Integrand size = 29, antiderivative size = 180 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=-\frac {(2 A b-a B) x}{a^3}+\frac {2 b \left (3 a^2 A b-2 A b^3-2 a^3 B+a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^3 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {\left (a^2 A-2 A b^2+a b B\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]
-(2*A*b-B*a)*x/a^3+2*b*(3*A*a^2*b-2*A*b^3-2*B*a^3+B*a*b^2)*arctanh((a-b)^( 1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^3/(a-b)^(3/2)/(a+b)^(3/2)/d+(A*a^2- 2*A*b^2+B*a*b)*sin(d*x+c)/a^2/(a^2-b^2)/d+b*(A*b-B*a)*sin(d*x+c)/a/(a^2-b^ 2)/d/(a+b*sec(d*x+c))
Time = 2.26 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.23 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\frac {(b+a \cos (c+d x)) (A+B \sec (c+d x)) \left ((-2 A b+a B) (c+d x) (b+a \cos (c+d x)) \sec (c+d x)+\frac {2 b \left (-3 a^2 A b+2 A b^3+2 a^3 B-a b^2 B\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x)) \sec (c+d x)}{\left (a^2-b^2\right )^{3/2}}+\frac {a b^2 (-A b+a B) \tan (c+d x)}{(a-b) (a+b)}+a A (b+a \cos (c+d x)) \tan (c+d x)\right )}{a^3 d (B+A \cos (c+d x)) (a+b \sec (c+d x))^2} \]
((b + a*Cos[c + d*x])*(A + B*Sec[c + d*x])*((-2*A*b + a*B)*(c + d*x)*(b + a*Cos[c + d*x])*Sec[c + d*x] + (2*b*(-3*a^2*A*b + 2*A*b^3 + 2*a^3*B - a*b^ 2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d *x])*Sec[c + d*x])/(a^2 - b^2)^(3/2) + (a*b^2*(-(A*b) + a*B)*Tan[c + d*x]) /((a - b)*(a + b)) + a*A*(b + a*Cos[c + d*x])*Tan[c + d*x]))/(a^3*d*(B + A *Cos[c + d*x])*(a + b*Sec[c + d*x])^2)
Time = 1.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {3042, 4518, 25, 3042, 4592, 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.
| \(\displaystyle \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4518 |
| \(\displaystyle \frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {\cos (c+d x) \left (A a^2+b B a-(A b-a B) \sec (c+d x) a-2 A b^2+b (A b-a B) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) \left (A a^2+b B a-(A b-a B) \sec (c+d x) a-2 A b^2+b (A b-a B) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {A a^2+b B a-(A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a-2 A b^2+b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\frac {\left (a^2 A+a b B-2 A b^2\right ) \sin (c+d x)}{a d}-\frac {\int \frac {\left (a^2-b^2\right ) (2 A b-a B)-a b (A b-a B) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2 A+a b B-2 A b^2\right ) \sin (c+d x)}{a d}-\frac {\int \frac {\left (a^2-b^2\right ) (2 A b-a B)-a b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\frac {\left (a^2 A+a b B-2 A b^2\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) (2 A b-a B)}{a}-\frac {b \left (-2 a^3 B+3 a^2 A b+a b^2 B-2 A b^3\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2 A+a b B-2 A b^2\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) (2 A b-a B)}{a}-\frac {b \left (-2 a^3 B+3 a^2 A b+a b^2 B-2 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\frac {\left (a^2 A+a b B-2 A b^2\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) (2 A b-a B)}{a}-\frac {\left (-2 a^3 B+3 a^2 A b+a b^2 B-2 A b^3\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2 A+a b B-2 A b^2\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) (2 A b-a B)}{a}-\frac {\left (-2 a^3 B+3 a^2 A b+a b^2 B-2 A b^3\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\left (a^2 A+a b B-2 A b^2\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) (2 A b-a B)}{a}-\frac {2 \left (-2 a^3 B+3 a^2 A b+a b^2 B-2 A b^3\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b (A b-a B) \sin (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\frac {\left (a^2 A+a b B-2 A b^2\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) (2 A b-a B)}{a}-\frac {2 b \left (-2 a^3 B+3 a^2 A b+a b^2 B-2 A b^3\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a}}{a \left (a^2-b^2\right )}\) |
