Integrand size = 31, antiderivative size = 538 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\frac {\left (a^2 A+20 A b^2-8 a b B\right ) x}{2 a^6}-\frac {b^2 \left (40 a^6 A b-84 a^4 A b^3+69 a^2 A b^5-20 A b^7-20 a^7 B+35 a^5 b^2 B-28 a^3 b^4 B+8 a b^6 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 (a-b)^{7/2} (a+b)^{7/2} d}-\frac {\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \] Output:
1/2*(A*a^2+20*A*b^2-8*B*a*b)*x/a^6-b^2*(40*A*a^6*b-84*A*a^4*b^3+69*A*a^2*b ^5-20*A*b^7-20*B*a^7+35*B*a^5*b^2-28*B*a^3*b^4+8*B*a*b^6)*arctanh((a-b)^(1 /2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^6/(a-b)^(7/2)/(a+b)^(7/2)/d-1/6*(24* A*a^6*b-146*A*a^4*b^3+167*A*a^2*b^5-60*A*b^7-6*B*a^7+65*B*a^5*b^2-68*B*a^3 *b^4+24*B*a*b^6)*sin(d*x+c)/a^5/(a^2-b^2)^3/d+1/2*(A*a^6-23*A*a^4*b^2+27*A *a^2*b^4-10*A*b^6+12*B*a^5*b-11*B*a^3*b^3+4*B*a*b^5)*cos(d*x+c)*sin(d*x+c) /a^4/(a^2-b^2)^3/d+1/3*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))^3+1/6*b*(10*A*a^2*b-5*A*b^3-7*B*a^3+2*B*a*b^2)*cos(d*x+c)*si n(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2+1/6*b*(48*A*a^4*b-53*A*a^2*b ^3+20*A*b^5-27*B*a^5+20*B*a^3*b^2-8*B*a*b^4)*cos(d*x+c)*sin(d*x+c)/a^3/(a^ 2-b^2)^3/d/(a+b*sec(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(1452\) vs. \(2(538)=1076\).
Time = 5.89 (sec) , antiderivative size = 1452, normalized size of antiderivative = 2.70 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx =\text {Too large to display} \] Input:
Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]
Output:
((-96*b^2*(-40*a^6*A*b + 84*a^4*A*b^3 - 69*a^2*A*b^5 + 20*A*b^7 + 20*a^7*B - 35*a^5*b^2*B + 28*a^3*b^4*B - 8*a*b^6*B)*ArcTanh[((-a + b)*Tan[(c + d*x )/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (72*a^10*A*b*c + 1272*a^8*A*b^ 3*c - 3288*a^6*A*b^5*c + 1512*a^4*A*b^7*c + 1392*a^2*A*b^9*c - 960*A*b^11* c - 576*a^9*b^2*B*c + 1344*a^7*b^4*B*c - 576*a^5*b^6*B*c - 576*a^3*b^8*B*c + 384*a*b^10*B*c + 72*a^10*A*b*d*x + 1272*a^8*A*b^3*d*x - 3288*a^6*A*b^5* d*x + 1512*a^4*A*b^7*d*x + 1392*a^2*A*b^9*d*x - 960*A*b^11*d*x - 576*a^9*b ^2*B*d*x + 1344*a^7*b^4*B*d*x - 576*a^5*b^6*B*d*x - 576*a^3*b^8*B*d*x + 38 4*a*b^10*B*d*x + 36*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(a^2*A + 20*A*b^2 - 8*a* b*B)*(c + d*x)*Cos[c + d*x] + 72*a^2*b*(a^2 - b^2)^3*(a^2*A + 20*A*b^2 - 8 *a*b*B)*(c + d*x)*Cos[2*(c + d*x)] + 12*a^11*A*c*Cos[3*(c + d*x)] + 204*a^ 9*A*b^2*c*Cos[3*(c + d*x)] - 684*a^7*A*b^4*c*Cos[3*(c + d*x)] + 708*a^5*A* b^6*c*Cos[3*(c + d*x)] - 240*a^3*A*b^8*c*Cos[3*(c + d*x)] - 96*a^10*b*B*c* Cos[3*(c + d*x)] + 288*a^8*b^3*B*c*Cos[3*(c + d*x)] - 288*a^6*b^5*B*c*Cos[ 3*(c + d*x)] + 96*a^4*b^7*B*c*Cos[3*(c + d*x)] + 12*a^11*A*d*x*Cos[3*(c + d*x)] + 204*a^9*A*b^2*d*x*Cos[3*(c + d*x)] - 684*a^7*A*b^4*d*x*Cos[3*(c + d*x)] + 708*a^5*A*b^6*d*x*Cos[3*(c + d*x)] - 240*a^3*A*b^8*d*x*Cos[3*(c + d*x)] - 96*a^10*b*B*d*x*Cos[3*(c + d*x)] + 288*a^8*b^3*B*d*x*Cos[3*(c + d* x)] - 288*a^6*b^5*B*d*x*Cos[3*(c + d*x)] + 96*a^4*b^7*B*d*x*Cos[3*(c + d*x )] + 6*a^11*A*Sin[c + d*x] - 270*a^9*A*b^2*Sin[c + d*x] + 750*a^7*A*b^4...
