Integrand size = 31, antiderivative size = 261 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\frac {\left (a^2 A+6 A b^2-4 a b B\right ) x}{2 a^4}-\frac {2 b^2 \left (4 a^2 A b-3 A b^3-3 a^3 B+2 a b^2 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{3/2} (a+b)^{3/2} d}-\frac {\left (2 a^2 A b-3 A b^3-a^3 B+2 a b^2 B\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}+\frac {\left (a^2 A-3 A b^2+2 a b B\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \] Output:
1/2*(A*a^2+6*A*b^2-4*B*a*b)*x/a^4-2*b^2*(4*A*a^2*b-3*A*b^3-3*B*a^3+2*B*a*b ^2)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^4/(a-b)^(3/2)/(a +b)^(3/2)/d-(2*A*a^2*b-3*A*b^3-B*a^3+2*B*a*b^2)*sin(d*x+c)/a^3/(a^2-b^2)/d +1/2*(A*a^2-3*A*b^2+2*B*a*b)*cos(d*x+c)*sin(d*x+c)/a^2/(a^2-b^2)/d+b*(A*b- B*a)*cos(d*x+c)*sin(d*x+c)/a/(a^2-b^2)/d/(a+b*sec(d*x+c))
Time = 1.45 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\frac {2 \left (a^2 A+6 A b^2-4 a b B\right ) (c+d x)-\frac {8 b^2 \left (-4 a^2 A b+3 A b^3+3 a^3 B-2 a b^2 B\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+4 a (-2 A b+a B) \sin (c+d x)-\frac {4 a b^3 (-A b+a B) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))}+a^2 A \sin (2 (c+d x))}{4 a^4 d} \] Input:
Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]
Output:
(2*(a^2*A + 6*A*b^2 - 4*a*b*B)*(c + d*x) - (8*b^2*(-4*a^2*A*b + 3*A*b^3 + 3*a^3*B - 2*a*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]) /(a^2 - b^2)^(3/2) + 4*a*(-2*A*b + a*B)*Sin[c + d*x] - (4*a*b^3*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + a^2*A*Sin[2*(c + d*x)])/(4*a^4*d)
Time = 1.74 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {3042, 4518, 25, 3042, 4592, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4518 |
| \(\displaystyle \frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {\cos ^2(c+d x) \left (A a^2+2 b B a-(A b-a B) \sec (c+d x) a-3 A b^2+2 b (A b-a B) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) \left (A a^2+2 b B a-(A b-a B) \sec (c+d x) a-3 A b^2+2 b (A b-a B) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {A a^2+2 b B a-(A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a-3 A b^2+2 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\int \frac {\cos (c+d x) \left (-b \left (A a^2+2 b B a-3 A b^2\right ) \sec ^2(c+d x)-a \left (A a^2-2 b B a+A b^2\right ) \sec (c+d x)+2 \left (-B a^3+2 A b a^2+2 b^2 B a-3 A b^3\right )\right )}{a+b \sec (c+d x)}dx}{2 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\int \frac {-b \left (A a^2+2 b B a-3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a \left (A a^2-2 b B a+A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (-B a^3+2 A b a^2+2 b^2 B a-3 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right ) \sin (c+d x)}{a d}-\frac {\int \frac {\left (a^2-b^2\right ) \left (A a^2-4 b B a+6 A b^2\right )+a b \left (A a^2+2 b B a-3 A b^2\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}}{2 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right ) \sin (c+d x)}{a d}-\frac {\int \frac {\left (a^2-b^2\right ) \left (A a^2-4 b B a+6 A b^2\right )+a b \left (A a^2+2 b B a-3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{2 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) \left (a^2 A-4 a b B+6 A b^2\right )}{a}-\frac {2 b^2 \left (-3 a^3 B+4 a^2 A b+2 a b^2 B-3 A b^3\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}}{a}}{2 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) \left (a^2 A-4 a b B+6 A b^2\right )}{a}-\frac {2 b^2 \left (-3 a^3 B+4 a^2 A b+2 a b^2 B-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a}}{2 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) \left (a^2 A-4 a b B+6 A b^2\right )}{a}-\frac {2 b \left (-3 a^3 B+4 a^2 A b+2 a b^2 B-3 A b^3\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}}{a}}{2 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) \left (a^2 A-4 a b B+6 A b^2\right )}{a}-\frac {2 b \left (-3 a^3 B+4 a^2 A b+2 a b^2 B-3 A b^3\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}}{a}}{2 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) \left (a^2 A-4 a b B+6 A b^2\right )}{a}-\frac {4 b \left (-3 a^3 B+4 a^2 A b+2 a b^2 B-3 A b^3\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a}}{2 a}}{a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {2 \left (a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right ) \sin (c+d x)}{a d}-\frac {\frac {x \left (a^2-b^2\right ) \left (a^2 A-4 a b B+6 A b^2\right )}{a}-\frac {4 b^2 \left (-3 a^3 B+4 a^2 A b+2 a b^2 B-3 A b^3\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}}{a}}{2 a}}{a \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]
Output:
(b*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) + (((a^2*A - 3*A*b^2 + 2*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - (-((((a^2 - b^2)*(a^2*A + 6*A*b^2 - 4*a*b*B)*x)/a - (4*b^2*(4*a^2*A*b - 3*A*b^3 - 3*a^3*B + 2*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) + (2*(2*a^2*A*b - 3*A*b^3 - a^ 3*B + 2*a*b^2*B)*Sin[c + d*x])/(a*d))/(2*a))/(a*(a^2 - b^2))
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 = 0.70 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-2 A a b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} A \,a^{2}-2 A a b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (A \,a^{2}+6 A \,b^{2}-4 B a b \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b^{2} \left (-\frac {a b \left (A b -B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (4 A \,a^{2} b -3 A \,b^{3}-3 B \,a^{3}+2 B a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4}}}{d}\) | \(270\) |
| default | \(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-2 A a b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} A \,a^{2}-2 A a b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (A \,a^{2}+6 A \,b^{2}-4 B a b \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b^{2} \left (-\frac {a b \left (A b -B a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (4 A \,a^{2} b -3 A \,b^{3}-3 B \,a^{3}+2 B a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{4}}}{d}\) | \(270\) |
| risch | \(\frac {A x}{2 a^{2}}+\frac {3 x A \,b^{2}}{a^{4}}-\frac {2 x B b}{a^{3}}+\frac {i {\mathrm e}^{i \left (d x +c \right )} A b}{a^{3} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{2 a^{2} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} A b}{a^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{2 a^{2} d}-\frac {2 i b^{3} \left (-A b +B a \right ) \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a^{4} \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 {i A \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{2} d}+\frac {i A \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{2} d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{\sqrt {a^{2}-b^{2}}\, a}\right ) A \,b^{3}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}-\frac {3 