Integrand size = 43, antiderivative size = 507 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=-\frac {\left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a^4 \left (a^2-b^2\right ) d}+\frac {\left (21 A b^5+2 a^5 B+16 a^3 b^2 B-15 a b^4 B-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 a^5 \left (a^2-b^2\right ) d}-\frac {b^2 \left (7 A b^4+7 a^3 b B-5 a b^3 B-3 a^2 b^2 (3 A-C)-5 a^4 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^5 (a-b) (a+b)^2 d}-\frac {\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))} \]
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Time = 1.66 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {4185, 4189, 4191, 3934, 2884, 3872, 3856, 2719, 2720} \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac {\sin (c+d x) \left (-\left (a^2 (2 A-5 C)\right )-5 a b B+7 A b^2\right )}{5 a^2 d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x)}+\frac {\sin (c+d x) \left (2 a^3 B-a^2 (4 A b-3 b C)-5 a b^2 B+7 A b^3\right )}{3 a^3 d \left (a^2-b^2\right ) \sqrt {\sec (c+d x)}}-\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-2 a^4 (3 A+5 C)+20 a^3 b B-3 a^2 b^2 (8 A-5 C)-25 a b^3 B+35 A b^4\right )}{5 a^4 d \left (a^2-b^2\right )}-\frac {b^2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-5 a^4 C+7 a^3 b B-3 a^2 b^2 (3 A-C)-5 a b^3 B+7 A b^4\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a^5 d (a-b) (a+b)^2}+\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (2 a^5 B-4 a^4 b (A+3 C)+16 a^3 b^2 B-a^2 b^3 (20 A-9 C)-15 a b^4 B+21 A b^5\right )}{3 a^5 d \left (a^2-b^2\right )} \]
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Rule 2719
Rule 2720
Rule 2884
Rule 3856
Rule 3872
Rule 3934
Rule 4185
Rule 4189
Rule 4191
Rubi steps \begin{align*} \text {integral}& = \frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac {\int \frac {\frac {1}{2} \left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right )+a (A b-a B+b C) \sec (c+d x)-\frac {5}{2} \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{a \left (a^2-b^2\right )} \\ & = -\frac {\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}+\frac {2 \int \frac {\frac {5}{4} \left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right )+\frac {1}{2} a \left (2 A b^2-5 a b B+a^2 (3 A+5 C)\right ) \sec (c+d x)-\frac {3}{4} b \left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{5 a^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac {4 \int \frac {\frac {3}{8} \left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right )+\frac {1}{4} a \left (14 A b^3-5 a^3 B-10 a b^2 B+a^2 b (A+15 C)\right ) \sec (c+d x)-\frac {5}{8} b \left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{15 a^3 \left (a^2-b^2\right )} \\ & = -\frac {\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac {4 \int \frac {\frac {3}{8} a \left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right )-\left (\frac {3}{8} b \left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right )-\frac {1}{4} a^2 \left (14 A b^3-5 a^3 B-10 a b^2 B+a^2 b (A+15 C)\right )\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )}-\frac {\left (b^2 \left (7 A b^4+7 a^3 b B-5 a b^3 B-3 a^2 b^2 (3 A-C)-5 a^4 C\right )\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )} \\ & = -\frac {\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}+\frac {\left (21 A b^5+2 a^5 B+16 a^3 b^2 B-15 a b^4 B-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)\right ) \int \sqrt {\sec (c+d x)} \, dx}{6 a^5 \left (a^2-b^2\right )}-\frac {\left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{10 a^4 \left (a^2-b^2\right )}-\frac {\left (b^2 \left (7 A b^4+7 a^3 b B-5 a b^3 B-3 a^2 b^2 (3 A-C)-5 a^4 C\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{2 a^5 \left (a^2-b^2\right )} \\ & = -\frac {b^2 \left (7 A b^4+7 a^3 b B-5 a b^3 B-3 a^2 b^2 (3 A-C)-5 a^4 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^5 (a-b) (a+b)^2 d}-\frac {\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))}+\frac {\left (\left (21 A b^5+2 a^5 B+16 a^3 b^2 B-15 a b^4 B-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{6 a^5 \left (a^2-b^2\right )}-\frac {\left (\left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{10 a^4 \left (a^2-b^2\right )} \\ & = -\frac {\left (35 A b^4+20 a^3 b B-25 a b^3 B-3 a^2 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 a^4 \left (a^2-b^2\right ) d}+\frac {\left (21 A b^5+2 a^5 B+16 a^3 b^2 B-15 a b^4 B-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 a^5 \left (a^2-b^2\right ) d}-\frac {b^2 \left (7 A b^4+7 a^3 b B-5 a b^3 B-3 a^2 b^2 (3 A-C)-5 a^4 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^5 (a-b) (a+b)^2 d}-\frac {\left (7 A b^2-5 a b B-a^2 (2 A-5 C)\right ) \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (7 A b^3+2 a^3 B-5 a b^2 B-a^2 (4 A b-3 b C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}}+\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))} \\ \end{align*}
Time = 14.38 (sec) , antiderivative size = 971, normalized size of antiderivative = 1.92 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\frac {(b+a \cos (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {2 \left (-18 a^4 A-32 a^2 A b^2+35 A b^4+40 a^3 b B-25 a b^3 B-30 a^4 C+15 a^2 b^2 C\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (4 a^3 A b+56 a A b^3-20 a^4 B-40 a^2 b^2 B+60 a^3 b C\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (a+b \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (-18 a^4 A-72 a^2 A b^2+105 A b^4+60 a^3 b B-75 a b^3 B-30 a^4 C+45 a^2 b^2 C\right ) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-2 a (a-2 b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 a^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 b^2 \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}\right )}{30 a^3 (-a+b) (a+b) d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^2}+\frac {(b+a \cos (c+d x))^2 \sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {\left (a^4 A-a^2 A b^2+10 A b^4-10 a b^3 B+10 a^2 b^2 C\right ) \sin (c+d x)}{5 a^4 \left (a^2-b^2\right )}-\frac {2 \left (A b^5 \sin (c+d x)-a b^4 B \sin (c+d x)+a^2 b^3 C \sin (c+d x)\right )}{a^4 \left (a^2-b^2\right ) (b+a \cos (c+d x))}+\frac {2 (-2 A b+a B) \sin (2 (c+d x))}{3 a^3}+\frac {A \sin (3 (c+d x))}{5 a^2}\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1376\) vs. \(2(559)=1118\).
Time = 3.53 (sec) , antiderivative size = 1377, normalized size of antiderivative = 2.72
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\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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