Integrand size = 41, antiderivative size = 266 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 (9 A b+9 a B+7 b C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 (7 a A+5 b B+5 a C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 (9 A b+9 a B+7 b C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (7 a A+5 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 (9 A b+9 a B+7 b C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 (b B+a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 b C \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \]
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Time = 0.36 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4161, 4132, 3853, 3856, 2719, 4131, 2720} \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (9 a B+9 A b+7 b C)}{45 d}+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (7 a A+5 a C+5 b B)}{21 d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} (9 a B+9 A b+7 b C)}{15 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (7 a A+5 a C+5 b B)}{21 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) (9 a B+9 A b+7 b C)}{15 d}+\frac {2 (a C+b B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 4131
Rule 4132
Rule 4161
Rubi steps \begin{align*} \text {integral}& = \frac {2 b C \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2}{9} \int \sec ^{\frac {5}{2}}(c+d x) \left (\frac {9 a A}{2}+\frac {1}{2} (9 A b+9 a B+7 b C) \sec (c+d x)+\frac {9}{2} (b B+a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 b C \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2}{9} \int \sec ^{\frac {5}{2}}(c+d x) \left (\frac {9 a A}{2}+\frac {9}{2} (b B+a C) \sec ^2(c+d x)\right ) \, dx+\frac {1}{9} (9 A b+9 a B+7 b C) \int \sec ^{\frac {7}{2}}(c+d x) \, dx \\ & = \frac {2 (9 A b+9 a B+7 b C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 (b B+a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 b C \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{7} (7 a A+5 b B+5 a C) \int \sec ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{15} (9 A b+9 a B+7 b C) \int \sec ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 (9 A b+9 a B+7 b C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (7 a A+5 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 (9 A b+9 a B+7 b C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 (b B+a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 b C \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{21} (7 a A+5 b B+5 a C) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} (-9 A b-9 a B-7 b C) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {2 (9 A b+9 a B+7 b C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (7 a A+5 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 (9 A b+9 a B+7 b C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 (b B+a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 b C \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {1}{21} \left ((7 a A+5 b B+5 a C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left ((-9 A b-9 a B-7 b C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {2 (9 A b+9 a B+7 b C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 (7 a A+5 b B+5 a C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 (9 A b+9 a B+7 b C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (7 a A+5 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 (9 A b+9 a B+7 b C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 (b B+a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 b C \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.57 (sec) , antiderivative size = 1262, normalized size of antiderivative = 4.74 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sqrt {2} A b e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^3(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{5 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {2 \sqrt {2} a B e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^3(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{5 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {14 \sqrt {2} b C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^3(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{45 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {4 a A \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{3 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {20 b B \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{21 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {20 a C \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{21 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4 (9 A b+9 a B+7 b C) \cos (d x) \csc (c)}{15 d}+\frac {4 b C \sec (c) \sec ^4(c+d x) \sin (d x)}{9 d}+\frac {4 \sec (c) \sec ^3(c+d x) (7 b C \sin (c)+9 b B \sin (d x)+9 a C \sin (d x))}{63 d}+\frac {4 \sec (c) \sec (c+d x) (63 A b \sin (c)+63 a B \sin (c)+49 b C \sin (c)+105 a A \sin (d x)+75 b B \sin (d x)+75 a C \sin (d x))}{315 d}+\frac {4 \sec (c) \sec ^2(c+d x) (45 b B \sin (c)+45 a C \sin (c)+63 A b \sin (d x)+63 a B \sin (d x)+49 b C \sin (d x))}{315 d}+\frac {4 (7 a A+5 b B+5 a C) \tan (c)}{21 d}\right )}{(b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {5}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1019\) vs. \(2(286)=572\).
Time = 52.80 (sec) , antiderivative size = 1020, normalized size of antiderivative = 3.83
method | result | size |
default | \(\text {Expression too large to display}\) | \(1020\) |
parts | \(\text {Expression too large to display}\) | \(1213\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.17 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {15 \, \sqrt {2} {\left (i \, {\left (7 \, A + 5 \, C\right )} a + 5 i \, B b\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 5 \, C\right )} a - 5 i \, B b\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (9 i \, B a + i \, {\left (9 \, A + 7 \, C\right )} b\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-9 i \, B a - i \, {\left (9 \, A + 7 \, C\right )} b\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (21 \, {\left (9 \, B a + {\left (9 \, A + 7 \, C\right )} b\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left ({\left (7 \, A + 5 \, C\right )} a + 5 \, B b\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (9 \, B a + {\left (9 \, A + 7 \, C\right )} b\right )} \cos \left (d x + c\right )^{2} + 35 \, C b + 45 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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\[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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