Integrand size = 43, antiderivative size = 342 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=-\frac {2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a^4 d}+\frac {2 \left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 a^5 d}-\frac {2 b^3 \left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^5 (a+b) d}+\frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \sqrt {\sec (c+d x)}} \]
2/7*A*sin(d*x+c)/a/d/sec(d*x+c)^(5/2)-2/5*(A*b-B*a)*sin(d*x+c)/a^2/d/sec(d *x+c)^(3/2)+2/21*(7*A*b^2-7*B*a*b+a^2*(5*A+7*C))*sin(d*x+c)/a^3/d/sec(d*x+ c)^(1/2)-2/5*(5*A*b^3-3*B*a^3-5*B*a*b^2+a^2*b*(3*A+5*C))*(cos(1/2*d*x+1/2* c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d *x+c)^(1/2)*sec(d*x+c)^(1/2)/a^4/d+2/21*(21*A*b^4-7*B*a^3*b-21*B*a*b^3+7*a ^2*b^2*(A+3*C)+a^4*(5*A+7*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2 *c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2 )/a^5/d-2*b^3*(A*b^2-a*(B*b-C*a))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x +1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*a/(a+b),2^(1/2))*cos(d*x+c)^(1/2)* sec(d*x+c)^(1/2)/a^5/(a+b)/d
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx \]
Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*(a + b*Sec[c + d*x])),x]
Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*(a + b*Sec[c + d*x])), x]
Time = 2.68 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 4592, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 3042, 3284}
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.
| \(\displaystyle \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \int \frac {-5 A b \sec ^2(c+d x)-a (5 A+7 C) \sec (c+d x)+7 (A b-a B)}{2 \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))}dx}{7 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-5 A b \sec ^2(c+d x)-a (5 A+7 C) \sec (c+d x)+7 (A b-a B)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))}dx}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\int \frac {-5 A b \csc \left (c+d x+\frac {\pi }{2}\right )^2-a (5 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right )+7 (A b-a B)}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{7 a}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {-21 b (A b-a B) \sec ^2(c+d x)+a (4 A b+21 a B) \sec (c+d x)+5 \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right )}{2 \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}dx}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-21 b (A b-a B) \sec ^2(c+d x)+a (4 A b+21 a B) \sec (c+d x)+5 \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right )}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}dx}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-21 b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a (4 A b+21 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+5 \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {-5 b \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \sec ^2(c+d x)+a \left (-5 (5 A+7 C) a^2-28 b B a+28 A b^2\right ) \sec (c+d x)+21 \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right )}{2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-5 b \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \sec ^2(c+d x)+a \left (-5 (5 A+7 C) a^2-28 b B a+28 A b^2\right ) \sec (c+d x)+21 \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right )}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))}dx}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-5 b \left ((5 A+7 C) a^2-7 b B a+7 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (-5 (5 A+7 C) a^2-28 b B a+28 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+21 \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 4594 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)}dx}{a^2}+\frac {\int \frac {21 a \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right )-5 \left ((5 A+7 C) a^4-7 b B a^3+7 b^2 (A+3 C) a^2-21 b^3 B a+21 A b^4\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\int \frac {21 a \left (-3 B a^3+b (3 A+5 C) a^2-5 b^2 B a+5 A b^3\right )-5 \left ((5 A+7 C) a^4-7 b B a^3+7 b^2 (A+3 C) a^2-21 b^3 B a+21 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {21 a \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-5 \left (a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right ) \int \sqrt {\sec (c+d x)}dx}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {21 a \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-5 \left (a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {21 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right ) \int \sqrt {\cos (c+d x)}dx-5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {21 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\frac {42 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}-5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \left (A b^2-a (b B-a C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\frac {42 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}-\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right )}{d}}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 4336 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a^2}+\frac {\frac {42 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}-\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right )}{d}}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {105 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2}+\frac {\frac {42 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}-\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right )}{d}}{a^2}}{3 a}}{5 a}}{7 a}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {2 A \sin (c+d x)}{7 a d \sec ^{\frac {5}{2}}(c+d x)}-\frac {\frac {14 (A b-a B) \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {10 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{3 a d \sqrt {\sec (c+d x)}}-\frac {\frac {210 b^3 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a^2 d (a+b)}+\frac {\frac {42 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right )}{d}-\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right )}{d}}{a^2}}{3 a}}{5 a}}{7 a}\) |
(2*A*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((14*(A*b - a*B)*Sin[c + d *x])/(5*a*d*Sec[c + d*x]^(3/2)) - (-1/3*(((42*a*(5*A*b^3 - 3*a^3*B - 5*a*b ^2*B + a^2*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqr t[Sec[c + d*x]])/d - (10*(21*A*b^4 - 7*a^3*b*B - 21*a*b^3*B + 7*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sq rt[Sec[c + d*x]])/d)/a^2 + (210*b^3*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d *x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d))/a + (10*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(3*a *d*Sqrt[Sec[c + d*x]]))/(5*a))/(7*a)
3.11.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]] Int[ 1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
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[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*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] && LeQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2) Int[(d*Csc[e + f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2 Int[(a*A - (A*b - a *B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1094\) vs. \(2(392)=784\).
Time = 3.17 (sec) , antiderivative size = 1095, normalized size of antiderivative = 3.20
int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2)/(a+b*sec(d*x+c)),x,me thod=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(8/105*A/a*(60* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-258*sin(1/2*d*x+1/2*c)^6*cos(1/2*d *x+1/2*c)+448*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4-167*sin(1/2*d*x+1/2* c)^2*cos(1/2*d*x+1/2*c)+85*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2 *c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-168*EllipticE(cos(1/2 *d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^ 2)^(1/2))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+4/5/a^2*(4* A*a+A*b-B*a)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(4*sin(1 /2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-14*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c )^4+6*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-5*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 9*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*( sin(1/2*d*x+1/2*c)^2)^(1/2))+2*b^3*(A*b^2-B*a*b+C*a^2)/a^4/(a^2-a*b)*(sin( 1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+ 1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a- b),2^(1/2))+4/3/a^3*(6*A*a^2+3*A*a*b+A*b^2-3*B*a^2-B*a*b+C*a^2)*(2*cos(1/2 *d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+2 *(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(c os(1/2*d*x+1/2*c),2^(1/2))-3*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin( 1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2))/(-2*sin(1/2*d*x...
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\text {Timed out} \]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2)/(a+b*sec(d*x+c) ),x, algorithm="fricas")
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\text {Timed out} \]
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2)/(a+b*sec(d*x+c) ),x, algorithm="maxima")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)/((b*sec(d*x + c) + a)*se c(d*x + c)^(7/2)), x)
\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
integrate((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2)/(a+b*sec(d*x+c) ),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)/((b*sec(d*x + c) + a)*se c(d*x + c)^(7/2)), x)
Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]