Time = 4.11 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.08, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.710, Rules used = {3042, 4518, 25, 3042, 4588, 25, 3042, 4588, 25, 3042, 4592, 27, 3042, 4592, 27, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4518 |
| \(\displaystyle \frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\int -\frac {\cos ^2(c+d x) \left (3 A a^2+2 b B a-3 (A b-a B) \sec (c+d x) a-5 A b^2+4 b (A b-a B) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) \left (3 A a^2+2 b B a-3 (A b-a B) \sec (c+d x) a-5 A b^2+4 b (A b-a B) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 A a^2+2 b B a-3 (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a-5 A b^2+4 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\int -\frac {\cos ^2(c+d x) \left (3 b \left (-7 B a^3+10 A b a^2+2 b^2 B a-5 A b^3\right ) \sec ^2(c+d x)-2 a \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \sec (c+d x)+2 \left (3 A a^4+9 b B a^3-18 A b^2 a^2-4 b^3 B a+10 A b^4\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^2(c+d x) \left (3 b \left (-7 B a^3+10 A b a^2+2 b^2 B a-5 A b^3\right ) \sec ^2(c+d x)-2 a \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \sec (c+d x)+2 \left (3 A a^4+9 b B a^3-18 A b^2 a^2-4 b^3 B a+10 A b^4\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 b \left (-7 B a^3+10 A b a^2+2 b^2 B a-5 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (3 A a^4+9 b B a^3-18 A b^2 a^2-4 b^3 B a+10 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\frac {\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int -\frac {\cos ^2(c+d x) \left (2 b \left (-27 B a^5+48 A b a^4+20 b^2 B a^3-53 A b^3 a^2-8 b^4 B a+20 A b^5\right ) \sec ^2(c+d x)-a \left (-6 B a^5+18 A b a^4-7 b^2 B a^3-8 A b^3 a^2-2 b^4 B a+5 A b^5\right ) \sec (c+d x)+6 \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right )\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\cos ^2(c+d x) \left (2 b \left (-27 B a^5+48 A b a^4+20 b^2 B a^3-53 A b^3 a^2-8 b^4 B a+20 A b^5\right ) \sec ^2(c+d x)-a \left (-6 B a^5+18 A b a^4-7 b^2 B a^3-8 A b^3 a^2-2 b^4 B a+5 A b^5\right ) \sec (c+d x)+6 \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right )\right )}{a+b \sec (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 b \left (-27 B a^5+48 A b a^4+20 b^2 B a^3-53 A b^3 a^2-8 b^4 B a+20 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a \left (-6 B a^5+18 A b a^4-7 b^2 B a^3-8 A b^3 a^2-2 b^4 B a+5 A b^5\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+6 \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\int \frac {2 \cos (c+d x) \left (-6 B a^7+24 A b a^6+65 b^2 B a^5-146 A b^3 a^4-68 b^4 B a^3+167 A b^5 a^2+24 b^6 B a-\left (3 A a^6-18 b B a^5+27 A b^2 a^4+7 b^3 B a^3-25 A b^4 a^2-4 b^5 B a+10 A b^6\right ) \sec (c+d x) a-60 A b^7-3 b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{2 a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\int \frac {\cos (c+d x) \left (-6 B a^7+24 A b a^6+65 b^2 B a^5-146 A b^3 a^4-68 b^4 B a^3+167 A b^5 a^2+24 b^6 B a-\left (3 A a^6-18 b B a^5+27 A b^2 a^4+7 b^3 B a^3-25 A b^4 a^2-4 b^5 B a+10 A b^6\right ) \sec (c+d x) a-60 A b^7-3 b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\int \frac {-6 B a^7+24 A b a^6+65 b^2 B a^5-146 A b^3 a^4-68 b^4 B a^3+167 A b^5 a^2+24 b^6 B a-\left (3 A a^6-18 b B a^5+27 A b^2 a^4+7 b^3 B a^3-25 A b^4 a^2-4 b^5 B a+10 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a-60 A b^7-3 b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \sin (c+d x)}{a d}-\frac {\int \frac {3 \left (\left (A a^2-8 b B a+20 A b^2\right ) \left (a^2-b^2\right )^3+a b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \sin (c+d x)}{a d}-\frac {3 \int \frac {\left (A