Integrand size = 45, antiderivative size = 378 \[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=-\frac {2 \left (2 A b^2+a b B-a^2 (3 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3 a^2 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (2 A b^3+3 a^3 B+a b^2 B-2 a^2 b (3 A+2 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac {2 \left (A b^4+2 a^3 b B+2 a b^3 B+a^4 C-5 a^2 b^2 (A+C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 a b \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}} \] Output:
-2/3*(2*A*b^2+B*a*b-a^2*(3*A+C))*((b+a*cos(d*x+c))/(a+b))^(1/2)*InverseJac obiAM(1/2*d*x+1/2*c,2^(1/2)*(a/(a+b))^(1/2))*sec(d*x+c)^(1/2)/a^2/(a^2-b^2 )/d/(a+b*sec(d*x+c))^(1/2)-2/3*(2*A*b^3+3*B*a^3+B*a*b^2-2*a^2*b*(3*A+2*C)) *EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1 /2)/a^2/(a^2-b^2)^2/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)-2/3* (A*b^2-a*(B*b-C*a))*sec(d*x+c)^(1/2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x +c))^(3/2)+2/3*(A*b^4+2*B*a^3*b+2*B*a*b^3+a^4*C-5*a^2*b^2*(A+C))*sec(d*x+c )^(1/2)*sin(d*x+c)/a/b/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 7.76 (sec) , antiderivative size = 5040, normalized size of antiderivative = 13.33 \[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]
Output:
Result too large to show
Time = 2.79 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.422, Rules used = {3042, 4586, 27, 3042, 4588, 27, 3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4586 |
| \(\displaystyle -\frac {2 \int -\frac {A b^2-3 (b B-a (A+C)) \sec (c+d x) b-\left (-C a^2-2 b B a+2 A b^2+3 b^2 C\right ) \sec ^2(c+d x)-a (b B-a C)}{2 \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A b^2-3 (b B-a (A+C)) \sec (c+d x) b-\left (-C a^2-2 b B a+2 A b^2+3 b^2 C\right ) \sec ^2(c+d x)-a (b B-a C)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {A b^2-3 (b B-a (A+C)) \csc \left (c+d x+\frac {\pi }{2}\right ) b+\left (C a^2+2 b B a-2 A b^2-3 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a (b B-a C)}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4588 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {2 \int \frac {b \left (3 B a^3-2 b (3 A+2 C) a^2+b^2 B a+2 A b^3\right )+a b \left (-\left ((3 A+C) a^2\right )+4 b B a-b^2 (A+3 C)\right ) \sec (c+d x)}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {b \left (3 B a^3-2 b (3 A+2 C) a^2+b^2 B a+2 A b^3\right )+a b \left (-\left ((3 A+C) a^2\right )+4 b B a-b^2 (A+3 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {b \left (3 B a^3-2 b (3 A+2 C) a^2+b^2 B a+2 A b^3\right )+a b \left (-\left ((3 A+C) a^2\right )+4 b B a-b^2 (A+3 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {\frac {2 b \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-\left (a^2 (3 A+C)\right )+a b B+2 A b^2\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a \left (a^2-b^2\right )}}{3 b \left (a^2-b^2\right )}-\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}\) |
Input:
Int[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Se c[c + d*x])^(5/2),x]
Output:
(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b ^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (-(((2*b*(a^2 - b^2)*(2*A*b^2 + a*b*B - a^2*(3*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b* (2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/( a + b)]*Sqrt[Sec[c + d*x]]))/(a*(a^2 - b^2))) + (2*(A*b^4 + 2*a^3*b*B + 2* a*b^3*B + a^4*C - 5*a^2*b^2*(A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*( a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]))/(3*b*(a^2 - b^2))
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[(2/(d*S qrt[a + b]))*EllipticF[(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[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[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_)]*(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[(-d)*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) ), x] + Simp[d/(b*(a^2 - b^2)*(m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)*( d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C }, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 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*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc [e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Sim p[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f *x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x ] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(2758\) vs. \(2(359)=718\).
Time = 28.18 (sec) , antiderivative size = 2759, normalized size of antiderivative = 7.30
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2759\) |
| parts | \(\text {Expression too large to display}\) | \(2938\) |
Input:
int(sec(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(5/2 ),x,method=_RETURNVERBOSE)
Output:
-2/3/d/(a-b)/(a+b)^2/((a-b)/(a+b))^(1/2)/a^2*(6*A*((a-b)/(a+b))^(1/2)*a^3* b*cos(d*x+c)*sin(d*x+c)+A*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c ))/(cos(d*x+c)+1))^(1/2)*a^3*b*EllipticE(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+ cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(6*cos(d*x+c)^3+12*cos(d*x+c)^2+6*cos(d* x+c))+(-2*cos(d*x+c)^2-4*cos(d*x+c)-2)*A*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b) *(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*b^4*EllipticE(((a-b)/(a+b))^(1/2)* (-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-cos(d*x+c)+5)*sin(d*x+c)* A*((a-b)/(a+b))^(1/2)*a^2*b^2+(-1-3*cos(d*x+c))*sin(d*x+c)*A*((a-b)/(a+b)) ^(1/2)*a*b^3+(cos(d*x+c)-2)*sin(d*x+c)*B*((a-b)/(a+b))^(1/2)*a^3*b+(3*cos( d*x+c)-1)*sin(d*x+c)*C*((a-b)/(a+b))^(1/2)*a^3*b+A*(1/(cos(d*x+c)+1))^(1/2 )*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4*EllipticF(((a-b)/(a+ b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(3*cos(d*x+c)^3+6 *cos(d*x+c)^2+3*cos(d*x+c))+B*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d *x+c))/(cos(d*x+c)+1))^(1/2)*a^4*EllipticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c )+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(3*cos(d*x+c)^3+6*cos(d*x+c)^2+3*cos(d *x+c))+C*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1) )^(1/2)*a^4*EllipticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b) /(a-b))^(1/2))*(cos(d*x+c)^3+2*cos(d*x+c)^2+cos(d*x+c))+B*(1/(cos(d*x+c)+1 ))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^4*EllipticE(((a -b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-3*cos...