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}-\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {a^{2}-b^{2}}\, b}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\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}-\sqrt {a^{2}-b^{2}}\, b}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{\sqrt {a^{2}-b^{2}}\, a}\right ) A \,b^{3}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}+\frac {3 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{4}}+\frac {3 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {a^{2}-b^{2}}\, b}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\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}+\sqrt {a^{2}-b^{2}}\, b}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}\) | \(918\) |
Input:
int(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^2,x,method=_RETURNVERBO SE)
Output:
1/d*(2/a^4*(((-1/2*A*a^2-2*A*a*b+B*a^2)*tan(1/2*d*x+1/2*c)^3+(1/2*A*a^2-2* A*a*b+B*a^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^2+1/2*(A*a^2+6*A *b^2-4*B*a*b)*arctan(tan(1/2*d*x+1/2*c)))+2*b^2/a^4*(-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) -(4*A*a^2*b-3*A*b^3-3*B*a^3+2*B*a*b^2)/(a+b)/(a-b)/((a+b)*(a-b))^(1/2)*arc tanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))))
Time = 0.15 (sec) , antiderivative size = 971, normalized size of antiderivative = 3.72 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^2,x, algorithm="f ricas")
Output:
[1/2*((A*a^7 - 4*B*a^6*b + 4*A*a^5*b^2 + 8*B*a^4*b^3 - 11*A*a^3*b^4 - 4*B* a^2*b^5 + 6*A*a*b^6)*d*x*cos(d*x + c) + (A*a^6*b - 4*B*a^5*b^2 + 4*A*a^4*b ^3 + 8*B*a^3*b^4 - 11*A*a^2*b^5 - 4*B*a*b^6 + 6*A*b^7)*d*x - (3*B*a^3*b^3 - 4*A*a^2*b^4 - 2*B*a*b^5 + 3*A*b^6 + (3*B*a^4*b^2 - 4*A*a^3*b^3 - 2*B*a^2 *b^4 + 3*A*a*b^5)*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*B*a^6*b - 4*A*a^5*b^2 - 6*B*a^4*b^3 + 10*A*a^3*b^4 + 4*B*a^2*b^5 - 6*A *a*b^6 + (A*a^7 - 2*A*a^5*b^2 + A*a^3*b^4)*cos(d*x + c)^2 + (2*B*a^7 - 3*A *a^6*b - 4*B*a^5*b^2 + 6*A*a^4*b^3 + 2*B*a^3*b^4 - 3*A*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/((a^9 - 2*a^7*b^2 + a^5*b^4)*d*cos(d*x + c) + (a^8*b - 2 *a^6*b^3 + a^4*b^5)*d), 1/2*((A*a^7 - 4*B*a^6*b + 4*A*a^5*b^2 + 8*B*a^4*b^ 3 - 11*A*a^3*b^4 - 4*B*a^2*b^5 + 6*A*a*b^6)*d*x*cos(d*x + c) + (A*a^6*b - 4*B*a^5*b^2 + 4*A*a^4*b^3 + 8*B*a^3*b^4 - 11*A*a^2*b^5 - 4*B*a*b^6 + 6*A*b ^7)*d*x + 2*(3*B*a^3*b^3 - 4*A*a^2*b^4 - 2*B*a*b^5 + 3*A*b^6 + (3*B*a^4*b^ 2 - 4*A*a^3*b^3 - 2*B*a^2*b^4 + 3*A*a*b^5)*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))) + (2*B*a^6*b - 4*A*a^5*b^2 - 6*B*a^4*b^3 + 10*A*a^3*b^4 + 4*B*a^2*b^5 - 6* A*a*b^6 + (A*a^7 - 2*A*a^5*b^2 + A*a^3*b^4)*cos(d*x + c)^2 + (2*B*a^7 - 3* A*a^6*b - 4*B*a^5*b^2 + 6*A*a^4*b^3 + 2*B*a^3*b^4 - 3*A*a^2*b^5)*cos(d*...
\[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(cos(d*x+c)**2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))**2,x)
Output:
Integral((A + B*sec(c + d*x))*cos(c + d*x)**2/(a + b*sec(c + d*x))**2, x)
Exception generated. \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^2,x, algorithm="m axima")
Output:
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
Time = 0.19 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.30 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {4 \, {\left (3 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3} - 2 \, B a b^{4} + 3 \, A b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} - a^{4} b^{2}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {4 \, {\left (B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} + \frac {{\left (A a^{2} - 4 \, B a b + 6 \, A b^{2}\right )} {\left (d x + c\right )}}{a^{4}} - \frac {2 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}}}{2 \, d} \] Input:
