Integrand size = 43, antiderivative size = 250 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=-\frac {3 (7 A-7 B+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a d}+\frac {5 (9 A-7 B+7 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 a d}+\frac {(9 A-7 B+7 C) \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {(7 A-7 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {5 (9 A-7 B+7 C) \sin (c+d x)}{21 a d \sqrt {\sec (c+d x)}}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))} \] Output:
-3/5*(7*A-7*B+5*C)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))* sec(d*x+c)^(1/2)/a/d+5/21*(9*A-7*B+7*C)*cos(d*x+c)^(1/2)*InverseJacobiAM(1 /2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/a/d+1/7*(9*A-7*B+7*C)*sin(d*x+c)/a/ d/sec(d*x+c)^(5/2)-1/5*(7*A-7*B+5*C)*sin(d*x+c)/a/d/sec(d*x+c)^(3/2)+5/21* (9*A-7*B+7*C)*sin(d*x+c)/a/d/sec(d*x+c)^(1/2)-(A-B+C)*sin(d*x+c)/d/sec(d*x +c)^(5/2)/(a+a*sec(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.03 (sec) , antiderivative size = 1406, normalized size of antiderivative = 5.62 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx =\text {Too large to display} \] Input:
Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])),x]
Output:
(7*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]^2*Cos[c + d*x]*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]*(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])) - (7*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]^2*Cos[c + d*x]*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]* (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])) + (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]^2*Cos[c + d*x]*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]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(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 ])) + (30*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(7*d *(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) - (10*B*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c...
Time = 1.09 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.90, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.326, Rules used = {3042, 4572, 27, 3042, 4274, 3042, 4256, 3042, 4256, 3042, 4258, 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.
| \(\displaystyle \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )}dx\) |
| \(\Big \downarrow \) 4572 |
| \(\displaystyle \frac {\int \frac {a (9 A-7 B+7 C)-a (7 A-7 B+5 C) \sec (c+d x)}{2 \sec ^{\frac {7}{2}}(c+d x)}dx}{a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a (9 A-7 B+7 C)-a (7 A-7 B+5 C) \sec (c+d x)}{\sec ^{\frac {7}{2}}(c+d x)}dx}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a (9 A-7 B+7 C)-a (7 A-7 B+5 C) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \int \frac {1}{\sec ^{\frac {7}{2}}(c+d x)}dx-a (7 A-7 B+5 C) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)}dx}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx-a (7 A-7 B+5 C) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \left (\frac {5}{7} \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )-a (7 A-7 B+5 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \left (\frac {5}{7} \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )-a (7 A-7 B+5 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )-a (7 A-7 B+5 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )-a (7 A-7 B+5 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )-a (7 A-7 B+5 C) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )-a (7 A-7 B+5 C) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )-a (7 A-7 B+5 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {a (9 A-7 B+7 C) \left (\frac {2 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {5}{7} \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )-a (7 A-7 B+5 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )}{2 a^2}-\frac {(A-B+C) \sin (c+d x)}{d \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)}\) |
Input:
Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*(a + a*Sec [c + d*x])),x]
Output:
-(((A - B + C)*Sin[c + d*x])/(d*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x]))) + (-(a*(7*A - 7*B + 5*C)*((6*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]* Sqrt[Sec[c + d*x]])/(5*d) + (2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)))) + a*(9*A - 7*B + 7*C)*((2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (5*((2*Sq rt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2* Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])))/7))/(2*a^2)
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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
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*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(a*f*(2*m + 1))), x] - Simp[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)]
Time = 8.76 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.36
| method | result | size |
