Integrand size = 33, antiderivative size = 229 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {A x}{a^3}+\frac {\left (5 a^2 A b^3-2 A b^5+2 a^5 B+a^3 b^2 B-3 a^4 b (2 A+C)\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{5/2} (a+b)^{5/2} d}+\frac {\left (A b^2-a (b B-a C)\right ) \tan (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (2 A b^4+3 a^3 b B-a^4 C-a^2 b^2 (5 A+2 C)\right ) \tan (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
A*x/a^3+(5*a^2*A*b^3-2*A*b^5+2*a^5*B+a^3*b^2*B-3*a^4*b*(2*A+C))*arctanh((a -b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^3/(a-b)^(5/2)/(a+b)^(5/2)/d+1/ 2*(A*b^2-a*(B*b-C*a))*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^2-1/2*(2*A *b^4+3*B*a^3*b-a^4*C-a^2*b^2*(5*A+2*C))*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b* sec(d*x+c))
Result contains complex when optimal does not.
Time = 6.10 (sec) , antiderivative size = 793, normalized size of antiderivative = 3.46 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {4 i \left (5 a^2 A b^3-2 A b^5+2 a^5 B+a^3 b^2 B-3 a^4 b (2 A+C)\right ) \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (a \sin (c)+(-b+a \cos (c)) \tan \left (\frac {d x}{2}\right )\right )}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}\right ) (b+a \cos (c+d x))^2 (\cos (c)-i \sin (c))}{\left (a^2-b^2\right )^{5/2} \sqrt {(\cos (c)-i \sin (c))^2}}+\frac {\sec (c) \left (2 A \left (a^2-b^2\right )^2 \left (a^2+2 b^2\right ) d x \cos (c)+4 a A b \left (a^2-b^2\right )^2 d x \cos (d x)+4 a^5 A b d x \cos (2 c+d x)-8 a^3 A b^3 d x \cos (2 c+d x)+4 a A b^5 d x \cos (2 c+d x)+a^6 A d x \cos (c+2 d x)-2 a^4 A b^2 d x \cos (c+2 d x)+a^2 A b^4 d x \cos (c+2 d x)+a^6 A d x \cos (3 c+2 d x)-2 a^4 A b^2 d x \cos (3 c+2 d x)+a^2 A b^4 d x \cos (3 c+2 d x)-6 a^4 A b^2 \sin (c)-9 a^2 A b^4 \sin (c)+6 A b^6 \sin (c)+4 a^5 b B \sin (c)+7 a^3 b^3 B \sin (c)-2 a b^5 B \sin (c)-2 a^6 C \sin (c)-5 a^4 b^2 C \sin (c)-2 a^2 b^4 C \sin (c)+17 a^3 A b^3 \sin (d x)-8 a A b^5 \sin (d x)-11 a^4 b^2 B \sin (d x)+2 a^2 b^4 B \sin (d x)+5 a^5 b C \sin (d x)+4 a^3 b^3 C \sin (d x)-7 a^3 A b^3 \sin (2 c+d x)+4 a A b^5 \sin (2 c+d x)+5 a^4 b^2 B \sin (2 c+d x)-2 a^2 b^4 B \sin (2 c+d x)-3 a^5 b C \sin (2 c+d x)+6 a^4 A b^2 \sin (c+2 d x)-3 a^2 A b^4 \sin (c+2 d x)-4 a^5 b B \sin (c+2 d x)+a^3 b^3 B \sin (c+2 d x)+2 a^6 C \sin (c+2 d x)+a^4 b^2 C \sin (c+2 d x)\right )}{\left (a^2-b^2\right )^2}\right )}{2 a^3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) (a+b \sec (c+d x))^3} \]
((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2) *(((-4*I)*(5*a^2*A*b^3 - 2*A*b^5 + 2*a^5*B + a^3*b^2*B - 3*a^4*b*(2*A + C) )*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/( Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^2*(Cos[ c] - I*Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (Sec[c]* (2*A*(a^2 - b^2)^2*(a^2 + 2*b^2)*d*x*Cos[c] + 4*a*A*b*(a^2 - b^2)^2*d*x*Co s[d*x] + 4*a^5*A*b*d*x*Cos[2*c + d*x] - 8*a^3*A*b^3*d*x*Cos[2*c + d*x] + 4 *a*A*b^5*d*x*Cos[2*c + d*x] + a^6*A*d*x*Cos[c + 2*d*x] - 2*a^4*A*b^2*d*x*C os[c + 2*d*x] + a^2*A*b^4*d*x*Cos[c + 2*d*x] + a^6*A*d*x*Cos[3*c + 2*d*x] - 2*a^4*A*b^2*d*x*Cos[3*c + 2*d*x] + a^2*A*b^4*d*x*Cos[3*c + 2*d*x] - 6*a^ 4*A*b^2*Sin[c] - 9*a^2*A*b^4*Sin[c] + 6*A*b^6*Sin[c] + 4*a^5*b*B*Sin[c] + 7*a^3*b^3*B*Sin[c] - 2*a*b^5*B*Sin[c] - 2*a^6*C*Sin[c] - 5*a^4*b^2*C*Sin[c ] - 2*a^2*b^4*C*Sin[c] + 17*a^3*A*b^3*Sin[d*x] - 8*a*A*b^5*Sin[d*x] - 11*a ^4*b^2*B*Sin[d*x] + 2*a^2*b^4*B*Sin[d*x] + 5*a^5*b*C*Sin[d*x] + 4*a^3*b^3* C*Sin[d*x] - 7*a^3*A*b^3*Sin[2*c + d*x] + 4*a*A*b^5*Sin[2*c + d*x] + 5*a^4 *b^2*B*Sin[2*c + d*x] - 2*a^2*b^4*B*Sin[2*c + d*x] - 3*a^5*b*C*Sin[2*c + d *x] + 6*a^4*A*b^2*Sin[c + 2*d*x] - 3*a^2*A*b^4*Sin[c + 2*d*x] - 4*a^5*b*B* Sin[c + 2*d*x] + a^3*b^3*B*Sin[c + 2*d*x] + 2*a^6*C*Sin[c + 2*d*x] + a^4*b ^2*C*Sin[c + 2*d*x]))/(a^2 - b^2)^2))/(2*a^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)