(b*(A*b - a*B)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) + (-(( ((a^2 - b^2)*(2*A*b - a*B)*x)/a - (2*b*(3*a^2*A*b - 2*A*b^3 - 2*a^3*B + a* b^2*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b] *Sqrt[a + b]*d))/a) + ((a^2*A - 2*A*b^2 + a*b*B)*Sin[c + d*x])/(a*d))/(a*( a^2 - b^2))
3.4.25.3.1 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[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[b*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(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)*(d*Csc[e + f*x])^n*Simp[A*(a^2*(m + 1) - b^2*(m + n + 1)) + a*b*B*n - a*(A*b - a*B)*(m + 1)*Csc[e + f*x] + b*(A*b - a*B)*(m + n + 2) *Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A* b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && IL tQ[n, 0])
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Time = 1.12 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\left (2 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{3}}-\frac {2 b \left (-\frac {a b \left (A b -B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (3 A \,a^{2} b -2 A \,b^{3}-2 B \,a^{3}+B a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{3}}}{d}\) | \(213\) |
| default | \(\frac {-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\left (2 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{3}}-\frac {2 b \left (-\frac {a b \left (A b -B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (3 A \,a^{2} b -2 A \,b^{3}-2 B \,a^{3}+B a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{3}}}{d}\) | \(213\) |
| risch | \(-\frac {2 x A b}{a^{3}}+\frac {x B}{a^{2}}-\frac {i A \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{2} d}+\frac {i A \,{\mathrm e}^{-i \left (d x +c \right )}}{2 a^{2} d}+\frac {2 i b^{2} \left (-A b +B a \right ) \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a^{3} \left (a^{2}-b^{2}\right ) d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}-\frac {2 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B b}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}+\frac {2 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B b}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}\) | \(818\) |
1/d*(-2/a^3*(-A*a*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)+(2*A*b-B*a)* arctan(tan(1/2*d*x+1/2*c)))-2*b/a^3*(-a*b*(A*b-B*a)/(a^2-b^2)*tan(1/2*d*x+ 1/2*c)/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)-(3*A*a^2*b-2*A* b^3-2*B*a^3+B*a*b^2)/(a-b)/(a+b)/((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. 364 vs. \(2 (171) = 342\).
Time = 0.37 (sec) , antiderivative size = 788, normalized size of antiderivative = 4.38 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\left [\frac {2 \, {\left (B a^{6} - 2 \, A a^{5} b - 2 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3} + B a^{2} b^{4} - 2 \, A a b^{5}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (B a^{5} b - 2 \, A a^{4} b^{2} - 2 \, B a^{3} b^{3} + 4 \, A a^{2} b^{4} + B a b^{5} - 2 \, A b^{6}\right )} d x + {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - B a b^{4} + 2 \, A b^{5} + {\left (2 \, B a^{4} b - 3 \, A a^{3} b^{2} - B a^{2} b^{3} + 2 \, A a b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + 2 \, {\left (A a^{5} b + B a^{4} b^{2} - 3 \, A a^{3} b^{3} - B a^{2} b^{4} + 2 \, A a b^{5} + {\left (A a^{6} - 2 \, A a^{4} b^{2} + A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d\right )}}, \frac {{\left (B a^{6} - 2 \, A a^{5} b - 2 