Integrand size = 33, antiderivative size = 454 \[ \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (a-b) \sqrt {a+b} \left (10 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (161 A+93 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (10 a^3 C+21 b^3 (9 A+7 C)+15 a^2 b (21 A+11 C)-6 a b^2 (28 A+19 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^2 d}+\frac {4 a \left (84 A b^2-5 a^2 C+57 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac {2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d} \]
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Time = 1.16 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4168, 4087, 4090, 3917, 4089} \[ \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}+\frac {4 a \left (-5 a^2 C+84 A b^2+57 b^2 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{315 b d}+\frac {2 (a-b) \sqrt {a+b} \left (10 a^4 C-3 a^2 b^2 (161 A+93 C)-21 b^4 (9 A+7 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{315 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (10 a^3 C+15 a^2 b (21 A+11 C)-6 a b^2 (28 A+19 C)+21 b^3 (9 A+7 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{315 b^2 d}+\frac {2 C \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}-\frac {4 a C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d} \]
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Rule 3917
Rule 4087
Rule 4089
Rule 4090
Rule 4168
Rubi steps \begin{align*} \text {integral}& = \frac {2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {2 \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac {1}{2} b (9 A+7 C)-a C \sec (c+d x)\right ) \, dx}{9 b} \\ & = -\frac {4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {4 \int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {3}{4} a b (21 A+13 C)-\frac {1}{4} \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) \sec (c+d x)\right ) \, dx}{63 b} \\ & = -\frac {2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {8 \int \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {3}{8} b \left (7 b^2 (9 A+7 C)+5 a^2 (21 A+11 C)\right )+\frac {3}{4} a \left (84 A b^2-5 a^2 C+57 b^2 C\right ) \sec (c+d x)\right ) \, dx}{315 b} \\ & = \frac {4 a \left (84 A b^2-5 a^2 C+57 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac {2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {16 \int \frac {\sec (c+d x) \left (\frac {3}{16} a b \left (5 a^2 (63 A+31 C)+3 b^2 (119 A+87 C)\right )-\frac {3}{16} \left (10 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (161 A+93 C)\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{945 b} \\ & = \frac {4 a \left (84 A b^2-5 a^2 C+57 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac {2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d}+\frac {\left ((a-b) \left (10 a^3 C+21 b^3 (9 A+7 C)+15 a^2 b (21 A+11 C)-6 a b^2 (28 A+19 C)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 b}-\frac {\left (10 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (161 A+93 C)\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{315 b} \\ & = \frac {2 (a-b) \sqrt {a+b} \left (10 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (161 A+93 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}}}{315 b^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (10 a^3 C+21 b^3 (9 A+7 C)+15 a^2 b (21 A+11 C)-6 a b^2 (28 A+19 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}}}{315 b^2 d}+\frac {4 a \left (84 A b^2-5 a^2 C+57 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b d}-\frac {2 \left (10 a^2 C-7 b^2 (9 A+7 C)\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{315 b d}-\frac {4 a C (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{63 b d}+\frac {2 C (a+b \sec (c+d x))^{7/2} \tan (c+d x)}{9 b d} \\ \end{align*}
Time = 24.08 (sec) , antiderivative size = 710, normalized size of antiderivative = 1.56 \[ \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 \sqrt {2} \sqrt {\frac {\cos (c+d x)}{(1+\cos (c+d x))^2}} \sqrt {\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )} \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \left ((a+b) \left (\left (10 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (161 A+93 C)\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )+b \left (-10 a^3 C+21 b^3 (9 A+7 C)+15 a^2 b (21 A+11 C)+6 a b^2 (28 A+19 C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}} \sec (c+d x)+\left (10 a^4 C-21 b^4 (9 A+7 C)-3 a^2 b^2 (161 A+93 C)\right ) \cos (c+d x) (b+a \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 b^2 d \sqrt {\frac {1}{1+\cos (c+d x)}} (b+a \cos (c+d x))^3 (A+2 C+A \cos (2 c+2 d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} \sec ^{\frac {9}{2}}(c+d x)}+\frac {\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {4 \left (483 a^2 A b^2+189 A b^4-10 a^4 C+279 a^2 b^2 C+147 b^4 C\right ) \sin (c+d x)}{315 b^2}+\frac {4}{315} \sec ^2(c+d x) \left (63 A b^2 \sin (c+d x)+75 a^2 C \sin (c+d x)+49 b^2 C \sin (c+d x)\right )+\frac {4 \sec (c+d x) \left (231 a A b^2 \sin (c+d x)+5 a^3 C \sin (c+d x)+163 a b^2 C \sin (c+d x)\right )}{315 b}+\frac {76}{63} a b C \sec ^2(c+d x) \tan (c+d x)+\frac {4}{9} b^2 C \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2 (A+2 C+A \cos (2 c+2 d x))} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(5477\) vs. \(2(416)=832\).
Time = 60.90 (sec) , antiderivative size = 5478, normalized size of antiderivative = 12.07
method | result | size |
parts | \(\text {Expression too large to display}\) | \(5478\) |
default | \(\text {Expression too large to display}\) | \(5558\) |
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \sec {\left (c + d x \right )}\, dx \]
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Timed out. \[ \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
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Timed out. \[ \int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{\cos \left (c+d\,x\right )} \,d x \]
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