a^2-8 b B a+20 A b^2\right ) \left (a^2-b^2\right )^3+a b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \sec (c+d x)}{a+b \sec (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \sin (c+d x)}{a d}-\frac {3 \int \frac {\left (A a^2-8 b B a+20 A b^2\right ) \left (a^2-b^2\right )^3+a b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4407 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right )}{a}-\frac {b^2 \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx}{a}\right )}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right )}{a}-\frac {b^2 \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right )}{a}-\frac {b \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{a}\right )}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right )}{a}-\frac {b \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{a}\right )}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right )}{a}-\frac {2 b \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b (A b-a B) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \sin (c+d x) \cos (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \sin (c+d x) \cos (c+d x)}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \sin (c+d x)}{a d}-\frac {3 \left (\frac {x \left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right )}{a}-\frac {2 b^2 \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]
Output:
(b*(A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((b*(10*a^2*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*Cos[c + d*x] *Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((b*(48*a^4*A* b - 53*a^2*A*b^3 + 20*A*b^5 - 27*a^5*B + 20*a^3*b^2*B - 8*a*b^4*B)*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) + ((3*(a^6*A - 23*a^4*A*b^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b ^5*B)*Cos[c + d*x]*Sin[c + d*x])/(a*d) - ((-3*(((a^2 - b^2)^3*(a^2*A + 20* A*b^2 - 8*a*b*B)*x)/a - (2*b^2*(40*a^6*A*b - 84*a^4*A*b^3 + 69*a^2*A*b^5 - 20*A*b^7 - 20*a^7*B + 35*a^5*b^2*B - 28*a^3*b^4*B + 8*a*b^6*B)*ArcTanh[(S qrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d))) /a + ((24*a^6*A*b - 146*a^4*A*b^3 + 167*a^2*A*b^5 - 60*A*b^7 - 6*a^7*B + 6 5*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*Sin[c + d*x])/(a*d))/a)/(a*(a^2 - b^2)))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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*b^2 - a*b*B + a^2*C)*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] + Sim p[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*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x ] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[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 = 2.22 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.14
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-4 A a b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} A \,a^{2}-4 A a b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (A \,a^{2}+20 A \,b^{2}-8 B a b \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}+\frac {2 b^{2} \left (\frac {-\frac {\left (30 A \,a^{4} b +6 A \,a^{3} b^{2}-34 A \,a^{2} b^{3}-3 A a \,b^{4}+12 A \,b^{5}-20 B \,a^{5}-5 B \,a^{4} b +18 B \,a^{3} b^{2}+2 B \,a^{2} b^{3}-6 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b -53 A \,a^{2} b^{3}+18 A \,b^{5}-30 B \,a^{5}+29 B \,a^{3} b^{2}-9 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b -6 A \,a^{3} b^{2}-34 A \,a^{2} b^{3}+3 A a \,b^{4}+12 A \,b^{5}-20 B \,a^{5}+5 B \,a^{4} b +18 B \,a^{3} b^{2}-2 B \,a^{2} b^{3}-6 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (40 A \,a^{6} b -84 A \,a^{4} b^{3}+69 A \,a^{2} b^{5}-20 A \,b^{7}-20 B \,a^{7}+35 B \,a^{5} b^{2}-28 B \,a^{3} b^{4}+8 B a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}}{d}\) | \(615\) |
