Integrand size = 40, antiderivative size = 156 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {(4 B-7 C) \text {arctanh}(\sin (c+d x))}{2 a^2 d}+\frac {2 (5 B-8 C) \tan (c+d x)}{3 a^2 d}-\frac {(4 B-7 C) \sec (c+d x) \tan (c+d x)}{2 a^2 d}+\frac {(5 B-8 C) \sec ^2(c+d x) \tan (c+d x)}{3 a^2 d (1+\sec (c+d x))}+\frac {(B-C) \sec ^3(c+d x) \tan (c+d x)}{3 d (a+a \sec (c+d x))^2} \]
-1/2*(4*B-7*C)*arctanh(sin(d*x+c))/a^2/d+2/3*(5*B-8*C)*tan(d*x+c)/a^2/d-1/ 2*(4*B-7*C)*sec(d*x+c)*tan(d*x+c)/a^2/d+1/3*(5*B-8*C)*sec(d*x+c)^2*tan(d*x +c)/a^2/d/(1+sec(d*x+c))+1/3*(B-C)*sec(d*x+c)^3*tan(d*x+c)/d/(a+a*sec(d*x+ c))^2
Time = 1.70 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.83 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\sec ^2(c+d x) \left (-24 (4 B-7 C) \text {arctanh}(\sin (c+d x)) \cos ^4\left (\frac {1}{2} (c+d x)\right )+(28 B-37 C+6 (7 B-10 C) \cos (c+d x)+(28 B-43 C) \cos (2 (c+d x))+10 B \cos (3 (c+d x))-16 C \cos (3 (c+d x))) \sec (c+d x) \tan (c+d x)\right )}{12 a^2 d (1+\sec (c+d x))^2} \]
(Sec[c + d*x]^2*(-24*(4*B - 7*C)*ArcTanh[Sin[c + d*x]]*Cos[(c + d*x)/2]^4 + (28*B - 37*C + 6*(7*B - 10*C)*Cos[c + d*x] + (28*B - 43*C)*Cos[2*(c + d* x)] + 10*B*Cos[3*(c + d*x)] - 16*C*Cos[3*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x]))/(12*a^2*d*(1 + Sec[c + d*x])^2)
Time = 1.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3042, 4560, 3042, 4507, 3042, 4507, 3042, 4274, 3042, 4254, 24, 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.
\(\displaystyle \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a \sec (c+d x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
\(\Big \downarrow \) 4560 |
\(\displaystyle \int \frac {\sec ^4(c+d x) (B+C \sec (c+d x))}{(a \sec (c+d x)+a)^2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^4 \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}dx\) |
\(\Big \downarrow \) 4507 |
\(\displaystyle \frac {\int \frac {\sec ^3(c+d x) (3 a (B-C)-a (2 B-5 C) \sec (c+d x))}{\sec (c+d x) a+a}dx}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (3 a (B-C)-a (2 B-5 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 4507 |
\(\displaystyle \frac {\frac {\int \sec ^2(c+d x) \left (2 a^2 (5 B-8 C)-3 a^2 (4 B-7 C) \sec (c+d x)\right )dx}{a^2}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (2 a^2 (5 B-8 C)-3 a^2 (4 B-7 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{a^2}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle \frac {\frac {2 a^2 (5 B-8 C) \int \sec ^2(c+d x)dx-3 a^2 (4 B-7 C) \int \sec ^3(c+d x)dx}{a^2}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {2 a^2 (5 B-8 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx-3 a^2 (4 B-7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {\frac {-\frac {2 a^2 (5 B-8 C) \int 1d(-\tan (c+d x))}{d}-3 a^2 (4 B-7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\frac {\frac {2 a^2 (5 B-8 C) \tan (c+d x)}{d}-3 a^2 (4 B-7 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx}{a^2}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {\frac {\frac {2 a^2 (5 B-8 C) \tan (c+d x)}{d}-3 a^2 (4 B-7 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{a^2}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 a^2 (5 B-8 C) \tan (c+d x)}{d}-3 a^2 (4 B-7 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{a^2}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {\frac {\frac {2 a^2 (5 B-8 C) \tan (c+d x)}{d}-3 a^2 (4 B-7 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )}{a^2}+\frac {(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{d (\sec (c+d x)+1)}}{3 a^2}+\frac {(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}\) |
((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) + (((5* B - 8*C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(1 + Sec[c + d*x])) + ((2*a^2*(5* B - 8*C)*Tan[c + d*x])/d - 3*a^2*(4*B - 7*C)*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/a^2)/(3*a^2)
3.4.40.3.1 Defintions of rubi rules used
