Integrand size = 35, antiderivative size = 467 \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^5 d}+\frac {2 (a-b) \sqrt {a+b} \left (16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^4 d}+\frac {2 a \left (21 A b^2+8 a^2 C+13 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac {2 a C \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{63 b d}+\frac {2 C \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9 d} \] Output:
2/315*(a-b)*(a+b)^(1/2)*(16*a^4*C+6*a^2*b^2*(7*A+4*C)-21*b^4*(9*A+7*C))*co t(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+2/31 5*(a-b)*(a+b)^(1/2)*(16*a^3*C+12*a^2*b*C+6*a*b^2*(7*A+6*C)+21*b^3*(9*A+7*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*a*(21*A*b^2+8*C*a^2+13*C*b^2)*(a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^3 /d-2/315*(6*C*a^2-7*b^2*(9*A+7*C))*sec(d*x+c)*(a+b*sec(d*x+c))^(1/2)*tan(d *x+c)/b^2/d+2/63*a*C*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
Time = 19.40 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.45 \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \left (2 (a+b) \left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )+2 b (a+b) \left (-16 a^3 C+12 a^2 b C-6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )+\left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \cos (c+d x) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 b^4 d (b+a \cos (c+d x)) (A+2 C+A \cos (2 c+2 d x)) \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {5}{2}}(c+d x)}+\frac {\cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {4 \left (-42 a^2 A b^2+189 A b^4-16 a^4 C-24 a^2 b^2 C+147 b^4 C\right ) \sin (c+d x)}{315 b^4}+\frac {4 \sec ^2(c+d x) \left (63 A b^2 \sin (c+d x)-6 a^2 C \sin (c+d x)+49 b^2 C \sin (c+d x)\right )}{315 b^2}+\frac {4 \sec (c+d x) \left (21 a A b^2 \sin (c+d x)+8 a^3 C \sin (c+d x)+13 a b^2 C \sin (c+d x)\right )}{315 b^3}+\frac {4 a C \sec ^2(c+d x) \tan (c+d x)}{63 b}+\frac {4}{9} C \sec ^3(c+d x) \tan (c+d x)\right )}{d (A+2 C+A \cos (2 c+2 d x))} \] Input:
Integrate[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x ]
Output:
(4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + C*S ec[c + d*x]^2)*(2*(a + b)*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/(( a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^ 3*(9*A + 7*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d *x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Co s[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315 *b^4*d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d *x)/2]^2]*Sec[c + d*x]^(5/2)) + (Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*( A + C*Sec[c + d*x]^2)*((4*(-42*a^2*A*b^2 + 189*A*b^4 - 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]^2*(63*A*b^2*Si n[c + d*x] - 6*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b^2) + (4 *Sec[c + d*x]*(21*a*A*b^2*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*a*C*Sec[c + d*x]^2*Tan[c + d*x])/(63*b) + ( 4*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(A + 2*C + A*Cos[2*c + 2*d*x]))
Time = 2.34 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {3042, 4585, 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.
| \(\displaystyle \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4585 |
| \(\displaystyle \frac {2}{9} \int \frac {\sec ^3(c+d x) \left (a C \sec ^2(c+d x)+b (9 A+7 C) \sec (c+d x)+3 a (3 A+2 C)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {\sec ^3(c+d x) \left (a C \sec ^2(c+d x)+b (9 A+7 C) \sec (c+d x)+3 a (3 A+2 C)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a C \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (9 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (3 A+2 C)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 \int \frac {\sec ^2(c+d x) \left (4 C a^2+b (63 A+47 C) \sec (c+d x) a-\left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {\int \frac {\sec ^2(c+d x) \left (4 C a^2+b (63 A+47 C) \sec (c+d x) a-\left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (4 C a^2+b (63 A+47 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a+\left (7 b^2 (9 A+7 C)-6 a^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4580 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {2 \int -\frac {\sec (c+d x) \left (-3 a \left (8 C a^2+21 A b^2+13 b^2 C\right ) \sec ^2(c+d x)-b \left (2 C a^2+189 A b^2+147 b^2 C\right ) \sec (c+d x)+2 a \left (6 a^2 C-7 b^2 (9 A+7 C)\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{5 b}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {-\frac {\int \frac {\sec (c+d x) \left (-3 a \left (8 C a^2+21 A b^2+13 b^2 C\right ) \sec ^2(c+d x)-b \left (2 C a^2+189 A b^2+147 b^2 C\right ) \sec (c+d x)+2 a \left (6 a^2 C-7 b^2 (9 A+7 C)\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{5 b}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (-3 