Integrand size = 31, antiderivative size = 547 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=-\frac {b^2 \left (40 a^6 A b-84 a^4 A b^3+69 a^2 A b^5-20 A b^7-20 a^7 B+35 a^5 b^2 B-28 a^3 b^4 B+8 a b^6 B\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (a^2 A+20 A b^2-8 a b B\right ) \text {arctanh}(\sin (c+d x))}{2 a^6 d}-\frac {\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \]
-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)*arctan((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+1/2*(A*a^2+20*A*b^2-8*B*a*b)*arctanh(sin(d* x+c))/a^6/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+6 5*B*a^5*b^2-68*B*a^3*b^4+24*B*a*b^6)*tan(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)*s ec(d*x+c)*tan(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*b*(A*b-B*a)*sec(d*x+c)*tan(d*x+ c)/a/(a^2-b^2)/d/(a+b*cos(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)*sec(d*x+c)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2+1/6*b*(4 8*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)*sec(d*x+c )*tan(d*x+c)/a^3/(a^2-b^2)^3/d/(a+b*cos(d*x+c))
Time = 4.64 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\frac {\frac {96 b^2 \left (-40 a^6 A b+84 a^4 A b^3-69 a^2 A b^5+20 A b^7+20 a^7 B-35 a^5 b^2 B+28 a^3 b^4 B-8 a b^6 B\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}-48 \left (a^2 A+20 A b^2-8 a b B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 \left (a^2 A+20 A b^2-8 a b B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \left (24 a^{10} A-324 a^8 A b^2+1116 a^6 A b^4-830 a^4 A b^6-61 a^2 A b^8+180 A b^{10}+72 a^9 b B-438 a^7 b^3 B+305 a^5 b^5 B+28 a^3 b^7 B-72 a b^9 B+6 a \left (-20 a^8 A b-9 a^6 A b^3+309 a^4 A b^5-400 a^2 A b^7+150 A b^9+8 a^9 B-6 a^7 b^2 B-135 a^5 b^4 B+163 a^3 b^6 B-60 a b^8 B\right ) \cos (c+d x)+12 b \left (-21 a^8 A b+85 a^6 A b^3-55 a^4 A b^5-19 a^2 A b^7+20 A b^9+6 a^9 B-36 a^7 b^2 B+20 a^5 b^4 B+8 a^3 b^6 B-8 a b^8 B\right ) \cos (2 (c+d x))-138 a^7 A b^3 \cos (3 (c+d x))+738 a^5 A b^5 \cos (3 (c+d x))-840 a^3 A b^7 \cos (3 (c+d x))+300 a A b^9 \cos (3 (c+d x))+36 a^8 b^2 B \cos (3 (c+d x))-318 a^6 b^4 B \cos (3 (c+d x))+342 a^4 b^6 B \cos (3 (c+d x))-120 a^2 b^8 B \cos (3 (c+d x))-24 a^6 A b^4 \cos (4 (c+d x))+146 a^4 A b^6 \cos (4 (c+d x))-167 a^2 A b^8 \cos (4 (c+d x))+60 A b^{10} \cos (4 (c+d x))+6 a^7 b^3 B \cos (4 (c+d x))-65 a^5 b^5 B \cos (4 (c+d x))+68 a^3 b^7 B \cos (4 (c+d x))-24 a b^9 B \cos (4 (c+d x))\right ) \sec (c+d x) \tan (c+d x)}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}}{96 a^6 d} \]
((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) - 48*(a^2*A + 20*A*b^2 - 8*a*b*B )*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(a^2*A + 20*A*b^2 - 8*a*b* B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(24*a^10*A - 324*a^8*A* b^2 + 1116*a^6*A*b^4 - 830*a^4*A*b^6 - 61*a^2*A*b^8 + 180*A*b^10 + 72*a^9* b*B - 438*a^7*b^3*B + 305*a^5*b^5*B + 28*a^3*b^7*B - 72*a*b^9*B + 6*a*(-20 *a^8*A*b - 9*a^6*A*b^3 + 309*a^4*A*b^5 - 400*a^2*A*b^7 + 150*A*b^9 + 8*a^9 *B - 6*a^7*b^2*B - 135*a^5*b^4*B + 163*a^3*b^6*B - 60*a*b^8*B)*Cos[c + d*x ] + 12*b*(-21*a^8*A*b + 85*a^6*A*b^3 - 55*a^4*A*b^5 - 19*a^2*A*b^7 + 20*A* b^9 + 6*a^9*B - 36*a^7*b^2*B + 20*a^5*b^4*B + 8*a^3*b^6*B - 8*a*b^8*B)*Cos [2*(c + d*x)] - 138*a^7*A*b^3*Cos[3*(c + d*x)] + 738*a^5*A*b^5*Cos[3*(c + d*x)] - 840*a^3*A*b^7*Cos[3*(c + d*x)] + 300*a*A*b^9*Cos[3*(c + d*x)] + 36 *a^8*b^2*B*Cos[3*(c + d*x)] - 318*a^6*b^4*B*Cos[3*(c + d*x)] + 342*a^4*b^6 *B*Cos[3*(c + d*x)] - 120*a^2*b^8*B*Cos[3*(c + d*x)] - 24*a^6*A*b^4*Cos[4* (c + d*x)] + 146*a^4*A*b^6*Cos[4*(c + d*x)] - 167*a^2*A*b^8*Cos[4*(c + d*x )] + 60*A*b^10*Cos[4*(c + d*x)] + 6*a^7*b^3*B*Cos[4*(c + d*x)] - 65*a^5*b^ 5*B*Cos[4*(c + d*x)] + 68*a^3*b^7*B*Cos[4*(c + d*x)] - 24*a*b^9*B*Cos[4*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3 ))/(96*a^6*d)
Time = 3.91 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.08, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {3042, 3479, 3042, 3534, 3042, 3534, 3042, 3534, 27, 3042, 3534, 27, 3042, 3480, 3042, 3138, 218, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3479 |
