Integrand size = 43, antiderivative size = 517 \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (24 a^3 b B+57 a b^3 B-16 a^4 C-6 a^2 b^2 (7 A+4 C)+21 b^4 (9 A+7 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^5 d}-\frac {2 (a-b) \sqrt {a+b} \left (12 a^2 b (2 B-C)-16 a^3 C-6 a b^2 (7 A-3 B+6 C)-3 b^3 (63 A-25 B+49 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^4 d}-\frac {2 \left (12 a^2 b B-75 b^3 B-8 a^3 C-a b^2 (21 A+13 C)\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}+\frac {2 \left (63 A b^2+9 a b B-6 a^2 C+49 b^2 C\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac {2 (9 b B+a C) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{63 b d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9 d} \] Output:
-2/315*(a-b)*(a+b)^(1/2)*(24*B*a^3*b+57*B*a*b^3-16*a^4*C-6*a^2*b^2*(7*A+4* C)+21*b^4*(9*A+7*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^5/d-2/315*(a-b)*(a+b)^(1/2)*(12*a^2*b*(2*B-C)-16*a^3*C-6*a* b^2*(7*A-3*B+6*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^4/d-2/315*(12*B*a^2*b-75*B*b^3-8*a^3*C -a*b^2*(21*A+13*C))*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^3/d+2/315*(63*A*b^ 2+9*B*a*b-6*C*a^2+49*C*b^2)*sec(d*x+c)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b ^2/d+2/63*(9*B*b+C*a)*sec(d*x+c)^2*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b/d+2 /9*C*sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(4780\) vs. \(2(517)=1034\).
Time = 28.12 (sec) , antiderivative size = 4780, normalized size of antiderivative = 9.25 \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C* Sec[c + d*x]^2),x]
Output:
(Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d *x]^2)*((4*(-42*a^2*A*b^2 + 189*A*b^4 + 24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 24*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^4) + (4*Sec[c + d*x]^3*( 9*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/(63*b) + (4*Sec[c + d*x]^2*(63*A*b ^2*Sin[c + d*x] + 9*a*b*B*Sin[c + d*x] - 6*a^2*C*Sin[c + d*x] + 49*b^2*C*S in[c + d*x]))/(315*b^2) + (4*Sec[c + d*x]*(21*a*A*b^2*Sin[c + d*x] - 12*a^ 2*b*B*Sin[c + d*x] + 75*b^3*B*Sin[c + d*x] + 8*a^3*C*Sin[c + d*x] + 13*a*b ^2*C*Sin[c + d*x]))/(315*b^3) + (4*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*( A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*((4*a^2*A)/(15*b*Sq rt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*A*b)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (38*a*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[S ec[c + d*x]]) - (16*a^3*B)/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^4*C)/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*a^2*C)/(105*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*b*C )/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*A*Sqrt[Sec[c + d *x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*A*Sqrt[Sec[c + d*x]])/(15*b^2 *Sqrt[b + a*Cos[c + d*x]]) - (16*a^4*B*Sqrt[Sec[c + d*x]])/(105*b^3*Sqrt[b + a*Cos[c + d*x]]) - (34*a^2*B*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[ c + d*x]]) + (10*b*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - ( 8*a*C*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (32*a^5*C*Sqr...
Time = 2.56 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 4584, 27, 3042, 4590, 27, 3042, 4580, 27, 3042, 4570, 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.
