Integrand size = 35, antiderivative size = 421 \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\left (42 a A b+17 a^2 B+16 b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{24 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (6 a^2 A b+8 A b^3-a^3 B+12 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{8 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (30 a A b+3 a^2 B+16 b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 b d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {b B \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {(6 A b+7 a B) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (30 a A b+3 a^2 B+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 b d \sqrt {\cos (c+d x)}} \]
1/24*(42*A*a*b+17*B*a^2+16*B*b^2)*(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)*(a/(a+b))^(1/2))*((b+a*cos(d* x+c))/(a+b))^(1/2)/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+1/8*(6*A*a^2* b+8*A*b^3-B*a^3+12*B*a*b^2)*(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),2,2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c ))/(a+b))^(1/2)/b/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)+1/3*b*B*sin(d* x+c)*(a+b*sec(d*x+c))^(1/2)/d/cos(d*x+c)^(5/2)+1/12*(6*A*b+7*B*a)*sin(d*x+ c)*(a+b*sec(d*x+c))^(1/2)/d/cos(d*x+c)^(3/2)+1/24*(30*A*a*b+3*B*a^2+16*B*b ^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/b/d/cos(d*x+c)^(1/2)-1/24*(30*A*a*b+ 3*B*a^2+16*B*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elliptic E(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*cos(d*x+c)^(1/2)*(a+b*sec(d* x+c))^(1/2)/b/d/((b+a*cos(d*x+c))/(a+b))^(1/2)
Result contains complex when optimal does not.
Time = 34.78 (sec) , antiderivative size = 151903, normalized size of antiderivative = 360.81 \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Result too large to show} \]
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.
\(\displaystyle \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3434 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
\(\Big \downarrow \) 4514 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{3} \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (b (6 A b+7 a B) \sec ^2(c+d x)+2 \left (3 B a^2+6 A b a+2 b^2 B\right ) \sec (c+d x)+3 a (2 a A+b B)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (b (6 A b+7 a B) \sec ^2(c+d x)+2 \left (3 B a^2+6 A b a+2 b^2 B\right ) \sec (c+d x)+3 a (2 a A+b B)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (b (6 A b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (3 B a^2+6 A b a+2 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (2 a A+b B)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 4590 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (b \left (3 B a^2+30 A b a+16 b^2 B\right ) \sec ^2(c+d x)+2 b \left (12 A a^2+13 b B a+6 A b^2\right ) \sec (c+d x)+a b (6 A b+7 a B)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\sec (c+d x)} \left (b \left (3 B a^2+30 A b a+16 b^2 B\right ) \sec ^2(c+d x)+2 b \left (12 A a^2+13 b B a+6 A b^2\right ) \sec (c+d x)+a b (6 A b+7 a B)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (3 B a^2+30 A b a+16 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (12 A a^2+13 b B a+6 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a b (6 A b+7 a B)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 4590 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\int -\frac {-2 a (6 A b+7 a B) \sec (c+d x) b^2-3 \left (-B a^3+6 A b a^2+12 b^2 B a+8 A b^3\right ) \sec ^2(c+d x) b+a \left (3 B a^2+30 A b a+16 b^2 B\right ) b}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{b}+\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {-2 a (6 A b+7 a B) \sec (c+d x) b^2-3 \left (-B a^3+6 A b a^2+12 b^2 B a+8 A b^3\right ) \sec ^2(c+d x) b+a \left (3 B a^2+30 A b a+16 b^2 B\right ) b}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {-2 a (6 A b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right ) b^2-3 \left (-B a^3+6 A b a^2+12 b^2 B a+8 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b+a \left (3 B a^2+30 A b a+16 b^2 B\right ) b}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 4596 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (3 B a^2+30 A b a+16 b^2 B\right )-2 a b^2 (6 A b+7 a B) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx-3 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (3 B a^2+30 A b a+16 b^2 B\right )-2 a b^2 (6 A b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-3 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 4346 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (3 B a^2+30 A b a+16 b^2 B\right )-2 a b^2 (6 A b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (3 B a^2+30 A b