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^2,x, algorithm="g iac")
Output:
1/2*(4*(3*B*a^3*b^2 - 4*A*a^2*b^3 - 2*B*a*b^4 + 3*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^6 - a^4*b^2)*sqrt(-a^2 + b^2)) + 4*(B*a*b^3*tan(1/2*d*x + 1/2*c) - A*b^4*tan(1/2*d*x + 1/2*c))/((a^5 - a^3 *b^2)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)) + (A* a^2 - 4*B*a*b + 6*A*b^2)*(d*x + c)/a^4 - 2*(A*a*tan(1/2*d*x + 1/2*c)^3 - 2 *B*a*tan(1/2*d*x + 1/2*c)^3 + 4*A*b*tan(1/2*d*x + 1/2*c)^3 - A*a*tan(1/2*d *x + 1/2*c) - 2*B*a*tan(1/2*d*x + 1/2*c) + 4*A*b*tan(1/2*d*x + 1/2*c))/((t an(1/2*d*x + 1/2*c)^2 + 1)^2*a^3))/d
Time = 20.62 (sec) , antiderivative size = 6731, normalized size of antiderivative = 25.79 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^2*(A + B/cos(c + d*x)))/(a + b/cos(c + d*x))^2,x)
Output:
(b^2*atan(((b^2*((a + b)^3*(a - b)^3)^(1/2)*((8*tan(c/2 + (d*x)/2)*(A^2*a^ 10 + 72*A^2*b^10 - 72*A^2*a*b^9 - 2*A^2*a^9*b - 120*A^2*a^2*b^8 + 120*A^2* a^3*b^7 + 17*A^2*a^4*b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^ 3 + 11*A^2*a^8*b^2 + 32*B^2*a^2*b^8 - 32*B^2*a^3*b^7 - 64*B^2*a^4*b^6 + 64 *B^2*a^5*b^5 + 20*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 16*B^2*a^8*b^2 - 96*A*B*a *b^9 - 8*A*B*a^9*b + 96*A*B*a^2*b^8 + 176*A*B*a^3*b^7 - 176*A*B*a^4*b^6 - 40*A*B*a^5*b^5 + 64*A*B*a^6*b^4 - 40*A*B*a^7*b^3 + 16*A*B*a^8*b^2))/(a^8*b + a^9 - a^6*b^3 - a^7*b^2) + (b^2*((8*(2*A*a^15 - 12*A*a^8*b^7 + 6*A*a^9* b^6 + 28*A*a^10*b^5 - 14*A*a^11*b^4 - 16*A*a^12*b^3 + 6*A*a^13*b^2 + 8*B*a ^9*b^6 - 4*B*a^10*b^5 - 20*B*a^11*b^4 + 12*B*a^12*b^3 + 12*B*a^13*b^2 - 8* B*a^14*b))/(a^11*b + a^12 - a^9*b^3 - a^10*b^2) - (8*b^2*tan(c/2 + (d*x)/2 )*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^3 + 3*B*a^3 - 4*A*a^2*b - 2*B*a*b^2)* (8*a^13*b - 8*a^8*b^6 + 8*a^9*b^5 + 16*a^10*b^4 - 16*a^11*b^3 - 8*a^12*b^2 ))/((a^8*b + a^9 - a^6*b^3 - a^7*b^2)*(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8* b^2)))*((a + b)^3*(a - b)^3)^(1/2)*(3*A*b^3 + 3*B*a^3 - 4*A*a^2*b - 2*B*a* b^2))/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2))*(3*A*b^3 + 3*B*a^3 - 4*A*a ^2*b - 2*B*a*b^2)*1i)/(a^10 - a^4*b^6 + 3*a^6*b^4 - 3*a^8*b^2) + (b^2*((a + b)^3*(a - b)^3)^(1/2)*((8*tan(c/2 + (d*x)/2)*(A^2*a^10 + 72*A^2*b^10 - 7 2*A^2*a*b^9 - 2*A^2*a^9*b - 120*A^2*a^2*b^8 + 120*A^2*a^3*b^7 + 17*A^2*a^4 *b^6 - 26*A^2*a^5*b^5 + 23*A^2*a^6*b^4 - 20*A^2*a^7*b^3 + 11*A^2*a^8*b^...
Time = 0.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.67 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx=\frac {-4 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {-a^{2}+b^{2}}}\right ) b^{3}+\cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4}-\cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{2}-2 \sin \left (d x +c \right ) a^{3} b +2 \sin \left (d x +c \right ) a \,b^{3}+a^{4} c +a^{4} d x +a^{2} b^{2} c +a^{2} b^{2} d x -2 b^{4} c -2 b^{4} d x}{2 a^{3} d \left (a^{2}-b^{2}\right )} \] Input:
int(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^2,x)
Output:
( - 4*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/ sqrt( - a**2 + b**2))*b**3 + cos(c + d*x)*sin(c + d*x)*a**4 - cos(c + d*x) *sin(c + d*x)*a**2*b**2 - 2*sin(c + d*x)*a**3*b + 2*sin(c + d*x)*a*b**3 + a**4*c + a**4*d*x + a**2*b**2*c + a**2*b**2*d*x - 2*b**4*c - 2*b**4*d*x)/( 2*a**3*d*(a**2 - b**2))