Integrand size = 33, antiderivative size = 172 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {(2 A+5 C) \text {arctanh}(\sin (c+d x))}{a^2 d}+\frac {(5 A+12 C) \tan (c+d x)}{a^2 d}-\frac {(2 A+5 C) \sec (c+d x) \tan (c+d x)}{a^2 d}-\frac {2 (2 A+5 C) \sec ^3(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}-\frac {(A+C) \sec ^4(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2}+\frac {(5 A+12 C) \tan ^3(c+d x)}{3 a^2 d} \]
-(2*A+5*C)*arctanh(sin(d*x+c))/a^2/d+(5*A+12*C)*tan(d*x+c)/a^2/d-(2*A+5*C) *sec(d*x+c)*tan(d*x+c)/a^2/d-2/3*(2*A+5*C)*sec(d*x+c)^3*tan(d*x+c)/a^2/d/( 1+sec(d*x+c))-1/3*(A+C)*sec(d*x+c)^4*tan(d*x+c)/d/(a+a*sec(d*x+c))^2+1/3*( 5*A+12*C)*tan(d*x+c)^3/a^2/d
Leaf count is larger than twice the leaf count of optimal. \(623\) vs. \(2(172)=344\).
Time = 6.33 (sec) , antiderivative size = 623, normalized size of antiderivative = 3.62 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \left (A+C \sec ^2(c+d x)\right ) \left (192 (2 A+5 C) \cos ^3\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 )+\sec \left (\frac {c}{2}\right ) \sec (c) \sec ^3(c+d x) \left (-3 (8 A+C) \sin \left (\frac {d x}{2}\right )+(66 A+155 C) \sin \left (\frac {3 d x}{2}\right )-60 A \sin \left (c-\frac {d x}{2}\right )-153 C \sin \left (c-\frac {d x}{2}\right )+24 A \sin \left (c+\frac {d x}{2}\right )+21 C \sin \left (c+\frac {d x}{2}\right )-60 A \sin \left (2 c+\frac {d x}{2}\right )-135 C \sin \left (2 c+\frac {d x}{2}\right )-4 A \sin \left (c+\frac {3 d x}{2}\right )+25 C \sin \left (c+\frac {3 d x}{2}\right )+36 A \sin \left (2 c+\frac {3 d x}{2}\right )+45 C \sin \left (2 c+\frac {3 d x}{2}\right )-34 A \sin \left (3 c+\frac {3 d x}{2}\right )-85 C \sin \left (3 c+\frac {3 d x}{2}\right )+42 A \sin \left (c+\frac {5 d x}{2}\right )+99 C \sin \left (c+\frac {5 d x}{2}\right )+21 C \sin \left (2 c+\frac {5 d x}{2}\right )+24 A \sin \left (3 c+\frac {5 d x}{2}\right )+33 C \sin \left (3 c+\frac {5 d x}{2}\right )-18 A \sin \left (4 c+\frac {5 d x}{2}\right )-45 C \sin \left (4 c+\frac {5 d x}{2}\right )+24 A \sin \left (2 c+\frac {7 d x}{2}\right )+57 C \sin \left (2 c+\frac {7 d x}{2}\right )+3 A \sin \left (3 c+\frac {7 d x}{2}\right )+18 C \sin \left (3 c+\frac {7 d x}{2}\right )+15 A \sin \left (4 c+\frac {7 d x}{2}\right )+24 C \sin \left (4 c+\frac {7 d x}{2}\right )-6 A \sin \left (5 c+\frac {7 d x}{2}\right )-15 C \sin \left (5 c+\frac {7 d x}{2}\right )+10 A \sin \left (3 c+\frac {9 d x}{2}\right )+24 C \sin \left (3 c+\frac {9 d x}{2}\right )+3 A \sin \left (4 c+\frac {9 d x}{2}\right )+11 C \sin \left (4 c+\frac {9 d x}{2}\right )+7 A \sin \left (5 c+\frac {9 d x}{2}\right )+13 C \sin \left (5 c+\frac {9 d x}{2}\right )\right )\right )}{24 a^2 d (A+2 C+A \cos (2 (c+d x))) (1+\sec (c+d x))^2} \]
(Cos[(c + d*x)/2]*(A + C*Sec[c + d*x]^2)*(192*(2*A + 5*C)*Cos[(c + d*x)/2] ^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[ (c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(-3*(8*A + C)*Sin[(d*x)/2] + (66*A + 155*C)*Sin[(3*d*x)/2] - 60*A*Sin[c - (d*x)/2] - 153*C*Sin[c - ( d*x)/2] + 24*A*Sin[c + (d*x)/2] + 21*C*Sin[c + (d*x)/2] - 60*A*Sin[2*c + ( d*x)/2] - 135*C*Sin[2*c + (d*x)/2] - 4*A*Sin[c + (3*d*x)/2] + 25*C*Sin[c + (3*d*x)/2] + 36*A*Sin[2*c + (3*d*x)/2] + 45*C*Sin[2*c + (3*d*x)/2] - 34*A *Sin[3*c + (3*d*x)/2] - 85*C*Sin[3*c + (3*d*x)/2] + 42*A*Sin[c + (5*d*x)/2 ] + 99*C*Sin[c + (5*d*x)/2] + 21*C*Sin[2*c + (5*d*x)/2] + 24*A*Sin[3*c + ( 5*d*x)/2] + 33*C*Sin[3*c + (5*d*x)/2] - 18*A*Sin[4*c + (5*d*x)/2] - 45*C*S in[4*c + (5*d*x)/2] + 24*A*Sin[2*c + (7*d*x)/2] + 57*C*Sin[2*c + (7*d*x)/2 ] + 3*A*Sin[3*c + (7*d*x)/2] + 18*C*Sin[3*c + (7*d*x)/2] + 15*A*Sin[4*c + (7*d*x)/2] + 24*C*Sin[4*c + (7*d*x)/2] - 6*A*Sin[5*c + (7*d*x)/2] - 15*C*S in[5*c + (7*d*x)/2] + 10*A*Sin[3*c + (9*d*x)/2] + 24*C*Sin[3*c + (9*d*x)/2 ] + 3*A*Sin[4*c + (9*d*x)/2] + 11*C*Sin[4*c + (9*d*x)/2] + 7*A*Sin[5*c + ( 9*d*x)/2] + 13*C*Sin[5*c + (9*d*x)/2])))/(24*a^2*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^2)
Time = 1.01 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3042, 4573, 3042, 4507, 27, 3042, 4274, 3042, 4254, 2009, 4255, 3042, 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.
| \(\displaystyle \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a \sec (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^4 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