| \(\displaystyle \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4584 |
| \(\displaystyle \frac {2}{9} \int \frac {\sec ^3(c+d x) \left ((9 b B+a C) \sec ^2(c+d x)+(9 A b+7 C b+9 a B) \sec (c+d x)+3 a (3 A+2 C)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {\sec ^3(c+d x) \left ((9 b B+a C) \sec ^2(c+d x)+(9 A b+7 C b+9 a B) \sec (c+d x)+3 a (3 A+2 C)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left ((9 b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(9 A b+7 C b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (3 A+2 C)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 \int \frac {\sec ^2(c+d x) \left (\left (-6 C a^2+9 b B a+63 A b^2+49 b^2 C\right ) \sec ^2(c+d x)+b (63 a A+45 b B+47 a C) \sec (c+d x)+4 a (9 b B+a C)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {\int \frac {\sec ^2(c+d x) \left (\left (-6 C a^2+9 b B a+63 A b^2+49 b^2 C\right ) \sec ^2(c+d x)+b (63 a A+45 b B+47 a C) \sec (c+d x)+4 a (9 b B+a C)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\left (-6 C a^2+9 b B a+63 A b^2+49 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (63 a A+45 b B+47 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+4 a (9 b B+a C)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4580 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {2 \int \frac {\sec (c+d x) \left (-3 \left (-8 C a^3+12 b B a^2-b^2 (21 A+13 C) a-75 b^3 B\right ) \sec ^2(c+d x)+b \left (2 C a^2+207 b B a+189 A b^2+147 b^2 C\right ) \sec (c+d x)+2 a \left (-6 C a^2+9 b B a+63 A b^2+49 b^2 C\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\int \frac {\sec (c+d x) \left (-3 \left (-8 C a^3+12 b B a^2-b^2 (21 A+13 C) a-75 b^3 B\right ) \sec ^2(c+d x)+b \left (2 C a^2+207 b B a+189 A b^2+147 b^2 C\right ) \sec (c+d x)+2 a \left (-6 C a^2+9 b B a+63 A b^2+49 b^2 C\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (-3 \left (-8 C a^3+12 b B a^2-b^2 (21 A+13 C) a-75 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (2 C a^2+207 b B a+189 A b^2+147 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a \left (-6 C a^2+9 b B a+63 A b^2+49 b^2 C\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {2 \int \frac {3 \sec (c+d x) \left (b \left (-4 C a^3+6 b B a^2+3 b^2 (49 A+37 C) a+75 b^3 B\right )+\left (-16 C a^4+24 b B a^3-6 b^2 (7 A+4 C) a^2+57 b^3 B a+21 b^4 (9 A+7 C)\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{3 b}-\frac {2 \tan (c+d x) \left (-8 a^3 C+12 a^2 b B-a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\sec (c+d x) \left (b \left (-4 C a^3+6 b B a^2+3 b^2 (49 A+37 C) a+75 b^3 B\right )+\left (-16 C a^4+24 b B a^3-6 b^2 (7 A+4 C) a^2+57 b^3 B a+21 b^4 (9 A+7 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 \tan (c+d x) \left (-8 a^3 C+12 a^2 b B-a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b \left (-4 C a^3+6 b B a^2+3 b^2 (49 A+37 C) a+75 b^3 B\right )+\left (-16 C a^4+24 b B a^3-6 b^2 (7 A+4 C) a^2+57 b^3 B a+21 b^4 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \tan (c+d x) \left (-8 a^3 C+12 a^2 b B-a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4493 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx-(a-b) \left (-16 a^3 C+12 a^2 b (2 B-C)-6 a b^2 (7 A-3 B+6 C)-3 b^3 (63 A-25 B+49 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 \tan (c+d x) \left (-8 a^3 C+12 a^2 b B-a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-(a-b) \left (-16 a^3 C+12 a^2 b (2 B-C)-6 a b^2 (7 A-3 B+6 C)-3 b^3 (63 A-25 B+49 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \tan (c+d x) \left (-8 a^3 C+12 a^2 b B-a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-16 a^3 C+12 a^2 b (2 B-C)-6 a b^2 (7 A-3 B+6 C)-3 b^3 (63 A-25 B+49 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{b}-\frac {2 \tan (c+d x) \left (-8 a^3 C+12 a^2 b B-a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {2 \tan (c+d x) \sec (c+d x) \left (-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{5 b d}+\frac {\frac {-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-16 a^3 C+12 a^2 b (2 B-C)-6 a b^2 (7 A-3 B+6 C)-3 b^3 (63 A-25 B+49 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}}{b}-\frac {2 \tan (c+d x) \left (-8 a^3 C+12 a^2 b B-a b^2 (21 A+13 C)-75 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}}{7 b}+\frac {2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
