Integrand size = 31, antiderivative size = 194 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {(A-4 B) \text {arctanh}(\sin (c+d x))}{a^4 d}-\frac {(55 A-244 B) \tan (c+d x)}{105 a^4 d}+\frac {(25 A-88 B) \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac {(A-4 B) \tan (c+d x)}{a^4 d (1+\sec (c+d x))}+\frac {(A-B) \sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {(5 A-12 B) \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3} \] Output:
(A-4*B)*arctanh(sin(d*x+c))/a^4/d-1/105*(55*A-244*B)*tan(d*x+c)/a^4/d+1/10 5*(25*A-88*B)*sec(d*x+c)^2*tan(d*x+c)/a^4/d/(1+sec(d*x+c))^2-(A-4*B)*tan(d *x+c)/a^4/d/(1+sec(d*x+c))+1/7*(A-B)*sec(d*x+c)^4*tan(d*x+c)/d/(a+a*sec(d* x+c))^4+1/35*(5*A-12*B)*sec(d*x+c)^3*tan(d*x+c)/a/d/(a+a*sec(d*x+c))^3
Leaf count is larger than twice the leaf count of optimal. \(593\) vs. \(2(194)=388\).
Time = 6.80 (sec) , antiderivative size = 593, normalized size of antiderivative = 3.06 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {-26880 (A-4 B) \cos ^8\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sec (c) \sec (c+d x) \left (245 (17 A-44 B) \sin \left (\frac {d x}{2}\right )-7 (635 A-2684 B) \sin \left (\frac {3 d x}{2}\right )+4795 A \sin \left (c-\frac {d x}{2}\right )-20524 B \sin \left (c-\frac {d x}{2}\right )-4795 A \sin \left (c+\frac {d x}{2}\right )+14644 B \sin \left (c+\frac {d x}{2}\right )+4165 A \sin \left (2 c+\frac {d x}{2}\right )-16660 B \sin \left (2 c+\frac {d x}{2}\right )+2275 A \sin \left (c+\frac {3 d x}{2}\right )-4690 B \sin \left (c+\frac {3 d x}{2}\right )-4445 A \sin \left (2 c+\frac {3 d x}{2}\right )+14378 B \sin \left (2 c+\frac {3 d x}{2}\right )+2275 A \sin \left (3 c+\frac {3 d x}{2}\right )-9100 B \sin \left (3 c+\frac {3 d x}{2}\right )-2785 A \sin \left (c+\frac {5 d x}{2}\right )+11668 B \sin \left (c+\frac {5 d x}{2}\right )+735 A \sin \left (2 c+\frac {5 d x}{2}\right )-630 B \sin \left (2 c+\frac {5 d x}{2}\right )-2785 A \sin \left (3 c+\frac {5 d x}{2}\right )+9358 B \sin \left (3 c+\frac {5 d x}{2}\right )+735 A \sin \left (4 c+\frac {5 d x}{2}\right )-2940 B \sin \left (4 c+\frac {5 d x}{2}\right )-1015 A \sin \left (2 c+\frac {7 d x}{2}\right )+4228 B \sin \left (2 c+\frac {7 d x}{2}\right )+105 A \sin \left (3 c+\frac {7 d x}{2}\right )+315 B \sin \left (3 c+\frac {7 d x}{2}\right )-1015 A \sin \left (4 c+\frac {7 d x}{2}\right )+3493 B \sin \left (4 c+\frac {7 d x}{2}\right )+105 A \sin \left (5 c+\frac {7 d x}{2}\right )-420 B \sin \left (5 c+\frac {7 d x}{2}\right )-160 A \sin \left (3 c+\frac {9 d x}{2}\right )+664 B \sin \left (3 c+\frac {9 d x}{2}\right )+105 B \sin \left (4 c+\frac {9 d x}{2}\right )-160 A \sin \left (5 c+\frac {9 d x}{2}\right )+559 B \sin \left (5 c+\frac {9 d x}{2}\right )\right )}{1680 a^4 d (1+\cos (c+d x))^4} \] Input:
Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]
Output:
(-26880*(A - 4*B)*Cos[(c + d*x)/2]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x) /2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/ 2]*Sec[c]*Sec[c + d*x]*(245*(17*A - 44*B)*Sin[(d*x)/2] - 7*(635*A - 2684*B )*Sin[(3*d*x)/2] + 4795*A*Sin[c - (d*x)/2] - 20524*B*Sin[c - (d*x)/2] - 47 95*A*Sin[c + (d*x)/2] + 14644*B*Sin[c + (d*x)/2] + 4165*A*Sin[2*c + (d*x)/ 2] - 16660*B*Sin[2*c + (d*x)/2] + 2275*A*Sin[c + (3*d*x)/2] - 4690*B*Sin[c + (3*d*x)/2] - 4445*A*Sin[2*c + (3*d*x)/2] + 14378*B*Sin[2*c + (3*d*x)/2] + 2275*A*Sin[3*c + (3*d*x)/2] - 9100*B*Sin[3*c + (3*d*x)/2] - 2785*A*Sin[ c + (5*d*x)/2] + 11668*B*Sin[c + (5*d*x)/2] + 735*A*Sin[2*c + (5*d*x)/2] - 630*B*Sin[2*c + (5*d*x)/2] - 2785*A*Sin[3*c + (5*d*x)/2] + 9358*B*Sin[3*c + (5*d*x)/2] + 735*A*Sin[4*c + (5*d*x)/2] - 2940*B*Sin[4*c + (5*d*x)/2] - 1015*A*Sin[2*c + (7*d*x)/2] + 4228*B*Sin[2*c + (7*d*x)/2] + 105*A*Sin[3*c + (7*d*x)/2] + 315*B*Sin[3*c + (7*d*x)/2] - 1015*A*Sin[4*c + (7*d*x)/2] + 3493*B*Sin[4*c + (7*d*x)/2] + 105*A*Sin[5*c + (7*d*x)/2] - 420*B*Sin[5*c + (7*d*x)/2] - 160*A*Sin[3*c + (9*d*x)/2] + 664*B*Sin[3*c + (9*d*x)/2] + 1 05*B*Sin[4*c + (9*d*x)/2] - 160*A*Sin[5*c + (9*d*x)/2] + 559*B*Sin[5*c + ( 9*d*x)/2]))/(1680*a^4*d*(1 + Cos[c + d*x])^4)