| \(\Big \downarrow \) 4573 |
| \(\displaystyle -\frac {\int \frac {\sec ^4(c+d x) (a (A+4 C)-3 a (A+2 C) \sec (c+d x))}{\sec (c+d x) a+a}dx}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^4 \left (a (A+4 C)-3 a (A+2 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4507 |
| \(\displaystyle -\frac {\frac {\int 3 \sec ^3(c+d x) \left (2 a^2 (2 A+5 C)-a^2 (5 A+12 C) \sec (c+d x)\right )dx}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3 \int \sec ^3(c+d x) \left (2 a^2 (2 A+5 C)-a^2 (5 A+12 C) \sec (c+d x)\right )dx}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (2 a^2 (2 A+5 C)-a^2 (5 A+12 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle -\frac {\frac {3 \left (2 a^2 (2 A+5 C) \int \sec ^3(c+d x)dx-a^2 (5 A+12 C) \int \sec ^4(c+d x)dx\right )}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 \left (2 a^2 (2 A+5 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-a^2 (5 A+12 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^4dx\right )}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle -\frac {\frac {3 \left (\frac {a^2 (5 A+12 C) \int \left (\tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}+2 a^2 (2 A+5 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {3 \left (2 a^2 (2 A+5 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {a^2 (5 A+12 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle -\frac {\frac {3 \left (2 a^2 (2 A+5 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a^2 (5 A+12 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3 \left (2 a^2 (2 A+5 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a^2 (5 A+12 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\frac {3 \left (2 a^2 (2 A+5 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a^2 (5 A+12 C) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )}{a^2}+\frac {2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}-\frac {(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
-1/3*((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])^2) - (( 2*(2*A + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(d*(1 + Sec[c + d*x])) + (3*(2* a^2*(2*A + 5*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x]) /(2*d)) + (a^2*(5*A + 12*C)*(-Tan[c + d*x] - Tan[c + d*x]^3/3))/d))/a^2)/( 3*a^2)
3.2.30.3.1 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[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_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[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]
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_)]*(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]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-a) *(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*C sc[e + f*x])^n*Simp[b*C*n + A*b*(2*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, C, n}, x] && EqQ[ a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Time = 0.54 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.14
| method | result | size |
| parallelrisch | \(\frac {36 \left (\frac {5 C}{2}+A \right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-36 \left (\frac {5 C}{2}+A \right ) \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+5 \left (\left (\frac {14 A}{5}+\frac {33 C}{5}\right ) \cos \left (3 d x +3 c \right )+\left (\frac {26 A}{5}+12 C \right ) \cos \left (2 d x +2 c \right )+\left (A +\frac {12 C}{5}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {42 A}{5}+19 C \right ) \cos \left (d x +c \right )+\frac {21 A}{5}+\frac {52 C}{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{6 d \,a^{2} \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(196\) |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {10 C +2 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (10 C +4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {2 C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\left (-10 C -4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {10 C +2 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {2 C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {3 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{2 d \,a^{2}}\) | \(210\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} A}{3}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) A +9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {10 C +2 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (10 C +4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {2 C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\left (-10 C -4 A \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {10 C +2 A}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {2 C}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {3 C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{2 d \,a^{2}}\) | \(210\) |