Time = 1.22 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.394, Rules used = {3042, 4548, 25, 3042, 4548, 25, 3042, 4407, 3042, 4318, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int -\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)-2 a (A b+C b-a B) \sec (c+d x)+2 A \left (a^2-b^2\right )}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)-2 a (A b+C b-a B) \sec (c+d x)+2 A \left (a^2-b^2\right )}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (A b^2-a (b B-a C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a (A b+C b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )+2 A \left (a^2-b^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4548 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 A \left (a^2-b^2\right )^2+a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {2 A \left (a^2-b^2\right )^2+a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 A \left (a^2-b^2\right )^2+a \left (2 B a^3-b (4 A+3 C) a^2+b^2 B a+A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\frac {\frac {\left (2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}+\frac {2 A x \left (a^2-b^2\right )^2}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}+\frac {2 A x \left (a^2-b^2\right )^2}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\frac {\frac {\left (2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a b}+\frac {2 A x \left (a^2-b^2\right )^2}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\left (2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a b}+\frac {2 A x \left (a^2-b^2\right )^2}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\frac {2 \left (2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a b d}+\frac {2 A x \left (a^2-b^2\right )^2}{a}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\tan (c+d x) \left (A b^2-a (b B-a C)\right )}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\frac {\frac {2 A x \left (a^2-b^2\right )^2}{a}+\frac {2 \left (2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a \left (a^2-b^2\right )}-\frac {\tan (c+d x) \left (a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}\) |
((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d *x])^2) + (((2*A*(a^2 - b^2)^2*x)/a + (2*(5*a^2*A*b^3 - 2*A*b^5 + 2*a^5*B + a^3*b^2*B - 3*a^4*b*(2*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sq rt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d))/(a*(a^2 - b^2)) - ((2*A*b^4 + 3 *a^3*b*B - a^4*C - a^2*b^2*(5*A + 2*C))*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])))/(2*a*(a^2 - b^2))
3.10.20.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a Int[Csc[e + f* x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^( m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x ] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Time = 0.43 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.51
| method | result | size |
| derivativedivides | \(\frac {\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}+\frac {\frac {2 \left (-\frac {\left (6 A \,a^{2} b^{2}+a A \,b^{3}-2 A \,b^{4}-4 B \,a^{3} b -B \,a^{2} b^{2}+2 a^{4} C +a^{3} b C +2 C \,a^{2} b^{2}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {a \left (6 A \,a^{2} b^{2}-a A \,b^{3}-2 A \,b^{4}-4 B \,a^{3} b +B \,a^{2} b^{2}+2 a^{4} C -a^{3} b C +2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 A \,a^{4} b -5 a^{2} A \,b^{3}+2 A \,b^{5}-2 a^{5} B -a^{3} b^{2} B +3 a^{4} b C \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{3}}}{d}\) | \(345\) |
| default | \(\frac {\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}+\frac {\frac {2 \left (-\frac {\left (6 A \,a^{2} b^{2}+a A \,b^{3}-2 A \,b^{4}-4 B \,a^{3} b -B \,a^{2} b^{2}+2 a^{4} C +a^{3} b C +2 C \,a^{2} b^{2}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {a \left (6 A \,a^{2} b^{2}-a A \,b^{3}-2 A \,b^{4}-4 B \,a^{3} b +B \,a^{2} b^{2}+2 a^{4} C -a^{3} b C +2 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 A \,a^{4} b -5 a^{2} A \,b^{3}+2 A \,b^{5}-2 a^{5} B -a^{3} b^{2} B +3 a^{4} b C \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{a^{3}}}{d}\) | \(345\) |
| risch | \(\text {Expression too large to display}\) | \(1461\) |
1/d*(2*A/a^3*arctan(tan(1/2*d*x+1/2*c))+2/a^3*((-1/2*(6*A*a^2*b^2+A*a*b^3- 2*A*b^4-4*B*a^3*b-B*a^2*b^2+2*C*a^4+C*a^3*b+2*C*a^2*b^2)*a/(a-b)/(a^2+2*a* b+b^2)*tan(1/2*d*x+1/2*c)^3+1/2*a*(6*A*a^2*b^2-A*a*b^3-2*A*b^4-4*B*a^3*b+B *a^2*b^2+2*C*a^4-C*a^3*b+2*C*a^2*b^2)/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c))/(t an(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2-1/2*(6*A*a^4*b-5*A*a^2 *b^3+2*A*b^5-2*B*a^5-B*a^3*b^2+3*C*a^4*b)/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b) )^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (213) = 426\).