a \left (8 C a^2+21 A b^2+13 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-b \left (2 C a^2+189 A b^2+147 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a \left (6 a^2 C-7 b^2 (9 A+7 C)\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {1}{9} \left (\frac {-\frac {\frac {2 \int -\frac {3 \sec (c+d x) \left (a b \left (-4 C a^2+147 A b^2+111 b^2 C\right )-\left (16 C a^4+6 b^2 (7 A+4 C) a^2-21 b^4 (9 A+7 C)\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{3 b}-\frac {2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {-\frac {-\frac {\int \frac {\sec (c+d x) \left (a b \left (-4 C a^2+147 A b^2+111 b^2 C\right )-\left (16 C a^4+6 b^2 (7 A+4 C) a^2-21 b^4 (9 A+7 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {-\frac {-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a b \left (-4 C a^2+147 A b^2+111 b^2 C\right )+\left (-16 C a^4-6 b^2 (7 A+4 C) a^2+21 b^4 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4493 |
| \(\displaystyle \frac {1}{9} \left (\frac {-\frac {-\frac {(a-b) \left (16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-\left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {-\frac {-\frac {(a-b) \left (16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{9} \left (\frac {-\frac {-\frac {\frac {2 (a-b) \sqrt {a+b} \left (16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{9} \left (\frac {-\frac {2 \left (6 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}-\frac {-\frac {2 a \left (8 a^2 C+21 A b^2+13 b^2 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}-\frac {\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{b}}{5 b}}{7 b}+\frac {2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 C \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
Input:
Int[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + 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*a*C *Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d) + ((-2*(6*a ^2*C - 7*b^2*(9*A + 7*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]*(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)]*Sq rt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) + (2*(a - b)*Sqrt[a + b]*(1 6*a^3*C + 12*a^2*b*C + 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*Cot[c + d *x]*EllipticF[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*d))/b) - (2*a*(21*A*b^2 + 8*a^2*C + 13*b^2*C)*Sqrt[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_)]^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*Csc[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 + b*(A*(m + n + 1) + C*(m + n))*Csc[e + f*x ] + a*C*m*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, 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. \(2254\) vs. \(2(429)=858\).
Time = 74.08 (sec) , antiderivative size = 2255, normalized size of antiderivative = 4.83
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2255\) |
| parts | \(\text {Expression too large to display}\) | \(2267\) |
Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(A+C*sec(d*x+c)^2),x,method=_RETUR NVERBOSE)
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))+42*A*a^3*b^2*cos(d*x+c)*sin(d*x+c)+7*(-2 1*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+2*(12*cos(d*x+c)^2-cos(d*x+c)-1)*C*a^3*b^2*tan(d*x+c)+21* (-9*cos(d*x+c)^2-4*cos(d*x+c)-4)*a*A*b^4*tan(d*x+c)+21*sin(d*x+c)*(1-cos(d *x+c))*a^2*A*b^3+8*sin(d*x+c)*(1-cos(d*x+c))*a^4*b*C+4*(-cos(d*x+c)^2-2...
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(A+C*sec(d*x+c)^2),x, algori thm="fricas")
Output:
integral((C*sec(d*x + c)^5 + A*sec(d*x + c)^3)*sqrt(b*sec(d*x + c) + a), x )
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sqrt {a + b \sec {\left (c + d x \right )}} \sec ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(sec(d*x+c)**3*(a+b*sec(d*x+c))**(1/2)*(A+C*sec(d*x+c)**2),x)
Output:
Integral((A + C*sec(c + d*x)**2)*sqrt(a + b*sec(c + d*x))*sec(c + d*x)**3, x)
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(A+C*sec(d*x+c)^2),x, algori thm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + A)*sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^3, x)
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(A+C*sec(d*x+c)^2),x, algori thm="giac")
Output:
integrate((C*sec(d*x + c)^2 + A)*sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^3, x)
Timed out. \[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{{\cos \left (c+d\,x\right )}^3} \,d x \] Input:
int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(1/2))/cos(c + d*x)^3,x)
Output:
int(((A + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(1/2))/cos(c + d*x)^3, x)
\[ \int \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{3}d x \right ) a \] Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2)*(A+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)**3,x)*a