Integrand size = 41, antiderivative size = 470 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\frac {(b B-4 a C) \text {arctanh}(\sin (c+d x))}{b^5 d}-\frac {\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \] Output:
(B*b-4*C*a)*arctanh(sin(d*x+c))/b^5/d-(2*A*b^8+2*a^7*b*B-7*a^5*b^3*B+8*a^3 *b^5*B-8*a*b^7*B-8*a^8*C+28*a^6*b^2*C-35*a^4*b^4*C+a^2*b^6*(3*A+20*C))*arc tanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^5/(a+b)^(7/ 2)/d-1/6*(5*A*b^4+3*B*a^3*b-8*B*a*b^3-12*C*a^4+23*C*a^2*b^2-6*C*b^4)*tan(d *x+c)/b^4/(a^2-b^2)^2/d-1/3*(A*b^2-a*(B*b-C*a))*sec(d*x+c)^3*tan(d*x+c)/b/ (a^2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*(3*A*b^4+B*a^3*b-6*B*a*b^3-4*a^4*C+a^2* b^2*(2*A+9*C))*sec(d*x+c)^2*tan(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^ 2+1/2*a*(2*A*b^6-a^5*b*B+2*a^3*b^3*B-6*a*b^5*B+4*a^6*C-11*a^4*b^2*C+3*a^2* b^4*(A+4*C))*tan(d*x+c)/b^4/(a^2-b^2)^3/d/(a+b*sec(d*x+c))
Time = 6.83 (sec) , antiderivative size = 756, normalized size of antiderivative = 1.61 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\frac {(b+a \cos (c+d x)) \sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {48 \left (-2 A b^8-2 a^7 b B+7 a^5 b^3 B-8 a^3 b^5 B+8 a b^7 B+8 a^8 C-28 a^6 b^2 C+35 a^4 b^4 C-a^2 b^6 (3 A+20 C)\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) \cos (c+d x) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}-48 (b B-4 a C) \cos (c+d x) (b+a \cos (c+d x))^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 (b B-4 a C) \cos (c+d x) (b+a \cos (c+d x))^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2 b \left (6 a^4 A b^5+54 a^2 A b^7-30 a^7 b^2 B+90 a^5 b^4 B-120 a^3 b^6 B+120 a^8 b C-318 a^6 b^3 C+246 a^4 b^5 C+36 a^2 b^7 C-24 b^9 C+a \left (-18 a^7 b B+7 a^5 b^3 B+50 a^3 b^5 B-144 a b^7 B+5 a^4 b^4 (4 A-61 C)+72 b^8 (A-C)+72 a^8 C-28 a^6 b^2 C+a^2 b^6 (13 A+438 C)\right ) \cos (c+d x)+6 a^2 b \left (-5 a^5 b B+15 a^3 b^3 B-20 a b^5 B+3 b^6 (3 A-2 C)+20 a^6 C-57 a^4 b^2 C+a^2 b^4 (A+53 C)\right ) \cos (2 (c+d x))+4 a^5 A b^4 \cos (3 (c+d x))+11 a^3 A b^6 \cos (3 (c+d x))-6 a^8 b B \cos (3 (c+d x))+17 a^6 b^3 B \cos (3 (c+d x))-26 a^4 b^5 B \cos (3 (c+d x))+24 a^9 C \cos (3 (c+d x))-68 a^7 b^2 C \cos (3 (c+d x))+65 a^5 b^4 C \cos (3 (c+d x))-6 a^3 b^6 C \cos (3 (c+d x))\right ) \sin (c+d x)}{\left (-a^2+b^2\right )^3}\right )}{24 b^5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) (a+b \sec (c+d x))^4} \] Input:
Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b* Sec[c + d*x])^4,x]
Output:
((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^ 2)*((-48*(-2*A*b^8 - 2*a^7*b*B + 7*a^5*b^3*B - 8*a^3*b^5*B + 8*a*b^7*B + 8 *a^8*C - 28*a^6*b^2*C + 35*a^4*b^4*C - a^2*b^6*(3*A + 20*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*Cos[c + d*x]*(b + a*Cos[c + d*x])^ 3)/(a^2 - b^2)^(7/2) - 48*(b*B - 4*a*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])^ 3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(b*B - 4*a*C)*Cos[c + d*x] *(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (2*b*(6 *a^4*A*b^5 + 54*a^2*A*b^7 - 30*a^7*b^2*B + 90*a^5*b^4*B - 120*a^3*b^6*B + 120*a^8*b*C - 318*a^6*b^3*C + 246*a^4*b^5*C + 36*a^2*b^7*C - 24*b^9*C + a* (-18*a^7*b*B + 7*a^5*b^3*B + 50*a^3*b^5*B - 144*a*b^7*B + 5*a^4*b^4*(4*A - 61*C) + 72*b^8*(A - C) + 72*a^8*C - 28*a^6*b^2*C + a^2*b^6*(13*A + 438*C) )*Cos[c + d*x] + 6*a^2*b*(-5*a^5*b*B + 15*a^3*b^3*B - 20*a*b^5*B + 3*b^6*( 3*A - 2*C) + 20*a^6*C - 57*a^4*b^2*C + a^2*b^4*(A + 53*C))*Cos[2*(c + d*x) ] + 4*a^5*A*b^4*Cos[3*(c + d*x)] + 11*a^3*A*b^6*Cos[3*(c + d*x)] - 6*a^8*b *B*Cos[3*(c + d*x)] + 17*a^6*b^3*B*Cos[3*(c + d*x)] - 26*a^4*b^5*B*Cos[3*( c + d*x)] + 24*a^9*C*Cos[3*(c + d*x)] - 68*a^7*b^2*C*Cos[3*(c + d*x)] + 65 *a^5*b^4*C*Cos[3*(c + d*x)] - 6*a^3*b^6*C*Cos[3*(c + d*x)])*Sin[c + d*x])/ (-a^2 + b^2)^3))/(24*b^5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x) ])*(a + b*Sec[c + d*x])^4)
