Integrand size = 45, antiderivative size = 550 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{192 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{64 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{192 b d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Time = 2.71 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.311, Rules used = {4350, 4181, 4187, 4193, 3944, 2886, 2884, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {\sin (c+d x) \left (5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\sin (c+d x) \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)}}{192 b d \sqrt {\cos (c+d x)}}+\frac {\left (a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{192 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\sqrt {\cos (c+d x)} \left (15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{192 b d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {\left (-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{64 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {(5 a C+8 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{4 d \cos ^{\frac {3}{2}}(c+d x)} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 3941
Rule 3943
Rule 3944
Rule 4120
Rule 4181
Rule 4187
Rule 4193
Rule 4350
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (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 {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{4} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left (\frac {1}{2} a (8 A+C)+(4 A b+4 a B+3 b C) \sec (c+d x)+\frac {1}{2} (8 b B+5 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{12} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)} \left (\frac {1}{4} a (48 a A+8 b B+11 a C)+\frac {1}{2} \left (24 a^2 B+16 b^2 B+a b (48 A+31 C)\right ) \sec (c+d x)+\frac {3}{4} \left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {1}{24} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (\frac {1}{8} a \left (192 a^2 A+48 A b^2+104 a b B+59 a^2 C+36 b^2 C\right )+\frac {1}{4} \left (96 a^3 B+152 a b^2 B+12 b^3 (4 A+3 C)+a^2 b (288 A+161 C)\right ) \sec (c+d x)+\frac {1}{8} \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{16} a \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right )+\frac {1}{8} a b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (192 A+59 C)\right ) \sec (c+d x)+\frac {3}{16} \left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{24 b} \\ & = \frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{16} a \left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right )+\frac {1}{8} a b \left (104 a b B+12 b^2 (4 A+3 C)+a^2 (192 A+59 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{24 b}+\frac {\left (\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{128 b} \\ & = \frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\left (-264 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (108 A+71 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{384 b}+\frac {1}{384} \left (\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {\left (\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}} \, dx}{128 b \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \\ & = \frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{384 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{128 b \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-264 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (108 A+71 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{384 b \sqrt {b+a \cos (c+d x)}} \\ & = \frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{64 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{384 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-264 a^2 b B-128 b^3 B-15 a^3 C-4 a b^2 (108 A+71 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{384 b \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \\ & = \frac {\left (472 a^2 b B+128 b^3 B+4 a b^2 (132 A+89 C)+a^3 (384 A+133 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 )}{192 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (40 a^3 b B+160 a b^3 B-5 a^4 C+120 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{64 b d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{192 b d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {\left (16 A b^2+24 a b B+5 a^2 C+12 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{32 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (264 a^2 b B+128 b^3 B+15 a^3 C+4 a b^2 (108 A+71 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{192 b d \sqrt {\cos (c+d x)}}+\frac {(8 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{24 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{4 d \cos ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 43.38 (sec) , antiderivative size = 259225, normalized size of antiderivative = 471.32 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Result too large to show} \]
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Result contains complex when optimal does not.
Time = 21.06 (sec) , antiderivative size = 6597, normalized size of antiderivative = 11.99
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{\sqrt {\cos \left (c+d\,x\right )}} \,d x \]
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