Input:
Int[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(2*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + ((2*(9* b*B + a*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d) + ((2*(63*A*b^2 + 9*a*b*B - 6*a^2*C + 49*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec [c + d*x]]*Tan[c + d*x])/(5*b*d) + (((-2*(a - b)*Sqrt[a + b]*(24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 6*a^2*b^2*(7*A + 4*C) + 21*b^4*(9*A + 7*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]*(12*a^2*b*(2*B - C) - 16*a^3*C - 6*a*b^2*(7*A - 3*B + 6*C) - 3*b^3*(63*A - 25*B + 49*C))*Cot[c + d*x]*El lipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqr t[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))]) /(b*d))/b - (2*(12*a^2*b*B - 75*b^3*B - 8*a^3*C - a*b^2*(21*A + 13*C))*Sqr t[a + b*Sec[c + d*x]]*Tan[c + d*x])/(b*d))/(5*b))/(7*b))/9
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_)))/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]
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[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(m + n + 1) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a *B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LeQ[n, -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[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 )*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(3251\) vs. \(2(479)=958\).
Time = 105.50 (sec) , antiderivative size = 3252, normalized size of antiderivative = 6.29
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3252\) |
| parts | \(\text {Expression too large to display}\) | \(3297\) |
Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
Output:
-2/315/d/b^4*(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)*((-13*cos(d*x+c)^3+11*cos(d*x+c)^2+cos(d*x+c)+1)*C*a^2*b^3*tan(d*x+ c)*sec(d*x+c)+(-147*cos(d*x+c)^4-62*cos(d*x+c)^3-62*cos(d*x+c)^2-40*cos(d* x+c)-40)*C*a*b^4*tan(d*x+c)*sec(d*x+c)^2+189*(-cos(d*x+c)^2-2*cos(d*x+c)-1 )*A*(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)*b^5*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+16*( 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))+147*(-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)*b^5*El lipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+189*(cos(d*x+c)^2+2*co s(d*x+c)+1)*A*(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)*b^5*EllipticF(-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*(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)*b^5*EllipticF(-csc(d*x+c )+cot(d*x+c),((a-b)/(a+b))^(1/2))+3*(-25*cos(d*x+c)^3-44*cos(d*x+c)^2-18*c os(d*x+c)-18)*a*b^4*B*tan(d*x+c)*sec(d*x+c)+75*(cos(d*x+c)^2+2*cos(d*x+c)+ 1)*B*(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))-24* B*a^4*b*cos(d*x+c)*sin(d*x+c)+3*(-19*cos(d*x+c)^2+cos(d*x+c)+1)*B*a^2*b...
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
Output:
integral((C*sec(d*x + c)^5 + B*sec(d*x + c)^4 + A*sec(d*x + c)^3)*sqrt(b*s ec(d*x + c) + a), x)
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \sqrt {a + b \sec {\left (c + d x \right )}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(sec(d*x+c)**3*(a+b*sec(d*x+c))**(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+ c)**2),x)
Output:
Integral(sqrt(a + b*sec(c + d*x))*(A + B*sec(c + d*x) + C*sec(c + d*x)**2) *sec(c + d*x)**3, x)
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(b*sec(d*x + c) + a) *sec(d*x + c)^3, x)
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(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)*sqrt(b*sec(d*x + c) + a) *sec(d*x + c)^3, x)
Timed out. \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^3} \,d x \] Input:
int(((a + b/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/c os(c + d*x)^3,x)
Output:
int(((a + b/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/c os(c + d*x)^3, x)
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}d x \right ) a \] Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(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)*c + int(sqrt(sec(c + d*x)* b + a)*sec(c + d*x)**4,x)*b + int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**3 ,x)*a