {\int \frac {(15 A+41 B+253 C) a^3+(15 A-B-83 C) \cos (c+d x) a^3}{2 \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)}dx}{3 a^2}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {-\frac {\frac {\int \frac {(15 A+41 B+253 C) a^3+(15 A-B-83 C) \cos (c+d x) a^3}{\sqrt {\cos (c+d x)} (\cos (c+d x) a+a)}dx}{6 a^2}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {\frac {\int \frac {(15 A+41 B+253 C) a^3+(15 A-B-83 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}dx}{6 a^2}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3457 |
\(\displaystyle -\frac {-\frac {\frac {\frac {\int \frac {5 (3 A+4 B+17 C) a^4+21 (B+8 C) \cos (c+d x) a^4}{\sqrt {\cos (c+d x)}}dx}{a^2}-\frac {42 a^3 (B+8 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {\frac {\frac {\int \frac {5 (3 A+4 B+17 C) a^4+21 (B+8 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}-\frac {42 a^3 (B+8 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle -\frac {-\frac {\frac {\frac {5 a^4 (3 A+4 B+17 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 a^4 (B+8 C) \int \sqrt {\cos (c+d x)}dx}{a^2}-\frac {42 a^3 (B+8 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\frac {\frac {\frac {5 a^4 (3 A+4 B+17 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 a^4 (B+8 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}-\frac {42 a^3 (B+8 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {-\frac {\frac {\frac {5 a^4 (3 A+4 B+17 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {42 a^4 (B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}-\frac {42 a^3 (B+8 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle -\frac {-\frac {\frac {\frac {\frac {10 a^4 (3 A+4 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {42 a^4 (B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}}{a^2}-\frac {42 a^3 (B+8 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d (a \cos (c+d x)+a)}}{6 a^2}+\frac {(15 A-B-83 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d (\cos (c+d x)+1)^2}}{5 a^2}-\frac {2 a (5 A+2 B-9 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}}{14 a^2}-\frac {(A-B+C) \sin (c+d x) \sqrt {\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}\) |
-1/7*((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]) ^4) - ((-2*a*(5*A + 2*B - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (((15*A - B - 83*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/ (3*d*(1 + Cos[c + d*x])^2) + (((42*a^4*(B + 8*C)*EllipticE[(c + d*x)/2, 2] )/d + (10*a^4*(3*A + 4*B + 17*C)*EllipticF[(c + d*x)/2, 2])/d)/a^2 - (42*a ^3*(B + 8*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x])))/(6 *a^2))/(5*a^2))/(14*a^2)
3.13.43.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^( n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Simp[1/(a*(2*m + 1)*(b*c - a*d)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b *d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*(m + n + 2)*Sin[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] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(a*A - b*B + a*C)*Cos[e + f*x]*(a + b* Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(f*(b*c - a*d)*(2*m + 1))), x ] + Simp[1/(b*(b*c - a*d)*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[A*(a*c*(m + 1) - b*d*(2*m + n + 2)) + B*(b*c*m + a *d*(n + 1)) - C*(a*c*m + b*d*(n + 1)) + (d*(a*A - b*B)*(m + n + 2) + C*(b*c *(2*m + 1) - a*d*(m - n - 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c ^2 - d^2, 0] && LtQ[m, -2^(-1)]
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] :> Simp[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] /; Fr eeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(594\) vs. \(2(266)=532\).