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3} + B a^{2} b^{4} - 2 \, A a b^{5}\right )} d x \cos \left (d x + c\right ) + {\left (B a^{5} b - 2 \, A a^{4} b^{2} - 2 \, B a^{3} b^{3} + 4 \, A a^{2} b^{4} + B a b^{5} - 2 \, A b^{6}\right )} d x - {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - B a b^{4} + 2 \, A b^{5} + {\left (2 \, B a^{4} b - 3 \, A a^{3} b^{2} - B a^{2} b^{3} + 2 \, A a b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + {\left (A a^{5} b + B a^{4} b^{2} - 3 \, A a^{3} b^{3} - B a^{2} b^{4} + 2 \, A a b^{5} + {\left (A a^{6} - 2 \, A a^{4} b^{2} + A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{{\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d}\right ] \]
[1/2*(2*(B*a^6 - 2*A*a^5*b - 2*B*a^4*b^2 + 4*A*a^3*b^3 + B*a^2*b^4 - 2*A*a *b^5)*d*x*cos(d*x + c) + 2*(B*a^5*b - 2*A*a^4*b^2 - 2*B*a^3*b^3 + 4*A*a^2* b^4 + B*a*b^5 - 2*A*b^6)*d*x + (2*B*a^3*b^2 - 3*A*a^2*b^3 - B*a*b^4 + 2*A* b^5 + (2*B*a^4*b - 3*A*a^3*b^2 - B*a^2*b^3 + 2*A*a*b^4)*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*sqr t(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*(A*a^5*b + B*a^4*b^2 - 3*A*a^3*b^ 3 - B*a^2*b^4 + 2*A*a*b^5 + (A*a^6 - 2*A*a^4*b^2 + A*a^2*b^4)*cos(d*x + c) )*sin(d*x + c))/((a^8 - 2*a^6*b^2 + a^4*b^4)*d*cos(d*x + c) + (a^7*b - 2*a ^5*b^3 + a^3*b^5)*d), ((B*a^6 - 2*A*a^5*b - 2*B*a^4*b^2 + 4*A*a^3*b^3 + B* a^2*b^4 - 2*A*a*b^5)*d*x*cos(d*x + c) + (B*a^5*b - 2*A*a^4*b^2 - 2*B*a^3*b ^3 + 4*A*a^2*b^4 + B*a*b^5 - 2*A*b^6)*d*x - (2*B*a^3*b^2 - 3*A*a^2*b^3 - B *a*b^4 + 2*A*b^5 + (2*B*a^4*b - 3*A*a^3*b^2 - B*a^2*b^3 + 2*A*a*b^4)*cos(d *x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/(( a^2 - b^2)*sin(d*x + c))) + (A*a^5*b + B*a^4*b^2 - 3*A*a^3*b^3 - B*a^2*b^4 + 2*A*a*b^5 + (A*a^6 - 2*A*a^4*b^2 + A*a^2*b^4)*cos(d*x + c))*sin(d*x + c ))/((a^8 - 2*a^6*b^2 + a^4*b^4)*d*cos(d*x + c) + (a^7*b - 2*a^5*b^3 + a^3* b^5)*d)]
\[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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. 1107 vs. \(2 (171) = 342\).
Time = 0.44 (sec) , antiderivative size = 1107, normalized size of antiderivative = 6.15 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
((B*a^8 - 2*A*a^7*b - 3*B*a^7*b + 5*A*a^6*b^2 - 2*B*a^6*b^2 + 4*A*a^5*b^3 + 5*B*a^5*b^3 - 9*A*a^4*b^4 + B*a^4*b^4 - 2*A*a^3*b^5 - 2*B*a^3*b^5 + 4*A* a^2*b^6 - B*a^3*abs(-a^5 + a^3*b^2) + 2*A*a^2*b*abs(-a^5 + a^3*b^2) - B*a^ 2*b*abs(-a^5 + a^3*b^2) + A*a*b^2*abs(-a^5 + a^3*b^2) + B*a*b^2*abs(-a^5 + a^3*b^2) - 2*A*b^3*abs(-a^5 + a^3*b^2))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(tan(1/2*d*x + 1/2*c)/sqrt(-(a^4*b - a^2*b^3 + sqrt((a^5 + a^4*b - a^3*b^2 - a^2*b^3)*(a^5 - a^4*b - a^3*b^2 + a^2*b^3) + (a^4*b - a^2*b^3) ^2))/(a^5 - a^4*b - a^3*b^2 + a^2*b^3))))/(a^4*b*abs(-a^5 + a^3*b^2) - a^2 *b^3*abs(-a^5 + a^3*b^2) + (a^5 - a^3*b^2)^2) - ((2*a^2*b + a*b^2 - 2*b^3) *sqrt(-a^2 + b^2)*A*abs(-a^5 + a^3*b^2)*abs(-a + b) - (a^3 + a^2*b - a*b^2 )*sqrt(-a^2 + b^2)*B*abs(-a^5 + a^3*b^2)*abs(-a + b) + (2*a^7*b - 5*a^6*b^ 2 - 4*a^5*b^3 + 9*a^4*b^4 + 2*a^3*b^5 - 4*a^2*b^6)*sqrt(-a^2 + b^2)*A*abs( -a + b) - (a^8 - 3*a^7*b - 2*a^6*b^2 + 5*a^5*b^3 + a^4*b^4 - 2*a^3*b^5)*sq rt(-a^2 + b^2)*B*abs(-a + b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(t an(1/2*d*x + 1/2*c)/sqrt(-(a^4*b - a^2*b^3 - sqrt((a^5 + a^4*b - a^3*b^2 - a^2*b^3)*(a^5 - a^4*b - a^3*b^2 + a^2*b^3) + (a^4*b - a^2*b^3)^2))/(a^5 - a^4*b - a^3*b^2 + a^2*b^3))))/((a^5 - a^3*b^2)^2*(a^2 - 2*a*b + b^2) - (a ^6*b - 2*a^5*b^2 + 2*a^3*b^4 - a^2*b^5)*abs(-a^5 + a^3*b^2)) + 2*(A*a^3*ta n(1/2*d*x + 1/2*c)^3 - A*a^2*b*tan(1/2*d*x + 1/2*c)^3 - A*a*b^2*tan(1/2*d* x + 1/2*c)^3 - B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^3*tan(1/2*d*x + 1...