| default | \(-\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (225 A \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+441 A \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-175 B \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-441 B \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+175 C \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+315 C \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )-480 A \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (864 A +336 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-888 A -392 B -280 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (930 A -210 B +630 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-321 A +161 B -245 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{105 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(341\) |
Input:
int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2)/(a+a*sec(d*x+c)),x,me thod=_RETURNVERBOSE)
Output:
-1/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(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)*(22 5*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+441*A*EllipticE(cos(1/2*d*x+1/2* c),2^(1/2))-175*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-441*B*EllipticE(co s(1/2*d*x+1/2*c),2^(1/2))+175*C*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+315* C*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))-480*A*sin(1/2*d*x+1/2*c)^10+(864* A+336*B)*sin(1/2*d*x+1/2*c)^8+(-888*A-392*B-280*C)*sin(1/2*d*x+1/2*c)^6+(9 30*A-210*B+630*C)*sin(1/2*d*x+1/2*c)^4+(-321*A+161*B-245*C)*sin(1/2*d*x+1/ 2*c)^2)/a/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
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.31 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=-\frac {25 \, {\left (\sqrt {2} {\left (9 i \, A - 7 i \, B + 7 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (9 i \, A - 7 i \, B + 7 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 25 \, {\left (\sqrt {2} {\left (-9 i \, A + 7 i \, B - 7 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-9 i \, A + 7 i \, B - 7 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 63 \, {\left (\sqrt {2} {\left (7 i \, A - 7 i \, B + 5 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (7 i \, A - 7 i \, B + 5 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 63 \, {\left (\sqrt {2} {\left (-7 i \, A + 7 i \, B - 5 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-7 i \, A + 7 i \, B - 5 i \, C\right )}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (30 \, A \cos \left (d x + c\right )^{4} - 6 \, {\left (2 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (39 \, A - 14 \, B + 35 \, C\right )} \cos \left (d x + c\right )^{2} + 25 \, {\left (9 \, A - 7 \, B + 7 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{210 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \] Input:
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2)/(a+a*sec(d*x+c) ),x, algorithm="fricas")
Output:
-1/210*(25*(sqrt(2)*(9*I*A - 7*I*B + 7*I*C)*cos(d*x + c) + sqrt(2)*(9*I*A - 7*I*B + 7*I*C))*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c) ) + 25*(sqrt(2)*(-9*I*A + 7*I*B - 7*I*C)*cos(d*x + c) + sqrt(2)*(-9*I*A + 7*I*B - 7*I*C))*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 63*(sqrt(2)*(7*I*A - 7*I*B + 5*I*C)*cos(d*x + c) + sqrt(2)*(7*I*A - 7*I* B + 5*I*C))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 63*(sqrt(2)*(-7*I*A + 7*I*B - 5*I*C)*cos(d*x + c) + sqrt(2)*(-7*I*A + 7*I*B - 5*I*C))*weierstrassZeta(-4, 0, weierstrassPInver se(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(30*A*cos(d*x + c)^4 - 6*(2* A - 7*B)*cos(d*x + c)^3 + 2*(39*A - 14*B + 35*C)*cos(d*x + c)^2 + 25*(9*A - 7*B + 7*C)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(a*d*cos(d*x + c) + a*d)
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\text {Timed out} \] Input:
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)**(7/2)/(a+a*sec(d*x+ c)),x)
Output:
Timed out
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2)/(a+a*sec(d*x+c) ),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)/((a*sec(d*x + c) + a)*se c(d*x + c)^(7/2)), x)
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2)/(a+a*sec(d*x+c) ),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)/((a*sec(d*x + c) + a)*se c(d*x + c)^(7/2)), x)
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \] Input:
int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + a/cos(c + d*x))*(1/cos(c + d*x))^(7/2)),x)
Output:
int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/((a + a/cos(c + d*x))*(1/cos(c + d*x))^(7/2)), x)
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx=\frac {\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}+\sec \left (d x +c \right )^{4}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}+\sec \left (d x +c \right )^{3}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}+\sec \left (d x +c \right )^{2}}d x \right ) c}{a} \] Input:
int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2)/(a+a*sec(d*x+c)),x)
Output:
(int(sqrt(sec(c + d*x))/(sec(c + d*x)**5 + sec(c + d*x)**4),x)*a + int(sqr t(sec(c + d*x))/(sec(c + d*x)**4 + sec(c + d*x)**3),x)*b + int(sqrt(sec(c + d*x))/(sec(c + d*x)**3 + sec(c + d*x)**2),x)*c)/a