Time = 0.37 (sec) , antiderivative size = 1237, normalized size of antiderivative = 5.40 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
[1/4*(4*(A*a^8 - 3*A*a^6*b^2 + 3*A*a^4*b^4 - A*a^2*b^6)*d*x*cos(d*x + c)^2 + 8*(A*a^7*b - 3*A*a^5*b^3 + 3*A*a^3*b^5 - A*a*b^7)*d*x*cos(d*x + c) + 4* (A*a^6*b^2 - 3*A*a^4*b^4 + 3*A*a^2*b^6 - A*b^8)*d*x - (2*B*a^5*b^2 - 3*(2* A + C)*a^4*b^3 + B*a^3*b^4 + 5*A*a^2*b^5 - 2*A*b^7 + (2*B*a^7 - 3*(2*A + C )*a^6*b + B*a^5*b^2 + 5*A*a^4*b^3 - 2*A*a^2*b^5)*cos(d*x + c)^2 + 2*(2*B*a ^6*b - 3*(2*A + C)*a^5*b^2 + B*a^4*b^3 + 5*A*a^3*b^4 - 2*A*a*b^6)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c) ^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a ^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + 2*(C*a^7*b - 3*B*a^6*b^2 + (5*A + C)*a^5*b^3 + 3*B*a^4*b^4 - (7*A + 2*C)*a^3*b^5 + 2*A*a*b^7 + (2*C *a^8 - 4*B*a^7*b + (6*A - C)*a^6*b^2 + 5*B*a^5*b^3 - (9*A + C)*a^4*b^4 - B *a^3*b^5 + 3*A*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^11 - 3*a^9*b^2 + 3 *a^7*b^4 - a^5*b^6)*d*cos(d*x + c)^2 + 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*cos(d*x + c) + (a^9*b^2 - 3*a^7*b^4 + 3*a^5*b^6 - a^3*b^8)*d), 1/2*(2*(A*a^8 - 3*A*a^6*b^2 + 3*A*a^4*b^4 - A*a^2*b^6)*d*x*cos(d*x + c)^2 + 4*(A*a^7*b - 3*A*a^5*b^3 + 3*A*a^3*b^5 - A*a*b^7)*d*x*cos(d*x + c) + 2* (A*a^6*b^2 - 3*A*a^4*b^4 + 3*A*a^2*b^6 - A*b^8)*d*x + (2*B*a^5*b^2 - 3*(2* A + C)*a^4*b^3 + B*a^3*b^4 + 5*A*a^2*b^5 - 2*A*b^7 + (2*B*a^7 - 3*(2*A + C )*a^6*b + B*a^5*b^2 + 5*A*a^4*b^3 - 2*A*a^2*b^5)*cos(d*x + c)^2 + 2*(2*B*a ^6*b - 3*(2*A + C)*a^5*b^2 + B*a^4*b^3 + 5*A*a^3*b^4 - 2*A*a*b^6)*cos(d...
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (213) = 426\).