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_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_. )*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*((c_.) + csc[(e_.) + (f_.) *(x_)]*(d_.))^(n_.), x_Symbol] :> Simp[1/b^2 Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n*(b*B - a*C + b*C*Csc[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03
method | result | size |
parallelrisch | \(\frac {4 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (B -\frac {7 C}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-4 \left (1+\cos \left (2 d x +2 c \right )\right ) \left (B -\frac {7 C}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\frac {\left (2 B -\frac {43 C}{14}\right ) \cos \left (2 d x +2 c \right )}{3}+\frac {\left (5 B -8 C \right ) \cos \left (3 d x +3 c \right )}{21}+\left (B -\frac {10 C}{7}\right ) \cos \left (d x +c \right )+\frac {2 B}{3}-\frac {37 C}{42}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 d \,a^{2} \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(160\) |
derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {2 B -5 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (7 C -4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 B -5 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (4 B -7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{2 d \,a^{2}}\) | \(177\) |
default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} B}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} C}{3}+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) B -7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {2 B -5 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\left (7 C -4 B \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 B -5 C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\left (4 B -7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\frac {C}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}}{2 d \,a^{2}}\) | \(177\) |
norman | \(\frac {\frac {\left (B -C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 a d}+\frac {\left (9 B -13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}-\frac {\left (11 B -18 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{a d}+\frac {\left (11 B -17 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{6 a d}+\frac {\left (61 B -100 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a d}-\frac {\left (95 B -149 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4} a}+\frac {\left (4 B -7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{2} d}-\frac {\left (4 B -7 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{2} d}\) | \(228\) |
risch | \(\frac {i \left (12 B \,{\mathrm e}^{6 i \left (d x +c \right )}-21 C \,{\mathrm e}^{6 i \left (d x +c \right )}+36 B \,{\mathrm e}^{5 i \left (d x +c \right )}-63 C \,{\mathrm e}^{5 i \left (d x +c \right )}+56 B \,{\mathrm e}^{4 i \left (d x +c \right )}-98 C \,{\mathrm e}^{4 i \left (d x +c \right )}+84 B \,{\mathrm e}^{3 i \left (d x +c \right )}-126 C \,{\mathrm e}^{3 i \left (d x +c \right )}+64 B \,{\mathrm e}^{2 i \left (d x +c \right )}-97 C \,{\mathrm e}^{2 i \left (d x +c \right )}+48 B \,{\mathrm e}^{i \left (d x +c \right )}-75 C \,{\mathrm e}^{i \left (d x +c \right )}+20 B -32 C \right )}{3 d \,a^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{a^{2} d}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 a^{2} d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{a^{2} d}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 a^{2} d}\) | \(276\) |
1/2*(4*(1+cos(2*d*x+2*c))*(B-7/4*C)*ln(tan(1/2*d*x+1/2*c)-1)-4*(1+cos(2*d* x+2*c))*(B-7/4*C)*ln(tan(1/2*d*x+1/2*c)+1)+7*tan(1/2*d*x+1/2*c)*(1/3*(2*B- 43/14*C)*cos(2*d*x+2*c)+1/21*(5*B-8*C)*cos(3*d*x+3*c)+(B-10/7*C)*cos(d*x+c )+2/3*B-37/42*C)*sec(1/2*d*x+1/2*c)^2)/d/a^2/(1+cos(2*d*x+2*c))