Time = 3.45 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.12, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.439, Rules used = {3042, 4586, 3042, 4586, 3042, 4578, 25, 3042, 4570, 27, 3042, 4486, 3042, 4257, 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.
| \(\displaystyle \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 4586 |
| \(\displaystyle -\frac {\int \frac {\sec ^3(c+d x) \left (-\left (\left (4 C a^2-b B a+A b^2-3 b^2 C\right ) \sec ^2(c+d x)\right )+3 b (b B-a (A+C)) \sec (c+d x)+3 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \sec (c+d x))^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (\left (-4 C a^2+b B a-A b^2+3 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 b (b B-a (A+C)) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4586 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sec ^2(c+d x) \left (-\left (\left (-12 C a^4+3 b B a^3+23 b^2 C a^2-8 b^3 B a+5 A b^4-6 b^4 C\right ) \sec ^2(c+d x)\right )+2 b \left (C a^3+2 b B a^2-b^2 (5 A+6 C) a+3 b^3 B\right ) \sec (c+d x)+2 \left (-4 C a^4+b B a^3+b^2 (2 A+9 C) a^2-6 b^3 B a+3 A b^4\right )\right )}{(a+b \sec (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\left (12 C a^4-3 b B a^3-23 b^2 C a^2+8 b^3 B a-5 A b^4+6 b^4 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (C a^3+2 b B a^2-b^2 (5 A+6 C) a+3 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 \left (-4 C a^4+b B a^3+b^2 (2 A+9 C) a^2-6 b^3 B a+3 A b^4\right )\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4578 |
| \(\displaystyle -\frac {-\frac {\frac {\int -\frac {\sec (c+d x) \left (b \left (a^2-b^2\right ) \left (-12 C a^4+3 b B a^3+23 b^2 C a^2-8 b^3 B a+5 A b^4-6 b^4 C\right ) \sec ^2(c+d x)-\left (a^2-b^2\right ) \left (-12 C a^5+3 b B a^4+25 b^2 C a^3-4 b^3 B a^2-b^4 (5 A+18 C) a+6 b^5 B\right ) \sec (c+d x)+3 b \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )\right )}{a+b \sec (c+d x)}dx}{b^2 \left (a^2-b^2\right )}+\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {\sec (c+d x) \left (b \left (a^2-b^2\right ) \left (-12 C a^4+3 b B a^3+23 b^2 C a^2-8 b^3 B a+5 A b^4-6 b^4 C\right ) \sec ^2(c+d x)-\left (a^2-b^2\right ) \left (-12 C a^5+3 b B a^4+25 b^2 C a^3-4 b^3 B a^2-b^4 (5 A+18 C) a+6 b^5 B\right ) \sec (c+d x)+3 b \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )\right )}{a+b \sec (c+d x)}dx}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b \left (a^2-b^2\right ) \left (-12 C a^4+3 b B a^3+23 b^2 C a^2-8 b^3 B a+5 A b^4-6 b^4 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-\left (a^2-b^2\right ) \left (-12 C a^5+3 b B a^4+25 b^2 C a^3-4 b^3 B a^2-b^4 (5 A+18 C) a+6 b^5 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 b \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\int \frac {3 \sec (c+d x) \left (b^2 \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )-2 b \left (a^2-b^2\right )^3 (b B-4 a C) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{b}+\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \int \frac {\sec (c+d x) \left (b^2 \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )-2 b \left (a^2-b^2\right )^3 (b B-4 a C) \sec (c+d x)\right )}{a+b \sec (c+d x)}dx}{b}+\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b^2 \left (4 C a^6-b B a^5-11 b^2 C a^4+2 b^3 B a^3+3 b^4 (A+4 C) a^2-6 b^5 B a+2 A b^6\right )-2 b \left (a^2-b^2\right )^3 (b B-4 a C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4486 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (\left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)}dx-2 \left (a^2-b^2\right )^3 (b B-4 a C) \int \sec (c+d x)dx\right )}{b}+\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (\left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx-2 \left (a^2-b^2\right )^3 (b B-4 a