| \(\displaystyle \frac {\int \frac {\left (3 A a^2+2 b B a-3 (A b-a B) \cos (c+d x) a-5 A b^2+4 b (A b-a B) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 A a^2+2 b B a-3 (A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a-5 A b^2+4 b (A b-a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\int \frac {\left (3 b \left (-7 B a^3+10 A b a^2+2 b^2 B a-5 A b^3\right ) \cos ^2(c+d x)-2 a \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \cos (c+d x)+2 \left (3 A a^4+9 b B a^3-18 A b^2 a^2-4 b^3 B a+10 A b^4\right )\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 b \left (-7 B a^3+10 A b a^2+2 b^2 B a-5 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a \left (-3 B a^3+6 A b a^2-2 b^2 B a-A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 \left (3 A a^4+9 b B a^3-18 A b^2 a^2-4 b^3 B a+10 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (2 b \left (-27 B a^5+48 A b a^4+20 b^2 B a^3-53 A b^3 a^2-8 b^4 B a+20 A b^5\right ) \cos ^2(c+d x)-a \left (-6 B a^5+18 A b a^4-7 b^2 B a^3-8 A b^3 a^2-2 b^4 B a+5 A b^5\right ) \cos (c+d x)+6 \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right )\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 b \left (-27 B a^5+48 A b a^4+20 b^2 B a^3-53 A b^3 a^2-8 b^4 B a+20 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a \left (-6 B a^5+18 A b a^4-7 b^2 B a^3-8 A b^3 a^2-2 b^4 B a+5 A b^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+6 \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int -\frac {2 \left (-6 B a^7+24 A b a^6+65 b^2 B a^5-146 A b^3 a^4-68 b^4 B a^3+167 A b^5 a^2+24 b^6 B a-\left (3 A a^6-18 b B a^5+27 A b^2 a^4+7 b^3 B a^3-25 A b^4 a^2-4 b^5 B a+10 A b^6\right ) \cos (c+d x) a-60 A b^7-3 b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{2 a}+\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\int \frac {\left (-6 B a^7+24 A b a^6+65 b^2 B a^5-146 A b^3 a^4-68 b^4 B a^3+167 A b^5 a^2+24 b^6 B a-\left (3 A a^6-18 b B a^5+27 A b^2 a^4+7 b^3 B a^3-25 A b^4 a^2-4 b^5 B a+10 A b^6\right ) \cos (c+d x) a-60 A b^7-3 b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)}dx}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\int \frac {-6 B a^7+24 A b a^6+65 b^2 B a^5-146 A b^3 a^4-68 b^4 B a^3+167 A b^5 a^2+24 b^6 B a-\left (3 A a^6-18 b B a^5+27 A b^2 a^4+7 b^3 B a^3-25 A b^4 a^2-4 b^5 B a+10 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-60 A b^7-3 b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\int -\frac {3 \left (\left (A a^2-8 b B a+20 A b^2\right ) \left (a^2-b^2\right )^3+a b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}+\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{a d}-\frac {3 \int \frac {\left (\left (A a^2-8 b B a+20 A b^2\right ) \left (a^2-b^2\right )^3+a b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{a d}-\frac {3 \int \frac {\left (A a^2-8 b B a+20 A b^2\right ) \left (a^2-b^2\right )^3+a b \left (A a^6+12 b B a^5-23 A b^2 a^4-11 b^3 B a^3+27 A b^4 a^2+4 b^5 B a-10 A b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right ) \int \sec (c+d x)dx}{a}-\frac {b^2 \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \int \frac {1}{a+b \cos (c+d x)}dx}{a}\right )}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b^2 \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \int \frac {1}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}\right )}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b^2 \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \int \frac {1}{(a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {2 b^2 \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a}}{a \left (a^2-b^2\right )}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}}{3 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {\frac {3 \left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{a d}-\frac {\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{a d}-\frac {3 \left (\frac {\left (a^2-b^2\right )^3 \left (a^2 A-8 a b B+20 A b^2\right ) \text {arctanh}(\sin (c+d x))}{a d}-\frac {2 b^2 \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}\right )}{a}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}}{3 a \left (a^2-b^2\right )}\) |
(b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + ((b*(10*a^2*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*Sec[c + d*x] *Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Cos[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)*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[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)*Sec[c + d*x]*Tan[c + d*x])/(a*d) - ((-3*((-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)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[ a - b]*Sqrt[a + b]*d) + ((a^2 - b^2)^3*(a^2*A + 20*A*b^2 - 8*a*b*B)*ArcTan h[Sin[c + d*x]])/(a*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 + 65*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*Tan[c + d *x])/(a*d))/a)/(a*(a^2 - b^2)))/(2*a*(a^2 - b^2)))/(3*a*(a^2 - b^2))
3.3.80.3.1 Defintions of rubi rules used