Time = 0.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.46 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {3 \, {\left ({\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (4 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (5 \, B - 8 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (28 \, B - 43 \, C\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (B - C\right )} \cos \left (d x + c\right ) + 3 \, C\right )} \sin \left (d x + c\right )}{12 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2}\right )}} \]
integrate(sec(d*x+c)^3*(B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")
-1/12*(3*((4*B - 7*C)*cos(d*x + c)^4 + 2*(4*B - 7*C)*cos(d*x + c)^3 + (4*B - 7*C)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 3*((4*B - 7*C)*cos(d*x + c )^4 + 2*(4*B - 7*C)*cos(d*x + c)^3 + (4*B - 7*C)*cos(d*x + c)^2)*log(-sin( d*x + c) + 1) - 2*(4*(5*B - 8*C)*cos(d*x + c)^3 + (28*B - 43*C)*cos(d*x + c)^2 + 6*(B - C)*cos(d*x + c) + 3*C)*sin(d*x + c))/(a^2*d*cos(d*x + c)^4 + 2*a^2*d*cos(d*x + c)^3 + a^2*d*cos(d*x + c)^2)
\[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {B \sec ^{4}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \sec ^{5}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
(Integral(B*sec(c + d*x)**4/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x) + I ntegral(C*sec(c + d*x)**5/(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. 336 vs. \(2 (146) = 292\).
Time = 0.22 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.15 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {C {\left (\frac {6 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {21 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}}\right )} - B {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {12 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \]
integrate(sec(d*x+c)^3*(B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^2,x, algorithm="maxima")
-1/6*(C*(6*(3*sin(d*x + c)/(cos(d*x + c) + 1) - 5*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^2 - 2*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + a^2*sin(d* x + c)^4/(cos(d*x + c) + 1)^4) + (21*sin(d*x + c)/(cos(d*x + c) + 1) + sin (d*x + c)^3/(cos(d*x + c) + 1)^3)/a^2 - 21*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a^2 + 21*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^2) - B*((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/(c os(d*x + c) + 1)^2)*(cos(d*x + c) + 1))))/d
Time = 0.32 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.27 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {\frac {3 \, {\left (4 \, B - 7 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac {3 \, {\left (4 \, B - 7 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} + \frac {6 \, {\left (2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{2}} - \frac {B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 21 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
-1/6*(3*(4*B - 7*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^2 - 3*(4*B - 7*C) *log(abs(tan(1/2*d*x + 1/2*c) - 1))/a^2 + 6*(2*B*tan(1/2*d*x + 1/2*c)^3 - 5*C*tan(1/2*d*x + 1/2*c)^3 - 2*B*tan(1/2*d*x + 1/2*c) + 3*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*a^2) - (B*a^4*tan(1/2*d*x + 1/2*c) ^3 - C*a^4*tan(1/2*d*x + 1/2*c)^3 + 15*B*a^4*tan(1/2*d*x + 1/2*c) - 21*C*a ^4*tan(1/2*d*x + 1/2*c))/a^6)/d
Time = 15.96 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.06 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,\left (B-C\right )}{2\,a^2}+\frac {2\,B-4\,C}{2\,a^2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,B-5\,C\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,B-3\,C\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (B-C\right )}{6\,a^2\,d}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,B-7\,C\right )}{a^2\,d} \]
(tan(c/2 + (d*x)/2)*((3*(B - C))/(2*a^2) + (2*B - 4*C)/(2*a^2)))/d - (tan( c/2 + (d*x)/2)^3*(2*B - 5*C) - tan(c/2 + (d*x)/2)*(2*B - 3*C))/(d*(a^2*tan (c/2 + (d*x)/2)^4 - 2*a^2*tan(c/2 + (d*x)/2)^2 + a^2)) + (tan(c/2 + (d*x)/ 2)^3*(B - C))/(6*a^2*d) - (atanh(tan(c/2 + (d*x)/2))*(4*B - 7*C))/(a^2*d)