C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx\right )}{b}+\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (\left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 \left (a^2-b^2\right )^3 (b B-4 a C) \text {arctanh}(\sin (c+d x))}{d}\right )}{b}+\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 4318 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (\frac {\left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \int \frac {1}{\frac {a \cos (c+d x)}{b}+1}dx}{b}-\frac {2 \left (a^2-b^2\right )^3 (b B-4 a C) \text {arctanh}(\sin (c+d x))}{d}\right )}{b}+\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (\frac {\left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \int \frac {1}{\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{b}+1}dx}{b}-\frac {2 \left (a^2-b^2\right )^3 (b B-4 a C) \text {arctanh}(\sin (c+d x))}{d}\right )}{b}+\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle -\frac {-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {3 \left (\frac {2 \left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \int \frac {1}{\left (1-\frac {a}{b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )+\frac {a+b}{b}}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}-\frac {2 \left (a^2-b^2\right )^3 (b B-4 a C) \text {arctanh}(\sin (c+d x))}{d}\right )}{b}+\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}}{3 b \left (a^2-b^2\right )}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {-\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\frac {3 a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{b^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^2-b^2\right ) \tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{d}+\frac {3 \left (\frac {2 \left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d \sqrt {a-b} \sqrt {a+b}}-\frac {2 \left (a^2-b^2\right )^3 (b B-4 a C) \text {arctanh}(\sin (c+d x))}{d}\right )}{b}}{b^2 \left (a^2-b^2\right )}}{2 b \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}\) |
Input:
Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]
Output:
-1/3*((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(b*(a^2 - b^2)* d*(a + b*Sec[c + d*x])^3) - (-1/2*((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4* C + a^2*b^2*(2*A + 9*C))*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])) - ((3*((-2*(a^2 - b^2)^3*(b*B - 4*a*C)*ArcTan h[Sin[c + d*x]])/d + (2*(2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C)) *ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)))/b + ((a^2 - b^2)*(5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 2 3*a^2*b^2*C - 6*b^4*C)*Tan[c + d*x])/d)/(b^2*(a^2 - b^2)))/(2*b*(a^2 - b^2 )))/(3*b*(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[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[( e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[B/b Int[Csc[e + f*x], x], x] + Simp[(A*b - a*B)/b Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x ] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) )), x] + Simp[1/(b*(m + 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[ (e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x _Symbol] :> Simp[a*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x ])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - Simp[1/(b^2*(m + 1)*(a^2 - b^ 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*Simp[b*(m + 1)*((-a)*(b *B - a*C) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1 ]
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[(-d)*(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 - 1)/(b*f*(a^2 - b^2)*(m + 1)) ), x] + Simp[d/(b*(a^2 - b^2)*(m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*( d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C }, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
Time = 1.52 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.44
