Integrand size = 43, antiderivative size = 469 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^2 b \left (a^2-b^2\right )^2 d}-\frac {\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^2 (a-b)^2 b^2 (a+b)^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]
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Time = 1.29 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4183, 4191, 3934, 2884, 3872, 3856, 2719, 2720} \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=-\frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{4 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{4 a^2 b d \left (a^2-b^2\right )^2}-\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{4 a b^2 d \left (a^2-b^2\right )^2}-\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a^2 b^2 d (a-b)^2 (a+b)^3} \]
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Rule 2719
Rule 2720
Rule 2884
Rule 3856
Rule 3872
Rule 3934
Rule 4183
Rule 4191
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\frac {1}{2} \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \sec (c+d x)+\frac {1}{2} \left (A b^2-a b B-3 a^2 C+4 b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )} \\ & = -\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\frac {1}{4} \left (-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-a^2 b^2 (5 A+9 C)\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \sec (c+d x)+\frac {1}{4} \left (a^3 b B-7 a b^3 B+a^2 b^2 (3 A-5 C)+3 a^4 C+b^4 (3 A+8 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{2 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\int \frac {\frac {1}{4} a \left (-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-a^2 b^2 (5 A+9 C)\right )-\left (-a b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right )+\frac {1}{4} b \left (-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-a^2 b^2 (5 A+9 C)\right )\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{2 a^2 b^2 \left (a^2-b^2\right )^2}-\frac {\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 C)\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{8 a^2 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right ) \int \sqrt {\sec (c+d x)} \, dx}{8 a^2 b \left (a^2-b^2\right )^2}-\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{8 a b^2 \left (a^2-b^2\right )^2}-\frac {\left (\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 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}{8 a^2 b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 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)}}{4 a^2 (a-b)^2 b^2 (a+b)^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{8 a^2 b \left (a^2-b^2\right )^2}-\frac {\left (\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 a b^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^2 b \left (a^2-b^2\right )^2 d}-\frac {\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 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)}}{4 a^2 (a-b)^2 b^2 (a+b)^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1046\) vs. \(2(469)=938\).
Time = 14.39 (sec) , antiderivative size = 1046, normalized size of antiderivative = 2.23 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x))^3 \sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {2 \left (a^2 A b^2+5 A b^4+3 a^3 b B-9 a b^3 B+9 a^4 C-19 a^2 b^2 C+16 b^4 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 (-24 a A b^3+8 a^2 b^2 B+16 b^4 B+8 a^3 b C-32 a b^3 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 (-5 a^2 A b^2-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-9 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 )}{8 (a-b)^2 b^2 (a+b)^2 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3}+\frac {(b+a \cos (c+d x))^3 \sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {\left (5 a^2 A b^2+A b^4-a^3 b B-5 a b^3 B-3 a^4 C+9 a^2 b^2 C\right ) \sin (c+d x)}{2 a b^2 \left (-a^2+b^2\right )^2}+\frac {A b^2 \sin (c+d x)-a b B \sin (c+d x)+a^2 C \sin (c+d x)}{a \left (a^2-b^2\right ) (b+a \cos (c+d x))^2}+\frac {-7 a^2 A b^2 \sin (c+d x)+A b^4 \sin (c+d x)+3 a^3 b B \sin (c+d x)+3 a b^3 B \sin (c+d x)+a^4 C \sin (c+d x)-7 a^2 b^2 C \sin (c+d x)}{2 a b \left (-a^2+b^2\right )^2 (b+a \cos (c+d x))}\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1878\) vs. \(2(521)=1042\).
Time = 4.99 (sec) , antiderivative size = 1879, normalized size of antiderivative = 4.01
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Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
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