Integrand size = 25, antiderivative size = 110 \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 b^2 (7 A+5 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 d}+\frac {2 b (7 A+5 C) (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d} \]
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Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {4131, 3853, 3856, 2720} \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 b^2 (7 A+5 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 d}+\frac {2 b (7 A+5 C) \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 d}+\frac {2 C \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d} \]
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Rule 2720
Rule 3853
Rule 3856
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{7} (7 A+5 C) \int (b \sec (c+d x))^{5/2} \, dx \\ & = \frac {2 b (7 A+5 C) (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{21} \left (b^2 (7 A+5 C)\right ) \int \sqrt {b \sec (c+d x)} \, dx \\ & = \frac {2 b (7 A+5 C) (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{21} \left (b^2 (7 A+5 C) \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 b^2 (7 A+5 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {b \sec (c+d x)}}{21 d}+\frac {2 b (7 A+5 C) (b \sec (c+d x))^{3/2} \sin (c+d x)}{21 d}+\frac {2 C (b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d} \\ \end{align*}
Time = 1.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76 \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(b \sec (c+d x))^{7/2} \left (4 (7 A+5 C) \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 (7 A+11 C+(7 A+5 C) \cos (2 (c+d x))) \sin (c+d x)\right )}{42 b d} \]
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Result contains complex when optimal does not.
Time = 22.69 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.65
method | result | size |
default | \(\frac {2 b^{2} \sqrt {b \sec \left (d x +c \right )}\, \left (7 i A \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )+5 i C \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )+7 i A \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right )+5 i C \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right )+7 A \tan \left (d x +c \right )+5 C \tan \left (d x +c \right )+3 C \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}\right )}{21 d}\) | \(291\) |
parts | \(\frac {2 A \sqrt {b \sec \left (d x +c \right )}\, b^{2} \left (i \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )+i \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\tan \left (d x +c \right )\right )}{3 d}+\frac {2 i C \sqrt {b \sec \left (d x +c \right )}\, b^{2} \left (5 \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )+5 \operatorname {EllipticF}\left (i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-5 i \tan \left (d x +c \right )-3 i \tan \left (d x +c \right ) \sec \left (d x +c \right )^{2}\right )}{21 d}\) | \(305\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.29 \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {-i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} b^{\frac {5}{2}} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} b^{\frac {5}{2}} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left ({\left (7 \, A + 5 \, C\right )} b^{2} \cos \left (d x + c\right )^{2} + 3 \, C b^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{21 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \left (A + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \]
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\[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
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\[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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