| method | result | size |
| derivativedivides | \(\frac {-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-B b +4 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{5}}+\frac {\frac {2 \left (-\frac {\left (2 a^{2} A \,b^{4}+3 A \,b^{5} a +6 A \,b^{6}-2 a^{5} b B +B \,a^{4} b^{2}+6 a^{3} b^{3} B -4 B \,a^{2} b^{4}-12 a \,b^{5} B +6 a^{6} C -2 a^{5} C b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 C \,a^{2} 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 (a^{2} A \,b^{4}+9 A \,b^{6}-3 a^{5} b B +11 a^{3} b^{3} B -18 a \,b^{5} B +9 a^{6} C -29 a^{4} b^{2} C +30 C \,a^{2} 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 (2 a^{2} A \,b^{4}-3 A \,b^{5} a +6 A \,b^{6}-2 a^{5} b B -B \,a^{4} b^{2}+6 a^{3} b^{3} B +4 B \,a^{2} b^{4}-12 a \,b^{5} B +6 a^{6} C +2 a^{5} C b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 C \,a^{2} 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 )}\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 (3 a^{2} A \,b^{6}+2 A \,b^{8}+2 a^{7} b B -7 a^{5} b^{3} B +8 a^{3} b^{5} B -8 a \,b^{7} B -8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} 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 )}{\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 )}}}{b^{5}}-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (B b -4 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{5}}}{d}\) | \(679\) |
| default | \(\frac {-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-B b +4 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{5}}+\frac {\frac {2 \left (-\frac {\left (2 a^{2} A \,b^{4}+3 A \,b^{5} a +6 A \,b^{6}-2 a^{5} b B +B \,a^{4} b^{2}+6 a^{3} b^{3} B -4 B \,a^{2} b^{4}-12 a \,b^{5} B +6 a^{6} C -2 a^{5} C b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 C \,a^{2} 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 (a^{2} A \,b^{4}+9 A \,b^{6}-3 a^{5} b B +11 a^{3} b^{3} B -18 a \,b^{5} B +9 a^{6} C -29 a^{4} b^{2} C +30 C \,a^{2} 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 (2 a^{2} A \,b^{4}-3 A \,b^{5} a +6 A \,b^{6}-2 a^{5} b B -B \,a^{4} b^{2}+6 a^{3} b^{3} B +4 B \,a^{2} b^{4}-12 a \,b^{5} B +6 a^{6} C +2 a^{5} C b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 C \,a^{2} 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 )}\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 (3 a^{2} A \,b^{6}+2 A \,b^{8}+2 a^{7} b B -7 a^{5} b^{3} B +8 a^{3} b^{5} B -8 a \,b^{7} B -8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} 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 )}{\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 )}}}{b^{5}}-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (B b -4 C a \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{5}}}{d}\) | \(679\) |
| risch | \(\text {Expression too large to display}\) | \(3248\) |
Input:
int(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x,meth od=_RETURNVERBOSE)
Output:
1/d*(-C/b^4/(tan(1/2*d*x+1/2*c)-1)+1/b^5*(-B*b+4*C*a)*ln(tan(1/2*d*x+1/2*c )-1)+2/b^5*((-1/2*(2*A*a^2*b^4+3*A*a*b^5+6*A*b^6-2*B*a^5*b+B*a^4*b^2+6*B*a ^3*b^3-4*B*a^2*b^4-12*B*a*b^5+6*C*a^6-2*C*a^5*b-18*C*a^4*b^2+5*C*a^3*b^3+2 0*C*a^2*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* (A*a^2*b^4+9*A*b^6-3*B*a^5*b+11*B*a^3*b^3-18*B*a*b^5+9*C*a^6-29*C*a^4*b^2+ 30*C*a^2*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 *(2*A*a^2*b^4-3*A*a*b^5+6*A*b^6-2*B*a^5*b-B*a^4*b^2+6*B*a^3*b^3+4*B*a^2*b^ 4-12*B*a*b^5+6*C*a^6+2*C*a^5*b-18*C*a^4*b^2-5*C*a^3*b^3+20*C*a^2*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*(3*A*a^2*b^6+2*A*b^8+2*B*a^7*b-7*B*a^5 *b^3+8*B*a^3*b^5-8*B*a*b^7-8*C*a^8+28*C*a^6*b^2-35*C*a^4*b^4+20*C*a^2*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)))-C/b^4/(tan(1/2*d*x+1/2*c)+1)+(B*b-4*C*a)/b ^5*ln(tan(1/2*d*x+1/2*c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 1793 vs. \(2 (456) = 912\).
