Integrand size = 43, antiderivative size = 505 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (18 a^3 b B-246 a b^3 B-8 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (21 A+11 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (6 a^2 b (3 B-C)-8 a^3 C-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^3 d}-\frac {2 \left (18 a^2 b B-75 b^3 B-8 a^3 C-3 a b^2 (21 A+13 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac {2 \left (63 A b^2-18 a b B+8 a^2 C+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b^2 d}+\frac {2 (9 b B-4 a C) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b^2 d}+\frac {2 C \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{9 b d} \] Output:
2/315*(a-b)*(a+b)^(1/2)*(18*B*a^3*b-246*B*a*b^3-8*a^4*C-21*b^4*(9*A+7*C)-3 *a^2*b^2*(21*A+11*C))*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1 /2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c)) /(a-b))^(1/2)/b^4/d+2/315*(a-b)*(a+b)^(1/2)*(6*a^2*b*(3*B-C)-8*a^3*C-3*a*b ^2*(21*A-57*B+13*C)+3*b^3*(63*A-25*B+49*C))*cot(d*x+c)*EllipticF((a+b*sec( d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1 /2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b^3/d-2/315*(18*B*a^2*b-75*B*b^3-8*a^3 *C-3*a*b^2*(21*A+13*C))*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^2/d+2/315*(63* A*b^2-18*B*a*b+8*C*a^2+49*C*b^2)*(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/b^2/d+2 /63*(9*B*b-4*C*a)*(a+b*sec(d*x+c))^(5/2)*tan(d*x+c)/b^2/d+2/9*C*sec(d*x+c) *(a+b*sec(d*x+c))^(5/2)*tan(d*x+c)/b/d
Leaf count is larger than twice the leaf count of optimal. \(4336\) vs. \(2(505)=1010\).
Time = 24.91 (sec) , antiderivative size = 4336, normalized size of antiderivative = 8.59 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(63*a^2*A*b^2 + 189*A*b^4 - 18*a^3*b*B + 246*a*b^3*B + 8*a^4* C + 33*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^3) + (4*Sec[c + d*x]^3* (9*b*B*Sin[c + d*x] + 10*a*C*Sin[c + d*x]))/63 + (4*Sec[c + d*x]^2*(63*A*b ^2*Sin[c + d*x] + 72*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 49*b^2*C* Sin[c + d*x]))/(315*b) + (4*Sec[c + d*x]*(126*a*A*b^2*Sin[c + d*x] + 9*a^2 *b*B*Sin[c + d*x] + 75*b^3*B*Sin[c + d*x] - 4*a^3*C*Sin[c + d*x] + 88*a*b^ 2*C*Sin[c + d*x]))/(315*b^2) + (4*b*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d* (b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*((-2*a^2*A)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*A*b^2) /(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^3*B)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (164*a*b*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (22*a^2*C)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqr t[Sec[c + d*x]]) - (16*a^4*C)/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*b^2*C)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^3*A*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) + (2*a*A*b*Sqr t[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (62*a^2*B*Sqrt[Sec[c + d*x ]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*B*Sqrt[Sec[c + d*x]])/(35*b^2* Sqrt[b + a*Cos[c + d*x]]) + (10*b^2*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*C os[c + d*x]]) - (16*a^5*C*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c...
Time = 2.26 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 4580, 27, 3042, 4570, 27, 3042, 4490, 27, 3042, 4490, 27, 3042, 4493, 3042, 4319, 4492}
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.