| default | \(\frac {\frac {\frac {2 \left (\left (-\frac {1}{2} A \,a^{2}-4 A a b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {1}{2} A \,a^{2}-4 A a b +B \,a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (A \,a^{2}+20 A \,b^{2}-8 B a b \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6}}+\frac {2 b^{2} \left (\frac {-\frac {\left (30 A \,a^{4} b +6 A \,a^{3} b^{2}-34 A \,a^{2} b^{3}-3 A a \,b^{4}+12 A \,b^{5}-20 B \,a^{5}-5 B \,a^{4} b +18 B \,a^{3} b^{2}+2 B \,a^{2} b^{3}-6 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (45 A \,a^{4} b -53 A \,a^{2} b^{3}+18 A \,b^{5}-30 B \,a^{5}+29 B \,a^{3} b^{2}-9 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b -6 A \,a^{3} b^{2}-34 A \,a^{2} b^{3}+3 A a \,b^{4}+12 A \,b^{5}-20 B \,a^{5}+5 B \,a^{4} b +18 B \,a^{3} b^{2}-2 B \,a^{2} b^{3}-6 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (40 A \,a^{6} b -84 A \,a^{4} b^{3}+69 A \,a^{2} b^{5}-20 A \,b^{7}-20 B \,a^{7}+35 B \,a^{5} b^{2}-28 B \,a^{3} b^{4}+8 B a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{6}}}{d}\) | \(615\) |
| risch | \(\text {Expression too large to display}\) | \(2264\) |
Input:
int(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,x,method=_RETURNVERBO SE)
Output:
1/d*(2/a^6*(((-1/2*A*a^2-4*A*a*b+B*a^2)*tan(1/2*d*x+1/2*c)^3+(1/2*A*a^2-4* 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+20* A*b^2-8*B*a*b)*arctan(tan(1/2*d*x+1/2*c)))+2*b^2/a^6*((-1/2*(30*A*a^4*b+6* A*a^3*b^2-34*A*a^2*b^3-3*A*a*b^4+12*A*b^5-20*B*a^5-5*B*a^4*b+18*B*a^3*b^2+ 2*B*a^2*b^3-6*B*a*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2 *c)^5+2/3*(45*A*a^4*b-53*A*a^2*b^3+18*A*b^5-30*B*a^5+29*B*a^3*b^2-9*B*a*b^ 4)*a*b/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(30*A*a^4* b-6*A*a^3*b^2-34*A*a^2*b^3+3*A*a*b^4+12*A*b^5-20*B*a^5+5*B*a^4*b+18*B*a^3* b^2-2*B*a^2*b^3-6*B*a*b^4)*a*b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*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)^3-1/2*(40*A*a ^6*b-84*A*a^4*b^3+69*A*a^2*b^5-20*A*b^7-20*B*a^7+35*B*a^5*b^2-28*B*a^3*b^4 +8*B*a*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((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. 1416 vs. \(2 (516) = 1032\).
Time = 0.37 (sec) , antiderivative size = 2890, normalized size of antiderivative = 5.37 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,x, algorithm="f ricas")
Output:
[1/12*(6*(A*a^13 - 8*B*a^12*b + 16*A*a^11*b^2 + 32*B*a^10*b^3 - 74*A*a^9*b ^4 - 48*B*a^8*b^5 + 116*A*a^7*b^6 + 32*B*a^6*b^7 - 79*A*a^5*b^8 - 8*B*a^4* b^9 + 20*A*a^3*b^10)*d*x*cos(d*x + c)^3 + 18*(A*a^12*b - 8*B*a^11*b^2 + 16 *A*a^10*b^3 + 32*B*a^9*b^4 - 74*A*a^8*b^5 - 48*B*a^7*b^6 + 116*A*a^6*b^7 + 32*B*a^5*b^8 - 79*A*a^4*b^9 - 8*B*a^3*b^10 + 20*A*a^2*b^11)*d*x*cos(d*x + c)^2 + 18*(A*a^11*b^2 - 8*B*a^10*b^3 + 16*A*a^9*b^4 + 32*B*a^8*b^5 - 74*A *a^7*b^6 - 48*B*a^6*b^7 + 116*A*a^5*b^8 + 32*B*a^4*b^9 - 79*A*a^3*b^10 - 8 *B*a^2*b^11 + 20*A*a*b^12)*d*x*cos(d*x + c) + 6*(A*a^10*b^3 - 8*B*a^9*b^4 + 16*A*a^8*b^5 + 32*B*a^7*b^6 - 74*A*a^6*b^7 - 48*B*a^5*b^8 + 116*A*a^4*b^ 9 + 32*B*a^3*b^10 - 79*A*a^2*b^11 - 8*B*a*b^12 + 20*A*b^13)*d*x - 3*(20*B* a^7*b^5 - 40*A*a^6*b^6 - 35*B*a^5*b^7 + 84*A*a^4*b^8 + 28*B*a^3*b^9 - 69*A *a^2*b^10 - 8*B*a*b^11 + 20*A*b^12 + (20*B*a^10*b^2 - 40*A*a^9*b^3 - 35*B* a^8*b^4 + 84*A*a^7*b^5 + 28*B*a^6*b^6 - 69*A*a^5*b^7 - 8*B*a^4*b^8 + 20*A* a^3*b^9)*cos(d*x + c)^3 + 3*(20*B*a^9*b^3 - 40*A*a^8*b^4 - 35*B*a^7*b^5 + 84*A*a^6*b^6 + 28*B*a^5*b^7 - 69*A*a^4*b^8 - 8*B*a^3*b^9 + 20*A*a^2*b^10)* cos(d*x + c)^2 + 3*(20*B*a^8*b^4 - 40*A*a^7*b^5 - 35*B*a^6*b^6 + 84*A*a^5* b^7 + 28*B*a^4*b^8 - 69*A*a^3*b^9 - 8*B*a^2*b^10 + 20*A*a*b^11)*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*(6*B*a^10*b^3 - 24*A*a...