Time = 19.67 (sec) , antiderivative size = 3264, normalized size of antiderivative = 18.13 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
((2*tan(c/2 + (d*x)/2)^3*(A*a*b^2 - 2*A*b^3 - A*a^3 + A*a^2*b + B*a*b^2))/ (a^2*(a + b)*(a - b)) + (2*tan(c/2 + (d*x)/2)*(A*a^3 - 2*A*b^3 - A*a*b^2 + A*a^2*b + B*a*b^2))/(a^2*(a + b)*(a - b)))/(d*(a + b - tan(c/2 + (d*x)/2) ^4*(a - b) + 2*b*tan(c/2 + (d*x)/2)^2)) + (log(tan(c/2 + (d*x)/2) - 1i)*(2 *A*b - B*a)*1i)/(a^3*d) - (log(tan(c/2 + (d*x)/2) + 1i)*(A*b*2i - B*a*1i)) /(a^3*d) - (b*atan(((b*((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + B^2*a^8 - 8*A^ 2*a*b^7 - 2*B^2*a^7*b - 16*A^2*a^2*b^6 + 16*A^2*a^3*b^5 + 5*A^2*a^4*b^4 - 8*A^2*a^5*b^3 + 4*A^2*a^6*b^2 + 2*B^2*a^2*b^6 - 2*B^2*a^3*b^5 - 5*B^2*a^4* b^4 + 4*B^2*a^5*b^3 + 3*B^2*a^6*b^2 - 8*A*B*a*b^7 - 4*A*B*a^7*b + 8*A*B*a^ 2*b^6 + 18*A*B*a^3*b^5 - 16*A*B*a^4*b^4 - 8*A*B*a^5*b^3 + 8*A*B*a^6*b^2))/ (a^6*b + a^7 - a^4*b^3 - a^5*b^2) + (b*((32*(A*a^7*b^5 - 2*A*a^6*b^6 - B*a ^12 + 5*A*a^8*b^4 - 3*A*a^9*b^3 - 3*A*a^10*b^2 + B*a^7*b^5 - 3*B*a^9*b^3 + B*a^10*b^2 + 2*A*a^11*b + 2*B*a^11*b))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) - (32*b*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(2*A*b^3 + 2*B*a^3 - 3*A*a^2*b - B*a*b^2)*(2*a^11*b - 2*a^6*b^6 + 2*a^7*b^5 + 4*a^8*b^4 - 4*a ^9*b^3 - 2*a^10*b^2))/((a^6*b + a^7 - a^4*b^3 - a^5*b^2)*(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(2*A*b^3 + 2*B*a^3 - 3*A*a^2*b - B*a*b^2))/(a^9 - a^3*b^6 + 3*a^5*b^4 - 3*a^7*b^2))*((a + b)^3* (a - b)^3)^(1/2)*(2*A*b^3 + 2*B*a^3 - 3*A*a^2*b - B*a*b^2)*1i)/(a^9 - a^3* b^6 + 3*a^5*b^4 - 3*a^7*b^2) + (b*((32*tan(c/2 + (d*x)/2)*(8*A^2*b^8 + ...