| \(\displaystyle \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4580 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left ((9 b B-4 a C) \sec ^2(c+d x)+b (9 A+7 C) \sec (c+d x)+2 a C\right )dx}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left ((9 b B-4 a C) \sec ^2(c+d x)+b (9 A+7 C) \sec (c+d x)+2 a C\right )dx}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((9 b B-4 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (9 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a C\right )dx}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (3 b (15 b B-2 a C)+\left (8 C a^2-18 b B a+63 A b^2+49 b^2 C\right ) \sec (c+d x)\right )dx}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (3 b (15 b B-2 a C)+\left (8 C a^2-18 b B a+63 A b^2+49 b^2 C\right ) \sec (c+d x)\right )dx}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b (15 b B-2 a C)+\left (8 C a^2-18 b B a+63 A b^2+49 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {\frac {\frac {2}{5} \int \frac {3}{2} \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (b \left (-2 C a^2+57 b B a+63 A b^2+49 b^2 C\right )-\left (-8 C a^3+18 b B a^2-3 b^2 (21 A+13 C) a-75 b^3 B\right ) \sec (c+d x)\right )dx+\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \int \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (b \left (-2 C a^2+57 b B a+63 A b^2+49 b^2 C\right )-\left (-8 C a^3+18 b B a^2-3 b^2 (21 A+13 C) a-75 b^3 B\right ) \sec (c+d x)\right )dx+\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (-2 C a^2+57 b B a+63 A b^2+49 b^2 C\right )+\left (8 C a^3-18 b B a^2+3 b^2 (21 A+13 C) a+75 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {2}{3} \int \frac {\sec (c+d x) \left (b \left (2 C a^3+153 b B a^2+6 b^2 (42 A+31 C) a+75 b^3 B\right )-\left (-8 C a^4+18 b B a^3-3 b^2 (21 A+11 C) a^2-246 b^3 B a-21 b^4 (9 A+7 C)\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx-\frac {2 \tan (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \int \frac {\sec (c+d x) \left (b \left (2 C a^3+153 b B a^2+6 b^2 (42 A+31 C) a+75 b^3 B\right )-\left (-8 C a^4+18 b B a^3-3 b^2 (21 A+11 C) a^2-246 b^3 B a-21 b^4 (9 A+7 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx-\frac {2 \tan (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b \left (2 C a^3+153 b B a^2+6 b^2 (42 A+31 C) a+75 b^3 B\right )+\left (8 C a^4-18 b B a^3+3 b^2 (21 A+11 C) a^2+246 b^3 B a+21 b^4 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \tan (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 4493 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left ((a-b) \left (-8 a^3 C+6 a^2 b (3 B-C)-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-\left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx\right )-\frac {2 \tan (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left ((a-b) \left (-8 a^3 C+6 a^2 b (3 B-C)-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 \tan (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-8 a^3 C+6 a^2 b (3 B-C)-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 \tan (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )+\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {\frac {\frac {2 \tan (c+d x) \left (8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right ) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-8 a^3 C+6 a^2 b (3 B-C)-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}\right )-\frac {2 \tan (c+d x) \left (-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{3 d}\right )}{7 b}+\frac {2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}\) |
Input:
Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[ c + d*x]^2),x]
Output:
(2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d) + ((2*( 9*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d) + ((2*(63* A*b^2 - 18*a*b*B + 8*a^2*C + 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d) + (3*(((2*(a - b)*Sqrt[a + b]*(18*a^3*b*B - 246*a*b^3*B - 8*a^ 4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(21*A + 11*C))*Cot[c + d*x]*EllipticE [ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d ) + (2*(a - b)*Sqrt[a + b]*(6*a^2*b*(3*B - C) - 8*a^3*C - 3*a*b^2*(21*A - 57*B + 13*C) + 3*b^3*(63*A - 25*B + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[S qrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d))/3 - (2*( 18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)))/5)/(7*b))/(9*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*Cot[e + f*x]*(( a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[1/(m + 1) Int[Csc[e + f*x]* (a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1 ))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a* B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(A - B) Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[B Int[Csc[e + f*x]*((1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B} , x] && NeQ[a^2 - b^2, 0] && NeQ[A^2 - B^2, 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[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B* (m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] & & NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(3249\) vs. \(2(467)=934\).