\[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \] Input:
integrate(cos(d*x+c)**2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))**4,x)
Output:
Integral((A + B*sec(c + d*x))*cos(c + d*x)**2/(a + b*sec(c + d*x))**4, x)
Exception generated. \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,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
Leaf count of result is larger than twice the leaf count of optimal. 1052 vs. \(2 (516) = 1032\).
Time = 0.25 (sec) , antiderivative size = 1052, normalized size of antiderivative = 1.96 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,x, algorithm="g iac")
Output:
1/6*(6*(20*B*a^7*b^2 - 40*A*a^6*b^3 - 35*B*a^5*b^4 + 84*A*a^4*b^5 + 28*B*a ^3*b^6 - 69*A*a^2*b^7 - 8*B*a*b^8 + 20*A*b^9)*(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^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*sqrt (-a^2 + b^2)) + 2*(60*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 - 90*A*a^6*b^4*tan( 1/2*d*x + 1/2*c)^5 - 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 + 162*A*a^5*b^5* tan(1/2*d*x + 1/2*c)^5 - 24*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 48*A*a^4*b^ 6*tan(1/2*d*x + 1/2*c)^5 + 117*B*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 213*A*a^ 3*b^7*tan(1/2*d*x + 1/2*c)^5 - 24*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 + 48*A* a^2*b^8*tan(1/2*d*x + 1/2*c)^5 - 42*B*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 81* A*a*b^9*tan(1/2*d*x + 1/2*c)^5 + 18*B*a*b^9*tan(1/2*d*x + 1/2*c)^5 - 36*A* b^10*tan(1/2*d*x + 1/2*c)^5 - 120*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^3 + 180*A *a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 236*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^3 - 3 92*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 - 152*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^3 + 284*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 + 36*B*a*b^9*tan(1/2*d*x + 1/2*c)^ 3 - 72*A*b^10*tan(1/2*d*x + 1/2*c)^3 + 60*B*a^7*b^3*tan(1/2*d*x + 1/2*c) - 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c) + 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c) - 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c) - 24*B*a^5*b^5*tan(1/2*d*x + 1/2*c) + 4 8*A*a^4*b^6*tan(1/2*d*x + 1/2*c) - 117*B*a^4*b^6*tan(1/2*d*x + 1/2*c) + 21 3*A*a^3*b^7*tan(1/2*d*x + 1/2*c) - 24*B*a^3*b^7*tan(1/2*d*x + 1/2*c) + ...