Time = 1.54 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.13, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {3042, 4507, 3042, 4507, 3042, 4507, 3042, 4496, 25, 3042, 4274, 3042, 4254, 24, 4257}
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.
| \(\displaystyle \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a \sec (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^5 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 4507 |
| \(\displaystyle \frac {\int \frac {\sec ^4(c+d x) (4 a (A-B)-a (A-8 B) \sec (c+d x))}{(\sec (c+d x) a+a)^3}dx}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^4 \left (4 a (A-B)-a (A-8 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4507 |
| \(\displaystyle \frac {\frac {\int \frac {\sec ^3(c+d x) \left (3 a^2 (5 A-12 B)-2 a^2 (5 A-26 B) \sec (c+d x)\right )}{(\sec (c+d x) a+a)^2}dx}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (3 a^2 (5 A-12 B)-2 a^2 (5 A-26 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4507 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sec ^2(c+d x) \left (2 a^3 (25 A-88 B)-a^3 (55 A-244 B) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (2 a^3 (25 A-88 B)-a^3 (55 A-244 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4496 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -\sec (c+d x) \left (105 a^4 (A-4 B)-a^4 (55 A-244 B) \sec (c+d x)\right )dx}{a^2}-\frac {105 a^3 (A-4 B) \tan (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \sec (c+d x) \left (105 a^4 (A-4 B)-a^4 (55 A-244 B) \sec (c+d x)\right )dx}{a^2}-\frac {105 a^3 (A-4 B) \tan (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (105 a^4 (A-4 B)-a^4 (55 A-244 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}-\frac {105 a^3 (A-4 B) \tan (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (A-4 B) \int \sec (c+d x)dx-a^4 (55 A-244 B) \int \sec ^2(c+d x)dx}{a^2}-\frac {105 a^3 (A-4 B) \tan (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (A-4 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-a^4 (55 A-244 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2}-\frac {105 a^3 (A-4 B) \tan (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {a^4 (55 A-244 B) \int 1d(-\tan (c+d x))}{d}+105 a^4 (A-4 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a^2}-\frac {105 a^3 (A-4 B) \tan (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {\frac {\frac {105 a^4 (A-4 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {a^4 (55 A-244 B) \tan (c+d x)}{d}}{a^2}-\frac {105 a^3 (A-4 B) \tan (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {105 a^4 (A-4 B) \text {arctanh}(\sin (c+d x))}{d}-\frac {a^4 (55 A-244 B) \tan (c+d x)}{d}}{a^2}-\frac {105 a^3 (A-4 B) \tan (c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{3 d (\sec (c+d x)+1)^2}}{5 a^2}+\frac {a (5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}}{7 a^2}+\frac {(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
Input:
Int[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]
Output:
((A - B)*Sec[c + d*x]^4*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((a*( 5*A - 12*B)*Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (( (25*A - 88*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(1 + Sec[c + d*x])^2) + (( -105*a^3*(A - 4*B)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((105*a^4*(A - 4*B)*ArcTanh[Sin[c + d*x]])/d - (a^4*(55*A - 244*B)*Tan[c + d*x])/d)/a^2) /(3*a^2))/(5*a^2))/(7*a^2)
