Integrand size = 35, antiderivative size = 487 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\frac {(4 A b-5 a B) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{4 b^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (12 a A b-15 a^2 B-4 b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{4 b^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 a (A b-a B) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {\left (4 a A b-5 a^2 B+b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (12 a^2 A b-4 A b^3-15 a^3 B+7 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}} \]
2*a*(A*b-B*a)*sin(d*x+c)/b/(a^2-b^2)/d/cos(d*x+c)^(5/2)/(a+b*sec(d*x+c))^( 1/2)+1/4*(4*A*b-5*B*a)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ell ipticF(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)/b^2/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)-1/4*(12*A*a*b-15*B*a ^2-4*B*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^3/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)-1/2*(4*A*a*b-5*B*a^2+B*b^2)* sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/b^2/(a^2-b^2)/d/cos(d*x+c)^(3/2)+1/4*(12 *A*a^2*b-4*A*b^3-15*B*a^3+7*B*a*b^2)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/b^3 /(a^2-b^2)/d/cos(d*x+c)^(1/2)-1/4*(12*A*a^2*b-4*A*b^3-15*B*a^3+7*B*a*b^2)* (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)*(a/(a+b))^(1/2))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)/b^3/(a ^2-b^2)/d/((b+a*cos(d*x+c))/(a+b))^(1/2)
Result contains complex when optimal does not.
Time = 37.16 (sec) , antiderivative size = 141550, normalized size of antiderivative = 290.66 \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Result too large to show} \]
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.
| \(\displaystyle \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3434 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4517 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-5 B a^2+4 A b a+b^2 B\right ) \sec ^2(c+d x)\right )-b (A b-a B) \sec (c+d x)+3 a (A b-a B)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-5 B a^2+4 A b a+b^2 B\right ) \sec ^2(c+d x)\right )-b (A b-a B) \sec (c+d x)+3 a (A b-a B)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\left (5 B a^2-4 A b a-b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-b (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (A b-a B)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {\frac {\int -\frac {\sqrt {\sec (c+d x)} \left (-\left (\left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right ) \sec ^2(c+d x)\right )-2 b \left (-B a^2+2 A b a-b^2 B\right ) \sec (c+d x)+a \left (-5 B a^2+4 A b a+b^2 B\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\int \frac {\sqrt {\sec (c+d x)} \left (-\left (\left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right ) \sec ^2(c+d x)\right )-2 b \left (-B a^2+2 A b a-b^2 B\right ) \sec (c+d x)+a \left (-5 B a^2+4 A b a+b^2 B\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (15 B a^3-12 A b a^2-7 b^2 B a+4 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 b \left (-B a^2+2 A b a-b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-5 B a^2+4 A b a+b^2 B\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\int \frac {\left (a^2-b^2\right ) \left (-15 B a^2+12 A b a-4 b^2 B\right ) \sec ^2(c+d x)+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \sec (c+d x)+a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\int \frac {\left (a^2-b^2\right ) \left (-15 B a^2+12 A b a-4 b^2 B\right ) \sec ^2(c+d x)+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \sec (c+d x)+a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\int \frac {\left (a^2-b^2\right ) \left (-15 B a^2+12 A b a-4 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4596 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4346 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\frac {\left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\int \frac {a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\frac {\left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\int \frac {a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\frac {\left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\int \frac {a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\frac {\left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\int \frac {a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {\int \frac {a \left (-15 B a^3+12 A b a^2+7 b^2 B a-4 A b^3\right )+2 a b \left (-5 B a^2+4 A b a+b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-b \left (a^2-b^2\right ) (4 A b-5 a B) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-b \left (a^2-b^2\right ) (4 A b-5 a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-b \left (a^2-b^2\right ) (4 A b-5 a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-b \left (a^2-b^2\right ) (4 A b-5 a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-b \left (a^2-b^2\right ) (4 A b-5 a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-b \left (a^2-b^2\right ) (4 A b-5 a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-b \left (a^2-b^2\right ) (4 A b-5 a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-\frac {b \left (a^2-b^2\right ) (4 A b-5 a B) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-\frac {b \left (a^2-b^2\right ) (4 A b-5 a B) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-\frac {b \left (a^2-b^2\right ) (4 A b-5 a B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {-\frac {\frac {-\frac {b \left (a^2-b^2\right ) (4 A b-5 a B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^2-b^2\right ) \left (-15 a^2 B+12 a A b-4 b^2 B\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{2 b}-\frac {\left (-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right ) \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}{b d}}{4 b}-\frac {\left (-5 a^2 B+4 a A b+b^2 B\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{2 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (A b-a B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}\right )\) |
3.7.27.3.1 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[a*d^2*( A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 2)/(b*f*(m + 1)*(a^2 - b^2))), x] - Simp[d/(b*(m + 1)*(a^2 - b^2)) Int[( a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*Simp[a*d*(A*b - a*B)*( n - 2) + b*d*(A*b - a*B)*(m + 1)*Csc[e + f*x] - (a*A*b*d*(m + n) - d*B*(a^2 *(n - 1) + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f , A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[ n, 1]
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 = 1.48 (sec) , antiderivative size = 3532, normalized size of antiderivative = 7.25
1/4/d*(4*A*((a-b)/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*b^3*cos(d*x+c)*sin (d*x+c)-24*A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*EllipticF(((a -b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c)),(-(a+b)/(a-b))^(1/2))*a^2*b*cos(d *x+c)^2-5*B*((a-b)/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a^2*b*cos(d*x+c)^ 2*sin(d*x+c)+2*B*((a-b)/(a+b))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a*b^2*cos(d* x+c)^2*sin(d*x+c)+24*A*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))*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d* x+c)))^(1/2)*a^2*b*cos(d*x+c)^3-8*B*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))*(1/(a+b)*(b+a*cos(d*x+ c))/(1+cos(d*x+c)))^(1/2)*a*b^2*cos(d*x+c)^3-16*A*(1/(a+b)*(b+a*cos(d*x+c) )/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+ c)),(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)^2+12*A*(1/(a+b)*(b+a*cos(d*x+c) )/(1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+ c)),(-(a+b)/(a-b))^(1/2))*a^2*b*cos(d*x+c)^2+20*B*(1/(a+b)*(b+a*cos(d*x+c) )/(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+ c)),(-(a+b)/(a-b))^(1/2))*a^2*b*cos(d*x+c)^2+2*B*(1/(a+b)*(b+a*cos(d*x+c)) /(1+cos(d*x+c)))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c )),(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)^2+7*B*(1/(a+b)*(b+a*cos(d*x+c))/ (1+cos(d*x+c)))^(1/2)*EllipticE(((a-b)/(a+b))^(1/2)*(cot(d*x+c)-csc(d*x+c) ),(-(a+b)/(a-b))^(1/2))*a*b^2*cos(d*x+c)^2-8*B*(1/(a+b)*(b+a*cos(d*x+c)...
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]