a+16 b^2 B\right )-2 a b^2 (6 A b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (3 B a^2+30 A b a+16 b^2 B\right )-2 a b^2 (6 A b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (3 B a^2+30 A b a+16 b^2 B\right )-2 a b^2 (6 A b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int \frac {a b \left (3 B a^2+30 A b a+16 b^2 B\right )-2 a b^2 (6 A b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 4523 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx\right )+b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 4343 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\left (b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 4345 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{6} \left (\frac {\frac {\left (3 a^2 B+30 a A b+16 b^2 B\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{d}-\frac {-\frac {b^2 \left (17 a^2 B+42 a A b+16 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (3 a^2 B+30 a A b+16 b^2 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {6 b \left (a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}}{4 b}+\frac {(7 a B+6 A b) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {b B \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )\) |
3.7.6.3.1 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[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
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[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], 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[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]* (d_.) + (c_))^(n_.)*((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_.), x_Symbol] :> Sim p[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p Int[(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(g*Csc[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g , m, n, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && !(IntegerQ[m] && I ntegerQ[n])
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_)], x_Symbol] :> Simp[d*Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x ]]/Sqrt[a + b*Csc[e + f*x]]) Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]] ), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B* Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*(m + n))), x] + Simp[1/(m + n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n* Simp[a^2*A*(m + n) + a*b*B*n + (a*(2*A*b + a*B)*(m + n) + b^2*B*(m + n - 1) )*Csc[e + f*x] + b*(A*b*(m + n) + a*B*(2*m + n - 1))*Csc[e + f*x]^2, x], x] , x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && !(IGtQ[n, 1] && !IntegerQ[m])
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && 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[(-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]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[C/d^2 Int[(d*Csc[e + f*x])^(3/2)/Sqrt[a + b*C sc[e + f*x]], x], x] + Int[(A + B*Csc[e + f*x])/(Sqrt[d*Csc[e + f*x]]*Sqrt[ a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0]
Result contains complex when optimal does not.
Time = 16.11 (sec) , antiderivative size = 3577, normalized size of antiderivative = 8.50
1/24/d*(8*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*b^3*cos(d*x+c)*si n(d*x+c)+12*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a*b^2*cos(d*x+c )^3*sin(d*x+c)+14*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a^2*b*cos (d*x+c)^3*sin(d*x+c)+16*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*a*b ^2*cos(d*x+c)^3*sin(d*x+c)+42*A*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1))^(1/ 2)*a*b^2*cos(d*x+c)^2*sin(d*x+c)+17*B*((a-b)/(a+b))^(1/2)*(1/(cos(d*x+c)+1 ))^(1/2)*a^2*b*cos(d*x+c)^2*sin(d*x+c)+22*B*((a-b)/(a+b))^(1/2)*(1/(cos(d* x+c)+1))^(1/2)*a*b^2*cos(d*x+c)^2*sin(d*x+c)+22*B*((a-b)/(a+b))^(1/2)*(1/( cos(d*x+c)+1))^(1/2)*a*b^2*cos(d*x+c)*sin(d*x+c)+3*B*(1/(a+b)*(b+a*cos(d*x +c))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d *x+c)),(-(a+b)/(a-b))^(1/2))*a^3*cos(d*x+c)^3-16*B*(1/(a+b)*(b+a*cos(d*x+c ))/(cos(d*x+c)+1))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x +c)),(-(a+b)/(a-b))^(1/2))*b^3*cos(d*x+c)^3+8*B*((a-b)/(a+b))^(1/2)*(1/(co s(d*x+c)+1))^(1/2)*b^3*sin(d*x+c)+24*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+ c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b) /(a-b))^(1/2))*b^3*cos(d*x+c)^4-48*A*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c) +1))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(a+b)/(a -b),I/((a-b)/(a+b))^(1/2))*b^3*cos(d*x+c)^4-6*B*(1/(a+b)*(b+a*cos(d*x+c))/ (cos(d*x+c)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c) ),(-(a+b)/(a-b))^(1/2))*a^3*cos(d*x+c)^4+6*B*(1/(a+b)*(b+a*cos(d*x+c))/...
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^{3/2}} \,d x \]