| norman | \(\frac {\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{6 a d}-\frac {3 \left (3 A +7 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}+\frac {\left (5 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 a d}+\frac {2 \left (47 A +115 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 a d}+\frac {\left (61 A +143 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}-\frac {\left (77 A +185 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6 a d}-\frac {\left (217 A +521 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{6 a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5} a}+\frac {\left (2 A +5 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2} d}-\frac {\left (2 A +5 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2} d}\) | \(251\) |
| risch | \(\frac {2 i \left (6 A \,{\mathrm e}^{8 i \left (d x +c \right )}+15 C \,{\mathrm e}^{8 i \left (d x +c \right )}+18 A \,{\mathrm e}^{7 i \left (d x +c \right )}+45 C \,{\mathrm e}^{7 i \left (d x +c \right )}+34 A \,{\mathrm e}^{6 i \left (d x +c \right )}+85 C \,{\mathrm e}^{6 i \left (d x +c \right )}+60 A \,{\mathrm e}^{5 i \left (d x +c \right )}+135 C \,{\mathrm e}^{5 i \left (d x +c \right )}+60 A \,{\mathrm e}^{4 i \left (d x +c \right )}+153 C \,{\mathrm e}^{4 i \left (d x +c \right )}+66 A \,{\mathrm e}^{3 i \left (d x +c \right )}+155 C \,{\mathrm e}^{3 i \left (d x +c \right )}+42 A \,{\mathrm e}^{2 i \left (d x +c \right )}+99 C \,{\mathrm e}^{2 i \left (d x +c \right )}+24 A \,{\mathrm e}^{i \left (d x +c \right )}+57 C \,{\mathrm e}^{i \left (d x +c \right )}+10 A +24 C \right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{a^{2} d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a^{2} d}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{a^{2} d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a^{2} d}\) | \(324\) |
1/6*(36*(5/2*C+A)*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*ln(tan(1/2*d*x+1/2*c)-1) -36*(5/2*C+A)*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*ln(tan(1/2*d*x+1/2*c)+1)+5*( (14/5*A+33/5*C)*cos(3*d*x+3*c)+(26/5*A+12*C)*cos(2*d*x+2*c)+(A+12/5*C)*cos (4*d*x+4*c)+(42/5*A+19*C)*cos(d*x+c)+21/5*A+52/5*C)*tan(1/2*d*x+1/2*c)*sec (1/2*d*x+1/2*c)^2)/d/a^2/(cos(3*d*x+3*c)+3*cos(d*x+c))
Time = 0.26 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.38 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {3 \, {\left ({\left (2 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (2 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (2 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (2 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (2 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (2 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, {\left (5 \, A + 12 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (14 \, A + 33 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (A + 2 \, C\right )} \cos \left (d x + c\right )^{2} - C \cos \left (d x + c\right ) + C\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]
-1/6*(3*((2*A + 5*C)*cos(d*x + c)^5 + 2*(2*A + 5*C)*cos(d*x + c)^4 + (2*A + 5*C)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 3*((2*A + 5*C)*cos(d*x + c) ^5 + 2*(2*A + 5*C)*cos(d*x + c)^4 + (2*A + 5*C)*cos(d*x + c)^3)*log(-sin(d *x + c) + 1) - 2*(2*(5*A + 12*C)*cos(d*x + c)^4 + (14*A + 33*C)*cos(d*x + c)^3 + 3*(A + 2*C)*cos(d*x + c)^2 - C*cos(d*x + c) + C)*sin(d*x + c))/(a^2 *d*cos(d*x + c)^5 + 2*a^2*d*cos(d*x + c)^4 + a^2*d*cos(d*x + c)^3)
\[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {A \sec ^{4}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{6}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
(Integral(A*sec(c + d*x)**4/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x) + I ntegral(C*sec(c + d*x)**6/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x))/a**2
Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (166) = 332\).