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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(A*b^2 - a*b*B))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin [e + f*x])^(1 + n)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B) *(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n }, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Rat ionalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(I ntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0]) ))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 5.06 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.23
| method | result | size |
| derivativedivides | \(\frac {-\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 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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 (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+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 ) \arctan \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}}-\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -8 A b +2 B a}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (A \,a^{2}+20 A \,b^{2}-8 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{6}}+\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -8 A b +2 B a}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-A \,a^{2}-20 A \,b^{2}+8 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{6}}}{d}\) | \(672\) |
| default | \(\frac {-\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 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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 (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+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 ) \arctan \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}}-\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -8 A b +2 B a}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (A \,a^{2}+20 A \,b^{2}-8 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{6}}+\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -8 A b +2 B a}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-A \,a^{2}-20 A \,b^{2}+8 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{6}}}{d}\) | \(672\) |
| risch | \(\text {Expression too large to display}\) | \(3144\) |
1/d*(-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-b*ta n(1/2*d*x+1/2*c)^2+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)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/ 2)))-1/2*A/a^4/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(-A*a-8*A*b+2*B*a)/a^5/(tan(1/ 2*d*x+1/2*c)+1)+1/2*(A*a^2+20*A*b^2-8*B*a*b)/a^6*ln(tan(1/2*d*x+1/2*c)+1)+ 1/2*A/a^4/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(-A*a-8*A*b+2*B*a)/a^5/(tan(1/2*d*x +1/2*c)-1)+1/2/a^6*(-A*a^2-20*A*b^2+8*B*a*b)*ln(tan(1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (525) = 1050\).
Time = 113.16 (sec) , antiderivative size = 3819, normalized size of antiderivative = 6.98 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]
[1/12*(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)*cos(d*x + c)^5 + 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 - 6 9*A*a^3*b^9 - 8*B*a^2*b^10 + 20*A*a*b^11)*cos(d*x + c)^4 + 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)^3 + (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)^2)*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) + 3*((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)*cos(d*x + c)^5 + 3*(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)*cos(d*x + c)^4 + 3*(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)*cos(d*x + c)^3 + (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)*cos(d*x + c)^2)*log(si...
\[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\int \frac {\left (A + B \cos {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{4}}\, dx \]
Exception generated. \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Exception raised: ValueError} \]
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*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1090 vs. \(2 (525) = 1050\).
Time = 0.37 (sec) , antiderivative size = 1090, normalized size of antiderivative = 1.99 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]
-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*t an(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 + 39 2*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) - 1 62*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) + 48 *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) + 213 *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) + 4...
Time = 14.28 (sec) , antiderivative size = 14398, normalized size of antiderivative = 26.32 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx=\text {Too large to display} \]
((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)^4*(6*a*b^2 - 6*a^2*b - 2*a^3 + 10*b^3) - tan(c...