Time = 3.76 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.54
method | result | size |
default | \(-\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (60 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-168 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+80 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-84 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1344 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+340 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-672 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+60 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+248 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+1684 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-30 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-54 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-282 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-90 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-8 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-34 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+75 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-33 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-9 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} C -15 A +15 B -15 C \right )}{840 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(595\) |
int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+a*sec(d*x+c))^4,x, method=_RETURNVERBOSE)
-1/840*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(60*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)*EllipticF(cos(1/ 2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^7-168*B*cos(1/2*d*x+1/2*c)^10+80* 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)*EllipticF (cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^7-84*B*cos(1/2*d*x+1/2*c)^ 7*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticE (cos(1/2*d*x+1/2*c),2^(1/2))-1344*C*cos(1/2*d*x+1/2*c)^10+340*C*(sin(1/2*d *x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x +1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^7-672*C*cos(1/2*d*x+1/2*c)^7*(sin(1/2* d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticE(cos(1/2*d* x+1/2*c),2^(1/2))+60*A*cos(1/2*d*x+1/2*c)^8+248*B*cos(1/2*d*x+1/2*c)^8+168 4*C*cos(1/2*d*x+1/2*c)^8-30*A*cos(1/2*d*x+1/2*c)^6-54*B*cos(1/2*d*x+1/2*c) ^6-282*C*cos(1/2*d*x+1/2*c)^6-90*A*cos(1/2*d*x+1/2*c)^4-8*B*cos(1/2*d*x+1/ 2*c)^4-34*C*cos(1/2*d*x+1/2*c)^4+75*A*cos(1/2*d*x+1/2*c)^2-33*B*cos(1/2*d* x+1/2*c)^2-9*cos(1/2*d*x+1/2*c)^2*C-15*A+15*B-15*C)/a^4/cos(1/2*d*x+1/2*c) ^7/(-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
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.63 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=-\frac {2 \, {\left (21 \, {\left (B + 8 \, C\right )} \cos \left (d x + c\right )^{3} - {\left (15 \, A - 64 \, B - 587 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (60 \, A - 53 \, B - 724 \, C\right )} \cos \left (d x + c\right ) - 15 \, A - 20 \, B + 335 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 5 \, {\left (\sqrt {2} {\left (3 i \, A + 4 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (3 i \, A + 4 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (3 i \, A + 4 i \, B + 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (3 i \, A + 4 i \, B + 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (3 i \, A + 4 i \, B + 17 i \, 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 - 4 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-3 i \, A - 4 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-3 i \, A - 4 i \, B - 17 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-3 i \, A - 4 i \, B - 17 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-3 i \, A - 4 i \, B - 17 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (-i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (-i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (-i \, B - 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (-i \, B - 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-i \, B - 8 i \, 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 ) + 21 \, {\left (\sqrt {2} {\left (i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{4} + 4 \, \sqrt {2} {\left (i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{3} + 6 \, \sqrt {2} {\left (i \, B + 8 i \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, \sqrt {2} {\left (i \, B + 8 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (i \, B + 8 i \, 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 )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+a*sec(d*x+c) )^4,x, algorithm="fricas")
-1/420*(2*(21*(B + 8*C)*cos(d*x + c)^3 - (15*A - 64*B - 587*C)*cos(d*x + c )^2 - (60*A - 53*B - 724*C)*cos(d*x + c) - 15*A - 20*B + 335*C)*sqrt(cos(d *x + c))*sin(d*x + c) + 5*(sqrt(2)*(3*I*A + 4*I*B + 17*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(3*I*A + 4*I*B + 17*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(3*I*A + 4*I*B + 17*I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(3*I*A + 4*I*B + 17*I*C)*cos(d* x + c) + sqrt(2)*(3*I*A + 4*I*B + 17*I*C))*weierstrassPInverse(-4, 0, cos( d*x + c) + I*sin(d*x + c)) + 5*(sqrt(2)*(-3*I*A - 4*I*B - 17*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(-3*I*A - 4*I*B - 17*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(- 3*I*A - 4*I*B - 17*I*C)*cos(d*x + c)^2 + 4*sqrt(2)*(-3*I*A - 4*I*B - 17*I* C)*cos(d*x + c) + sqrt(2)*(-3*I*A - 4*I*B - 17*I*C))*weierstrassPInverse(- 4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*(sqrt(2)*(-I*B - 8*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(-I*B - 8*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(-I*B - 8*I*C )*cos(d*x + c)^2 + 4*sqrt(2)*(-I*B - 8*I*C)*cos(d*x + c) + sqrt(2)*(-I*B - 8*I*C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(sqrt(2)*(I*B + 8*I*C)*cos(d*x + c)^4 + 4*sqrt(2)*(I *B + 8*I*C)*cos(d*x + c)^3 + 6*sqrt(2)*(I*B + 8*I*C)*cos(d*x + c)^2 + 4*sq rt(2)*(I*B + 8*I*C)*cos(d*x + c) + sqrt(2)*(I*B + 8*I*C))*weierstrassZeta( -4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/(a^4*d* cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d *cos(d*x + c) + a^4*d)
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+a*sec(d*x+c) )^4,x, algorithm="maxima")
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2)/(a+a*sec(d*x+c) )^4,x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)/((a*sec(d*x + c) + a)^4* cos(d*x + c)^(5/2)), x)
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^4} \,d x \]