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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[csc[(e_.) + (f_.)*(x_)]^2*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*( csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-(A*b - a*B))*Cot [e + f*x]*((a + b*Csc[e + f*x])^m/(b*f*(2*m + 1))), x] + Simp[1/(b^2*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[A*b*m - a*B*m + b *B*(2*m + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && Ne Q[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
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] - Simp[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)] && G tQ[n, 0]
Time = 0.46 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.82
| method | result | size |
| parallelrisch | \(\frac {-3360 \cos \left (d x +c \right ) \left (A -4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+3360 \cos \left (d x +c \right ) \left (A -4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-80 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (\left (\frac {39 A}{2}-\frac {165 B}{2}\right ) \cos \left (2 d x +2 c \right )+\left (\frac {107 A}{16}-\frac {559 B}{20}\right ) \cos \left (3 d x +3 c \right )+\left (A -\frac {83 B}{20}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {529 A}{16}-\frac {2861 B}{20}\right ) \cos \left (d x +c \right )+\frac {37 A}{2}-\frac {418 B}{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3360 \cos \left (d x +c \right ) a^{4} d}\) | \(160\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A +\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (32 B -8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-32 B +8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(190\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} A}{7}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} B}{7}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} A +\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} B}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +49 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B +\left (32 B -8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (-32 B +8 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {8 B}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}}{8 d \,a^{4}}\) | \(190\) |
| risch | \(-\frac {2 i \left (105 A \,{\mathrm e}^{8 i \left (d x +c \right )}-420 B \,{\mathrm e}^{8 i \left (d x +c \right )}+735 A \,{\mathrm e}^{7 i \left (d x +c \right )}-2940 B \,{\mathrm e}^{7 i \left (d x +c \right )}+2275 A \,{\mathrm e}^{6 i \left (d x +c \right )}-9100 B \,{\mathrm e}^{6 i \left (d x +c \right )}+4165 A \,{\mathrm e}^{5 i \left (d x +c \right )}-16660 B \,{\mathrm e}^{5 i \left (d x +c \right )}+4795 A \,{\mathrm e}^{4 i \left (d x +c \right )}-20524 B \,{\mathrm e}^{4 i \left (d x +c \right )}+4445 A \,{\mathrm e}^{3 i \left (d x +c \right )}-18788 B \,{\mathrm e}^{3 i \left (d x +c \right )}+2785 A \,{\mathrm e}^{2 i \left (d x +c \right )}-11668 B \,{\mathrm e}^{2 i \left (d x +c \right )}+1015 \,{\mathrm e}^{i \left (d x +c \right )} A -4228 B \,{\mathrm e}^{i \left (d x +c \right )}+160 A -664 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a^{4} d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{4} d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a^{4} d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{4} d}\) | \(323\) |
Input:
int(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x,method=_RETURNVERBO SE)
Output:
1/3360*(-3360*cos(d*x+c)*(A-4*B)*ln(tan(1/2*d*x+1/2*c)-1)+3360*cos(d*x+c)* (A-4*B)*ln(tan(1/2*d*x+1/2*c)+1)-80*sec(1/2*d*x+1/2*c)^6*((39/2*A-165/2*B) *cos(2*d*x+2*c)+(107/16*A-559/20*B)*cos(3*d*x+3*c)+(A-83/20*B)*cos(4*d*x+4 *c)+(529/16*A-2861/20*B)*cos(d*x+c)+37/2*A-418/5*B)*tan(1/2*d*x+1/2*c))/co s(d*x+c)/a^4/d