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 1180, normalized size of antiderivative = 3.12 \[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c) )^(5/2),x, algorithm="fricas")
Output:
1/9*(sqrt(2)*(-3*I*(3*A + C)*a^4*b^2 + 6*I*B*a^3*b^3 + I*(9*A - C)*a^2*b^4 - 2*I*B*a*b^5 - 4*I*A*b^6 + (-3*I*(3*A + C)*a^6 + 6*I*B*a^5*b + I*(9*A - C)*a^4*b^2 - 2*I*B*a^3*b^3 - 4*I*A*a^2*b^4)*cos(d*x + c)^2 - 2*(3*I*(3*A + C)*a^5*b - 6*I*B*a^4*b^2 - I*(9*A - C)*a^3*b^3 + 2*I*B*a^2*b^4 + 4*I*A*a* b^5)*cos(d*x + c))*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8 /27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2* b)/a) + sqrt(2)*(3*I*(3*A + C)*a^4*b^2 - 6*I*B*a^3*b^3 - I*(9*A - C)*a^2*b ^4 + 2*I*B*a*b^5 + 4*I*A*b^6 + (3*I*(3*A + C)*a^6 - 6*I*B*a^5*b - I*(9*A - C)*a^4*b^2 + 2*I*B*a^3*b^3 + 4*I*A*a^2*b^4)*cos(d*x + c)^2 - 2*(-3*I*(3*A + C)*a^5*b + 6*I*B*a^4*b^2 + I*(9*A - C)*a^3*b^3 - 2*I*B*a^2*b^4 - 4*I*A* a*b^5)*cos(d*x + c))*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(2)*(3*I*B*a^4*b^2 - 2*I*(3*A + 2*C)*a^3*b^3 + I*B*a^2*b^4 + 2*I*A*a*b^5 + (3*I*B*a^6 - 2*I*(3*A + 2*C)*a^5*b + I*B*a^4*b^2 + 2*I*A* a^3*b^3)*cos(d*x + c)^2 + 2*(3*I*B*a^5*b - 2*I*(3*A + 2*C)*a^4*b^2 + I*B*a ^3*b^3 + 2*I*A*a^2*b^4)*cos(d*x + c))*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*si n(d*x + c) + 2*b)/a)) - 3*sqrt(2)*(-3*I*B*a^4*b^2 + 2*I*(3*A + 2*C)*a^3*b^ 3 - I*B*a^2*b^4 - 2*I*A*a*b^5 + (-3*I*B*a^6 + 2*I*(3*A + 2*C)*a^5*b - I...
\[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sqrt {\sec {\left (c + d x \right )}}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(sec(d*x+c)**(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+b*sec(d*x+ c))**(5/2),x)
Output:
Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*sqrt(sec(c + d*x))/(a + b*sec(c + d*x))**(5/2), x)
\[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {\sec \left (d x + c\right )}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c) )^(5/2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(sec(d*x + c))/(b*se c(d*x + c) + a)^(5/2), x)
\[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sqrt {\sec \left (d x + c\right )}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(sec(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c) )^(5/2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*sqrt(sec(d*x + c))/(b*se c(d*x + c) + a)^(5/2), x)
Timed out. \[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \] Input:
int(((1/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + b/cos(c + d*x))^(5/2),x)
Output:
int(((1/cos(c + d*x))^(1/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + b/cos(c + d*x))^(5/2), x)
\[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{3} b^{3}+3 \sec \left (d x +c \right )^{2} a \,b^{2}+3 \sec \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )}{\sec \left (d x +c \right )^{3} b^{3}+3 \sec \left (d x +c \right )^{2} a \,b^{2}+3 \sec \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{3} b^{3}+3 \sec \left (d x +c \right )^{2} a \,b^{2}+3 \sec \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) a \] Input:
int(sec(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^(5/2 ),x)
Output:
int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**2)/(sec(c + d*x)**3*b**3 + 3*sec(c + d*x)**2*a*b**2 + 3*sec(c + d*x)*a**2*b + a**3),x )*c + int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a)*sec(c + d*x))/(sec( c + d*x)**3*b**3 + 3*sec(c + d*x)**2*a*b**2 + 3*sec(c + d*x)*a**2*b + a**3 ),x)*b + int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a))/(sec(c + d*x)** 3*b**3 + 3*sec(c + d*x)**2*a*b**2 + 3*sec(c + d*x)*a**2*b + a**3),x)*a