Time = 25.19 (sec) , antiderivative size = 14438, normalized size of antiderivative = 26.84 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^2*(A + B/cos(c + d*x)))/(a + b/cos(c + d*x))^4,x)
Output:
((tan(c/2 + (d*x)/2)*(A*a^8 + 20*A*b^8 + 2*B*a^8 - 59*A*a^2*b^6 - 27*A*a^3 *b^5 + 57*A*a^4*b^4 + 21*A*a^5*b^3 - 11*A*a^6*b^2 - 4*B*a^2*b^6 + 24*B*a^3 *b^5 + 11*B*a^4*b^4 - 26*B*a^5*b^3 - 6*B*a^6*b^2 + 10*A*a*b^7 - 7*A*a^7*b - 8*B*a*b^7 + 2*B*a^7*b))/(a^5*(a + b)*(a - b)^3) + (2*tan(c/2 + (d*x)/2)^ 5*(9*A*a^10 + 180*A*b^10 - 611*A*a^2*b^8 + 740*A*a^4*b^6 - 324*A*a^6*b^4 + 36*A*a^8*b^2 + 248*B*a^3*b^7 - 320*B*a^5*b^5 + 132*B*a^7*b^3 - 72*B*a*b^9 - 18*B*a^9*b))/(3*a^5*(a + b)^3*(a - b)^3) + (tan(c/2 + (d*x)/2)^9*(A*a^8 + 20*A*b^8 - 2*B*a^8 - 59*A*a^2*b^6 + 27*A*a^3*b^5 + 57*A*a^4*b^4 - 21*A* a^5*b^3 - 11*A*a^6*b^2 + 4*B*a^2*b^6 + 24*B*a^3*b^5 - 11*B*a^4*b^4 - 26*B* a^5*b^3 + 6*B*a^6*b^2 - 10*A*a*b^7 + 7*A*a^7*b - 8*B*a*b^7 + 2*B*a^7*b))/( a^5*(a + b)^3*(a - b)) - (2*tan(c/2 + (d*x)/2)^3*(6*A*a^9 - 120*A*b^9 + 6* B*a^9 + 364*A*a^2*b^7 + 71*A*a^3*b^6 - 369*A*a^4*b^5 - 45*A*a^5*b^4 + 111* A*a^6*b^3 + 3*A*a^7*b^2 + 12*B*a^2*b^7 - 148*B*a^3*b^6 - 29*B*a^4*b^5 + 15 9*B*a^5*b^4 + 18*B*a^6*b^3 - 30*B*a^7*b^2 - 30*A*a*b^8 - 21*A*a^8*b + 48*B *a*b^8 - 6*B*a^8*b))/(3*a^5*(a + b)^2*(a - b)^3) - (2*tan(c/2 + (d*x)/2)^7 *(6*A*a^9 + 120*A*b^9 - 6*B*a^9 - 364*A*a^2*b^7 + 71*A*a^3*b^6 + 369*A*a^4 *b^5 - 45*A*a^5*b^4 - 111*A*a^6*b^3 + 3*A*a^7*b^2 + 12*B*a^2*b^7 + 148*B*a ^3*b^6 - 29*B*a^4*b^5 - 159*B*a^5*b^4 + 18*B*a^6*b^3 + 30*B*a^7*b^2 - 30*A *a*b^8 + 21*A*a^8*b - 48*B*a*b^8 - 6*B*a^8*b))/(3*a^5*(a + b)^3*(a - b)^2) )/(d*(tan(c/2 + (d*x)/2)^2*(9*a*b^2 + 3*a^2*b - a^3 + 5*b^3) + tan(c/2 ...
Time = 0.17 (sec) , antiderivative size = 1469, normalized size of antiderivative = 2.73 \[ \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(cos(d*x+c)^2*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,x)
Output:
( - 80*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b) /sqrt( - a**2 + b**2))*cos(c + d*x)*a**5*b**4 + 116*sqrt( - a**2 + b**2)*a tan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*a**3*b**6 - 48*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan( (c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*a*b**8 + 40*sqrt( - a** 2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b* *2))*sin(c + d*x)**2*a**6*b**3 - 58*sqrt( - a**2 + b**2)*atan((tan((c + d* x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*sin(c + d*x)**2*a**4*b **5 + 24*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)* b)/sqrt( - a**2 + b**2))*sin(c + d*x)**2*a**2*b**7 - 40*sqrt( - a**2 + b** 2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*a* *6*b**3 + 18*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x) /2)*b)/sqrt( - a**2 + b**2))*a**4*b**5 + 34*sqrt( - a**2 + b**2)*atan((tan ((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*a**2*b**7 - 24 *sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*b**9 - cos(c + d*x)*sin(c + d*x)**3*a**10 + 3*cos(c + d*x )*sin(c + d*x)**3*a**8*b**2 - 3*cos(c + d*x)*sin(c + d*x)**3*a**6*b**4 + c os(c + d*x)*sin(c + d*x)**3*a**4*b**6 + cos(c + d*x)*sin(c + d*x)*a**10 - 14*cos(c + d*x)*sin(c + d*x)*a**8*b**2 + 46*cos(c + d*x)*sin(c + d*x)*a**6 *b**4 - 51*cos(c + d*x)*sin(c + d*x)*a**4*b**6 + 18*cos(c + d*x)*sin(c ...