Time = 0.09 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.63 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {105 \, {\left ({\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (A - 4 \, B\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (A - 4 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (A - 4 \, B\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (8 \, {\left (20 \, A - 83 \, B\right )} \cos \left (d x + c\right )^{4} + {\left (535 \, A - 2236 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (155 \, A - 659 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (65 \, A - 296 \, B\right )} \cos \left (d x + c\right ) - 105 \, B\right )} \sin \left (d x + c\right )}{210 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \] Input:
integrate(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="f ricas")
Output:
1/210*(105*((A - 4*B)*cos(d*x + c)^5 + 4*(A - 4*B)*cos(d*x + c)^4 + 6*(A - 4*B)*cos(d*x + c)^3 + 4*(A - 4*B)*cos(d*x + c)^2 + (A - 4*B)*cos(d*x + c) )*log(sin(d*x + c) + 1) - 105*((A - 4*B)*cos(d*x + c)^5 + 4*(A - 4*B)*cos( d*x + c)^4 + 6*(A - 4*B)*cos(d*x + c)^3 + 4*(A - 4*B)*cos(d*x + c)^2 + (A - 4*B)*cos(d*x + c))*log(-sin(d*x + c) + 1) - 2*(8*(20*A - 83*B)*cos(d*x + c)^4 + (535*A - 2236*B)*cos(d*x + c)^3 + 4*(155*A - 659*B)*cos(d*x + c)^2 + 4*(65*A - 296*B)*cos(d*x + c) - 105*B)*sin(d*x + c))/(a^4*d*cos(d*x + c )^5 + 4*a^4*d*cos(d*x + c)^4 + 6*a^4*d*cos(d*x + c)^3 + 4*a^4*d*cos(d*x + c)^2 + a^4*d*cos(d*x + c))
\[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {A \sec ^{5}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \] Input:
integrate(sec(d*x+c)**5*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))**4,x)
Output:
(Integral(A*sec(c + d*x)**5/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x) + Integral(B*sec(c + d*x)**6/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x) )/a**4
Time = 0.04 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.68 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {B {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {3360 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )} - 5 \, A {\left (\frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {77 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac {168 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}\right )}}{840 \, d} \] Input:
integrate(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="m axima")
Output:
1/840*(B*(1680*sin(d*x + c)/((a^4 - a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^ 2)*(cos(d*x + c) + 1)) + (5145*sin(d*x + c)/(cos(d*x + c) + 1) + 805*sin(d *x + c)^3/(cos(d*x + c) + 1)^3 + 147*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 - 3360*log(sin(d*x + c)/(cos( d*x + c) + 1) + 1)/a^4 + 3360*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4 ) - 5*A*((315*sin(d*x + c)/(cos(d*x + c) + 1) + 77*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 3*sin(d*x + c)^7/( cos(d*x + c) + 1)^7)/a^4 - 168*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^ 4 + 168*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^4))/d
Time = 0.20 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.13 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {840 \, {\left (A - 4 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {840 \, {\left (A - 4 \, B\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} - \frac {1680 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{4}} - \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 805 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, B a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \] Input:
integrate(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x, algorithm="g iac")
Output:
1/840*(840*(A - 4*B)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^4 - 840*(A - 4*B )*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^4 - 1680*B*tan(1/2*d*x + 1/2*c)/((t an(1/2*d*x + 1/2*c)^2 - 1)*a^4) - (15*A*a^24*tan(1/2*d*x + 1/2*c)^7 - 15*B *a^24*tan(1/2*d*x + 1/2*c)^7 + 105*A*a^24*tan(1/2*d*x + 1/2*c)^5 - 147*B*a ^24*tan(1/2*d*x + 1/2*c)^5 + 385*A*a^24*tan(1/2*d*x + 1/2*c)^3 - 805*B*a^2 4*tan(1/2*d*x + 1/2*c)^3 + 1575*A*a^24*tan(1/2*d*x + 1/2*c) - 5145*B*a^24* tan(1/2*d*x + 1/2*c))/a^28)/d
Time = 11.49 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.22 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (A-4\,B\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {A-B}{20\,a^4}+\frac {3\,A-5\,B}{40\,a^4}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A-B}{2\,a^4}+\frac {3\,\left (3\,A-5\,B\right )}{8\,a^4}+\frac {2\,A-10\,B}{4\,a^4}-\frac {2\,A+10\,B}{8\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {A-B}{8\,a^4}+\frac {3\,A-5\,B}{12\,a^4}+\frac {2\,A-10\,B}{24\,a^4}\right )}{d}-\frac {2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^4\right )} \] Input:
int((A + B/cos(c + d*x))/(cos(c + d*x)^5*(a + a/cos(c + d*x))^4),x)
Output:
(2*atanh(tan(c/2 + (d*x)/2))*(A - 4*B))/(a^4*d) - (tan(c/2 + (d*x)/2)^5*(( A - B)/(20*a^4) + (3*A - 5*B)/(40*a^4)))/d - (tan(c/2 + (d*x)/2)*((A - B)/ (2*a^4) + (3*(3*A - 5*B))/(8*a^4) + (2*A - 10*B)/(4*a^4) - (2*A + 10*B)/(8 *a^4)))/d - (tan(c/2 + (d*x)/2)^7*(A - B))/(56*a^4*d) - (tan(c/2 + (d*x)/2 )^3*((A - B)/(8*a^4) + (3*A - 5*B)/(12*a^4) + (2*A - 10*B)/(24*a^4)))/d - (2*B*tan(c/2 + (d*x)/2))/(d*(a^4*tan(c/2 + (d*x)/2)^2 - a^4))
Time = 0.21 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.67 \[ \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx=\frac {-840 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +3360 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +840 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a -3360 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b +840 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -3360 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -840 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a +3360 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b -15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} a +15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} b -90 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} a +132 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} b -280 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a +658 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b -1190 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a +4340 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b +1575 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -6825 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{840 a^{4} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )} \] Input:
int(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+a*sec(d*x+c))^4,x)
Output:
( - 840*log(tan((c + d*x)/2) - 1)*tan((c + d*x)/2)**2*a + 3360*log(tan((c + d*x)/2) - 1)*tan((c + d*x)/2)**2*b + 840*log(tan((c + d*x)/2) - 1)*a - 3 360*log(tan((c + d*x)/2) - 1)*b + 840*log(tan((c + d*x)/2) + 1)*tan((c + d *x)/2)**2*a - 3360*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)**2*b - 840*l og(tan((c + d*x)/2) + 1)*a + 3360*log(tan((c + d*x)/2) + 1)*b - 15*tan((c + d*x)/2)**9*a + 15*tan((c + d*x)/2)**9*b - 90*tan((c + d*x)/2)**7*a + 132 *tan((c + d*x)/2)**7*b - 280*tan((c + d*x)/2)**5*a + 658*tan((c + d*x)/2)* *5*b - 1190*tan((c + d*x)/2)**3*a + 4340*tan((c + d*x)/2)**3*b + 1575*tan( (c + d*x)/2)*a - 6825*tan((c + d*x)/2)*b)/(840*a**4*d*(tan((c + d*x)/2)**2 - 1))