Integrand size = 43, antiderivative size = 540 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a b d}+\frac {\sqrt {a+b} \left (3 A b^2+4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a d}+\frac {\sqrt {a+b} \left (A b^3-8 a^3 B-6 a b^2 B-12 a^2 b (A+2 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a^2 d}+\frac {\left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac {(A b+2 a B) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d} \]
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Time = 1.46 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4179, 4189, 4143, 4006, 3869, 3917, 4089} \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sqrt {a+b} \cot (c+d x) \left (4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)+3 A b^2\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{24 a d}+\frac {(a-b) \sqrt {a+b} \cot (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{24 a b d}+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{24 a d}+\frac {\sqrt {a+b} \cot (c+d x) \left (-8 a^3 B-12 a^2 b (A+2 C)-6 a b^2 B+A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{8 a^2 d}+\frac {(2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{4 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d} \]
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Rule 3869
Rule 3917
Rule 4006
Rule 4089
Rule 4143
Rule 4179
Rule 4189
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {3}{2} (A b+2 a B)+(2 a A+3 b B+3 a C) \sec (c+d x)+\frac {1}{2} b (A+6 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(A b+2 a B) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac {1}{6} \int \frac {\cos (c+d x) \left (\frac {1}{4} \left (3 A b^2+30 a b B+4 a^2 (4 A+6 C)\right )+\frac {1}{2} \left (6 a^2 B+12 b^2 B+a b (13 A+24 C)\right ) \sec (c+d x)+\frac {1}{4} b (7 A b+6 a B+24 b C) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac {(A b+2 a B) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}-\frac {\int \frac {\frac {3}{8} \left (A b^3-8 a^3 B-6 a b^2 B-12 a^2 b (A+2 C)\right )-\frac {1}{4} a b (7 A b+6 a B+24 b C) \sec (c+d x)+\frac {1}{8} b \left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{6 a} \\ & = \frac {\left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac {(A b+2 a B) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}-\frac {\int \frac {\frac {3}{8} \left (A b^3-8 a^3 B-6 a b^2 B-12 a^2 b (A+2 C)\right )+\left (-\frac {1}{4} a b (7 A b+6 a B+24 b C)-\frac {1}{8} b \left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{6 a}-\frac {\left (b \left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{48 a} \\ & = \frac {(a-b) \sqrt {a+b} \left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a b d}+\frac {\left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac {(A b+2 a B) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d}-\frac {\left (A b^3-8 a^3 B-6 a b^2 B-12 a^2 b (A+2 C)\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx}{16 a}+\frac {\left (b \left (3 A b^2+4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{48 a} \\ & = \frac {(a-b) \sqrt {a+b} \left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a b d}+\frac {\sqrt {a+b} \left (3 A b^2+4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a d}+\frac {\sqrt {a+b} \left (A b^3-8 a^3 B-6 a b^2 B-12 a^2 b (A+2 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a^2 d}+\frac {\left (3 A b^2+30 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac {(A b+2 a B) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(5040\) vs. \(2(540)=1080\).
Time = 28.35 (sec) , antiderivative size = 5040, normalized size of antiderivative = 9.33 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(5617\) vs. \(2(495)=990\).
Time = 3.90 (sec) , antiderivative size = 5618, normalized size of antiderivative = 10.40
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Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{3} \,d x } \]
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\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^3\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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