Integrand size = 43, antiderivative size = 652 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a b d}+\frac {\sqrt {a+b} \left (15 A b^3+8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{192 a d}+\frac {\sqrt {a+b} \left (5 A b^4-160 a^3 b B-40 a b^3 B-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{64 a^2 d}+\frac {\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d} \]
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Time = 2.22 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4179, 4189, 4143, 4006, 3869, 3917, 4089} \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sin (c+d x) \cos (c+d x) \left (4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{32 d}+\frac {\sqrt {a+b} \cot (c+d x) \left (8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)+15 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{192 a d}+\frac {(a-b) \sqrt {a+b} \cot (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{192 a b d}+\frac {\sin (c+d x) \left (128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{192 a d}+\frac {\sqrt {a+b} \cot (c+d x) \left (-16 a^4 (3 A+4 C)-160 a^3 b B-120 a^2 b^2 (A+2 C)-40 a b^3 B+5 A b^4\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{64 a^2 d}+\frac {(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{24 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d} \]
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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 ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {1}{2} (5 A b+8 a B)+(3 a A+4 b B+4 a C) \sec (c+d x)+\frac {1}{2} b (A+8 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {3}{4} \left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right )+\frac {1}{2} \left (16 a^2 B+24 b^2 B+a b (31 A+48 C)\right ) \sec (c+d x)+\frac {1}{4} b (11 A b+8 a B+48 b C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {1}{24} \int \frac {\cos (c+d x) \left (\frac {1}{8} \left (15 A b^3+128 a^3 B+264 a b^2 B+8 a^2 \left (\frac {71 A b}{2}+54 b C\right )\right )+\frac {1}{4} \left (152 a^2 b B+96 b^3 B+12 a^3 (3 A+4 C)+a b^2 (161 A+288 C)\right ) \sec (c+d x)+\frac {1}{8} b \left (104 a b B+12 a^2 (3 A+4 C)+b^2 (59 A+192 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}-\frac {\int \frac {\frac {3}{16} \left (5 A b^4-160 a^3 b B-40 a b^3 B-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right )-\frac {1}{8} a b \left (104 a b B+12 a^2 (3 A+4 C)+b^2 (59 A+192 C)\right ) \sec (c+d x)+\frac {1}{16} b \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a} \\ & = \frac {\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}-\frac {\int \frac {\frac {3}{16} \left (5 A b^4-160 a^3 b B-40 a b^3 B-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right )+\left (-\frac {1}{16} b \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right )-\frac {1}{8} a b \left (104 a b B+12 a^2 (3 A+4 C)+b^2 (59 A+192 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a}-\frac {\left (b \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{384 a} \\ & = \frac {(a-b) \sqrt {a+b} \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 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}}}{192 a b d}+\frac {\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}-\frac {\left (5 A b^4-160 a^3 b B-40 a b^3 B-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx}{128 a}+\frac {\left (b \left (15 A b^3+8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{384 a} \\ & = \frac {(a-b) \sqrt {a+b} \left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 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}}}{192 a b d}+\frac {\sqrt {a+b} \left (15 A b^3+8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 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}}}{192 a d}+\frac {\sqrt {a+b} \left (5 A b^4-160 a^3 b B-40 a b^3 B-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 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}}}{64 a^2 d}+\frac {\left (15 A b^3+128 a^3 B+264 a b^2 B+4 a^2 b (71 A+108 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 a d}+\frac {\left (5 A b^2+24 a b B+4 a^2 (3 A+4 C)\right ) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {(5 A b+8 a B) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {A \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(5667\) vs. \(2(652)=1304\).
Time = 37.57 (sec) , antiderivative size = 5667, normalized size of antiderivative = 8.69 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(8027\) vs. \(2(603)=1206\).
Time = 519.44 (sec) , antiderivative size = 8028, normalized size of antiderivative = 12.31
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\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
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\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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