Time = 0.39 (sec) , antiderivative size = 606, normalized size of antiderivative = 2.65 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {{\left (2 \, B a^{5} - 6 \, A a^{4} b - 3 \, C a^{4} b + B a^{3} b^{2} + 5 \, A a^{2} b^{3} - 2 \, A b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {{\left (d x + c\right )} A}{a^{3}} - \frac {2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, B a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, C a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, B a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, A b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}}}{d} \]
((2*B*a^5 - 6*A*a^4*b - 3*C*a^4*b + B*a^3*b^2 + 5*A*a^2*b^3 - 2*A*b^5)*(pi *floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^7 - 2*a^5*b^2 + a^ 3*b^4)*sqrt(-a^2 + b^2)) + (d*x + c)*A/a^3 - (2*C*a^5*tan(1/2*d*x + 1/2*c) ^3 - 4*B*a^4*b*tan(1/2*d*x + 1/2*c)^3 - C*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 6 *A*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 + 3*B*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 + C *a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 5*A*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + B*a ^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 2*C*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 3*A*a *b^4*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^5*tan(1/2*d*x + 1/2*c)^3 - 2*C*a^5*tan (1/2*d*x + 1/2*c) + 4*B*a^4*b*tan(1/2*d*x + 1/2*c) - C*a^4*b*tan(1/2*d*x + 1/2*c) - 6*A*a^3*b^2*tan(1/2*d*x + 1/2*c) + 3*B*a^3*b^2*tan(1/2*d*x + 1/2 *c) - C*a^3*b^2*tan(1/2*d*x + 1/2*c) - 5*A*a^2*b^3*tan(1/2*d*x + 1/2*c) - B*a^2*b^3*tan(1/2*d*x + 1/2*c) - 2*C*a^2*b^3*tan(1/2*d*x + 1/2*c) + 3*A*a* b^4*tan(1/2*d*x + 1/2*c) + 2*A*b^5*tan(1/2*d*x + 1/2*c))/((a^6 - 2*a^4*b^2 + a^2*b^4)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^ 2))/d
Time = 28.75 (sec) , antiderivative size = 8147, normalized size of antiderivative = 35.58 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^3*(2*C*a^4 - 2*A*b^4 + 6*A*a^2*b^2 - B*a^2*b^2 + 2*C* a^2*b^2 + A*a*b^3 - 4*B*a^3*b + C*a^3*b))/((a^2*b - a^3)*(a + b)^2) - (tan (c/2 + (d*x)/2)*(2*A*b^4 - 2*C*a^4 - 6*A*a^2*b^2 - B*a^2*b^2 - 2*C*a^2*b^2 + A*a*b^3 + 4*B*a^3*b + C*a^3*b))/((a + b)*(a^4 - 2*a^3*b + a^2*b^2)))/(d *(2*a*b - tan(c/2 + (d*x)/2)^2*(2*a^2 - 2*b^2) + tan(c/2 + (d*x)/2)^4*(a^2 - 2*a*b + b^2) + a^2 + b^2)) + (2*A*atan(((A*((8*tan(c/2 + (d*x)/2)*(4*A^ 2*a^10 + 8*A^2*b^10 + 4*B^2*a^10 - 8*A^2*a*b^9 - 8*A^2*a^9*b - 32*A^2*a^2* b^8 + 32*A^2*a^3*b^7 + 57*A^2*a^4*b^6 - 48*A^2*a^5*b^5 - 52*A^2*a^6*b^4 + 32*A^2*a^7*b^3 + 24*A^2*a^8*b^2 + B^2*a^6*b^4 + 4*B^2*a^8*b^2 + 9*C^2*a^8* b^2 - 24*A*B*a^9*b - 12*B*C*a^9*b - 4*A*B*a^3*b^7 + 2*A*B*a^5*b^5 + 8*A*B* a^7*b^3 + 12*A*C*a^4*b^6 - 30*A*C*a^6*b^4 + 36*A*C*a^8*b^2 - 6*B*C*a^7*b^3 ))/(a^10*b + a^11 - a^4*b^7 - a^5*b^6 + 3*a^6*b^5 + 3*a^7*b^4 - 3*a^8*b^3 - 3*a^9*b^2) + (A*((8*(4*A*a^15 + 4*B*a^15 - 4*A*a^6*b^9 + 2*A*a^7*b^8 + 1 8*A*a^8*b^7 - 4*A*a^9*b^6 - 36*A*a^10*b^5 + 6*A*a^11*b^4 + 34*A*a^12*b^3 - 8*A*a^13*b^2 - 2*B*a^8*b^7 + 2*B*a^9*b^6 + 6*B*a^12*b^3 - 6*B*a^13*b^2 + 6*C*a^9*b^6 - 6*C*a^10*b^5 - 12*C*a^11*b^4 + 12*C*a^12*b^3 + 6*C*a^13*b^2 - 12*A*a^14*b - 4*B*a^14*b - 6*C*a^14*b))/(a^12*b + a^13 - a^6*b^7 - a^7*b ^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (A*tan(c/2 + (d*x) /2)*(8*a^15*b - 8*a^6*b^10 + 8*a^7*b^9 + 32*a^8*b^8 - 32*a^9*b^7 - 48*a^10 *b^6 + 48*a^11*b^5 + 32*a^12*b^4 - 32*a^13*b^3 - 8*a^14*b^2)*8i)/(a^3*(...