Time = 0.21 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.20 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {C {\left (\frac {4 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{2} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\frac {27 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {30 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {30 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )} + A {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \]
1/6*(C*(4*(9*sin(d*x + c)/(cos(d*x + c) + 1) - 20*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^2 - 3*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6) + (27*sin(d*x + c)/(cos(d*x + c) + 1) + sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 30*log(sin(d*x + c)/(co s(d*x + c) + 1) + 1)/a^2 + 30*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^2 ) + A*((15*sin(d*x + c)/(cos(d*x + c) + 1) + sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 12*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^2 + 12*log(sin (d*x + c)/(cos(d*x + c) + 1) - 1)/a^2 + 12*sin(d*x + c)/((a^2 - a^2*sin(d* x + c)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1))))/d
Time = 0.32 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {6 \, {\left (2 \, A + 5 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac {6 \, {\left (2 \, A + 5 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac {4 \, {\left (3 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 20 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{2}} - \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
-1/6*(6*(2*A + 5*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^2 - 6*(2*A + 5*C) *log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^2 + 4*(3*A*tan(1/2*d*x + 1/2*c)^5 + 15*C*tan(1/2*d*x + 1/2*c)^5 - 6*A*tan(1/2*d*x + 1/2*c)^3 - 20*C*tan(1/2*d* x + 1/2*c)^3 + 3*A*tan(1/2*d*x + 1/2*c) + 9*C*tan(1/2*d*x + 1/2*c))/((tan( 1/2*d*x + 1/2*c)^2 - 1)^3*a^2) - (A*a^4*tan(1/2*d*x + 1/2*c)^3 + C*a^4*tan (1/2*d*x + 1/2*c)^3 + 15*A*a^4*tan(1/2*d*x + 1/2*c) + 27*C*a^4*tan(1/2*d*x + 1/2*c))/a^6)/d
Time = 15.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.15 \[ \int \frac {\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {2\,\left (A+C\right )}{a^2}+\frac {A+5\,C}{2\,a^2}\right )}{d}-\frac {\left (2\,A+10\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-4\,A-\frac {40\,C}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A+6\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^2\right )}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,A+5\,C\right )}{a^2\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A+C\right )}{6\,a^2\,d} \]
(tan(c/2 + (d*x)/2)*((2*(A + C))/a^2 + (A + 5*C)/(2*a^2)))/d - (tan(c/2 + (d*x)/2)^5*(2*A + 10*C) - tan(c/2 + (d*x)/2)^3*(4*A + (40*C)/3) + tan(c/2 + (d*x)/2)*(2*A + 6*C))/(d*(3*a^2*tan(c/2 + (d*x)/2)^2 - 3*a^2*tan(c/2 + ( d*x)/2)^4 + a^2*tan(c/2 + (d*x)/2)^6 - a^2)) - (2*atanh(tan(c/2 + (d*x)/2) )*(2*A + 5*C))/(a^2*d) + (tan(c/2 + (d*x)/2)^3*(A + C))/(6*a^2*d)