Time = 128.17 (sec) , antiderivative size = 3644, normalized size of antiderivative = 7.75 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4, x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{4}}\, dx \] Input:
integrate(sec(d*x+c)**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))* *4,x)
Output:
Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*sec(c + d*x)**4/(a + b*s ec(c + d*x))**4, x)
Exception generated. \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4, x, algorithm="maxima")
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. 1264 vs. \(2 (456) = 912\).
Time = 0.37 (sec) , antiderivative size = 1264, normalized size of antiderivative = 2.69 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4, x, algorithm="giac")
Output:
1/3*(3*(8*C*a^8 - 2*B*a^7*b - 28*C*a^6*b^2 + 7*B*a^5*b^3 + 35*C*a^4*b^4 - 8*B*a^3*b^5 - 3*A*a^2*b^6 - 20*C*a^2*b^6 + 8*B*a*b^7 - 2*A*b^8)*(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*b^5 - 3*a^4*b^7 + 3*a^2 *b^9 - b^11)*sqrt(-a^2 + b^2)) - (18*C*a^9*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^ 8*b*tan(1/2*d*x + 1/2*c)^5 - 42*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 + 15*B*a^7* b^2*tan(1/2*d*x + 1/2*c)^5 - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^6 *b^3*tan(1/2*d*x + 1/2*c)^5 + 117*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*a ^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 45*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 24*C *a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*B *a^4*b^5*tan(1/2*d*x + 1/2*c)^5 - 105*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6 *A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 60*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 60*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 27*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 + 18*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 - 36*C*a^9*tan(1/2*d*x + 1/2*c)^3 + 12*B*a^8*b*tan(1/2*d*x + 1/2*c)^3 + 15 2*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 - 56*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 - 4*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 236*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 + 116*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 32*A*a^3*b^6*tan(1/2*d*x + 1/2*c )^3 + 120*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 - 72*B*a^2*b^7*tan(1/2*d*x + 1/ 2*c)^3 + 36*A*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^9*tan(1/2*d*x + 1/2...
Time = 27.40 (sec) , antiderivative size = 15959, normalized size of antiderivative = 33.96 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \] Input:
int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/(cos(c + d*x)^4*(a + b/cos(c + d*x))^4),x)
Output:
(atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8 *B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*a ^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a ^6*b^10 + 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^ 10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^14*b ^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^ 11 - 824*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9 *b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2 *a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15 *b - 16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 4 8*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 5 92*B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 - 948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^ 3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^1 1 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8) + (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^ 2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*A *a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*...
Time = 0.20 (sec) , antiderivative size = 6901, normalized size of antiderivative = 14.68 \[ \int \frac {\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x)
Output:
(144*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/s qrt( - a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**10*b*c - 36*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)*sin(c + d*x)**2*a**9*b**3 - 504*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)*sin(c + d*x)**2*a**8*b**3*c + 126*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)*sin( c + d*x)**2*a**7*b**5 + 630*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)*sin(c + d*x)**2*a **6*b**5*c - 198*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)*sin(c + d*x)**2*a**5*b**7 - 360*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - tan((c + d*x)/2)*b)/sq rt( - a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**4*b**7*c + 108*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)*sin(c + d*x)**2*a**3*b**9 - 144*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**10*b*c + 36*sqrt( - a**2 + b**2)*atan((tan((c + d*x)/2)*a - ta n((c + d*x)/2)*b)/sqrt( - a**2 + b**2))*cos(c + d*x)*a**9*b**3 + 456*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**8*b**3*c - 114*sqrt( - a**2 + b**2)*atan((ta...