Time = 116.38 (sec) , antiderivative size = 3250, normalized size of antiderivative = 6.44
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3250\) |
| parts | \(\text {Expression too large to display}\) | \(3288\) |
Input:
int(sec(d*x+c)^2*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
Output:
2/315/d/b^3*(a+b*sec(d*x+c))^(1/2)/(cos(d*x+c)^2*a+a*cos(d*x+c)+b*cos(d*x+ c)+b)*((33*cos(d*x+c)^2-cos(d*x+c)-1)*C*a^3*b^2*tan(d*x+c)+189*(cos(d*x+c) ^2+cos(d*x+c)+1)*a*A*b^4*tan(d*x+c)+63*sin(d*x+c)*(3+2*cos(d*x+c))*a^2*A*b ^3+7*(21*cos(d*x+c)^4+7*cos(d*x+c)^3+7*cos(d*x+c)^2+5*cos(d*x+c)+5)*C*b^5* tan(d*x+c)*sec(d*x+c)^3+3*(27+82*cos(d*x+c)^2+27*cos(d*x+c))*B*a^2*b^3*tan (d*x+c)+3*(25*cos(d*x+c)^3+107*cos(d*x+c)^2+39*cos(d*x+c)+39)*a*b^4*B*tan( d*x+c)*sec(d*x+c)+8*(cos(d*x+c)^2+2*cos(d*x+c)+1)*C*(1/(a+b)*(b+a*cos(d*x+ c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^5*EllipticE( -csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(88*cos(d*x+c)^3+121*cos(d*x+c )^2+53*cos(d*x+c)+53)*C*a^2*b^3*tan(d*x+c)*sec(d*x+c)+(147*cos(d*x+c)^4+13 7*cos(d*x+c)^3+137*cos(d*x+c)^2+85*cos(d*x+c)+85)*C*b^4*a*tan(d*x+c)*sec(d *x+c)^2+189*(cos(d*x+c)^2+2*cos(d*x+c)+1)*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1 /2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^5*EllipticE(-csc(d*x +c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+147*(cos(d*x+c)^2+2*cos(d*x+c)+1)*C*(c os(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^ (1/2)*b^5*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+189*(-cos( d*x+c)^2-2*cos(d*x+c)-1)*A*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a *cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^5*EllipticF(-csc(d*x+c)+cot(d*x+c),(( a-b)/(a+b))^(1/2))-18*B*a^4*b*cos(d*x+c)*sin(d*x+c)+9*sin(d*x+c)*(cos(d*x+ c)-1)*a^3*b^2*B+147*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*C*(cos(d*x+c)/(cos(d...
\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{2} \,d x } \] Input:
integrate(sec(d*x+c)^2*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
Output:
integral((C*b*sec(d*x + c)^5 + (C*a + B*b)*sec(d*x + c)^4 + A*a*sec(d*x + c)^2 + (B*a + A*b)*sec(d*x + c)^3)*sqrt(b*sec(d*x + c) + a), x)
\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \] Input:
integrate(sec(d*x+c)**2*(a+b*sec(d*x+c))**(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+ c)**2),x)
Output:
Integral((a + b*sec(c + d*x))**(3/2)*(A + B*sec(c + d*x) + C*sec(c + d*x)* *2)*sec(c + d*x)**2, x)
Timed out. \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^2*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
Output:
Timed out
\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{2} \,d x } \] Input:
integrate(sec(d*x+c)^2*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(3/ 2)*sec(d*x + c)^2, x)
Timed out. \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^2} \,d x \] Input:
int(((a + b/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/c os(c + d*x)^2,x)
Output:
int(((a + b/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/c os(c + d*x)^2, x)
\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) b c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) a c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) b^{2}+2 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}d x \right ) a b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{2}d x \right ) a^{2} \] Input:
int(sec(d*x+c)^2*(a+b*sec(d*x+c))^(3/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**5,x)*b*c + int(sqrt(sec(c + d*x )*b + a)*sec(c + d*x)**4,x)*a*c + int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x )**4,x)*b**2 + 2*int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**3,x)*a*b + int (sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**2,x)*a**2