Integrand size = 43, antiderivative size = 289 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 b B+4 a C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d} \]
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Time = 0.59 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4181, 4161, 4132, 3853, 3856, 2719, 4131, 2720} \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right )}{21 d}+\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right )}{5 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right )}{21 d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^2 B+10 a A b+6 a b C+3 b^2 B\right )}{5 d}+\frac {2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 4131
Rule 4132
Rule 4161
Rule 4181
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2}{7} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (\frac {1}{2} a (7 A+C)+\frac {1}{2} (7 A b+7 a B+5 b C) \sec (c+d x)+\frac {1}{2} (7 b B+4 a C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 b (7 b B+4 a C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} a^2 (7 A+C)+\frac {7}{4} \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sec (c+d x)+\frac {5}{4} \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 b (7 b B+4 a C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {4}{35} \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} a^2 (7 A+C)+\frac {5}{4} \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{5} \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 b B+4 a C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{5} \left (-10 a A b-5 a^2 B-3 b^2 B-6 a b C\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \int \sqrt {\sec (c+d x)} \, dx \\ & = \frac {2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 b B+4 a C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{5} \left (\left (-10 a A b-5 a^2 B-3 b^2 B-6 a b C\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (\left (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = -\frac {2 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \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 \left (10 a A b+5 a^2 B+3 b^2 B+6 a b C\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 \left (7 A b^2+14 a b B+4 a^2 C+5 b^2 C\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 b (7 b B+4 a C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{7 d} \\ \end{align*}
Time = 4.77 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.15 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {4 (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-21 \left (5 a^2 B+3 b^2 B+2 a b (5 A+3 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 \left (14 a b B+7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+210 a A b \sin (c+d x)+105 a^2 B \sin (c+d x)+63 b^2 B \sin (c+d x)+126 a b C \sin (c+d x)+35 A b^2 \tan (c+d x)+70 a b B \tan (c+d x)+35 a^2 C \tan (c+d x)+25 b^2 C \tan (c+d x)+21 b^2 B \sec (c+d x) \tan (c+d x)+42 a b C \sec (c+d x) \tan (c+d x)+15 b^2 C \sec ^2(c+d x) \tan (c+d x)\right )}{105 d (b+a \cos (c+d x))^2 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {7}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(919\) vs. \(2(313)=626\).
Time = 6.86 (sec) , antiderivative size = 920, normalized size of antiderivative = 3.18
method | result | size |
default | \(\text {Expression too large to display}\) | \(920\) |
parts | \(\text {Expression too large to display}\) | \(1172\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.24 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, \sqrt {2} {\left (7 i \, {\left (3 \, A + C\right )} a^{2} + 14 i \, B a b + i \, {\left (7 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, {\left (3 \, A + C\right )} a^{2} - 14 i \, B a b - i \, {\left (7 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, B a^{2} + 2 i \, {\left (5 \, A + 3 \, C\right )} a b + 3 i \, B b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-5 i \, B a^{2} - 2 i \, {\left (5 \, A + 3 \, C\right )} a b - 3 i \, B b^{2}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (21 \, {\left (5 \, B a^{2} + 2 \, {\left (5 \, A + 3 \, C\right )} a b + 3 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, C b^{2} + 5 \, {\left (7 \, C a^{2} + 14 \, B a b + {\left (7 \, A + 5 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} + 21 \, {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {\sec \left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {\sec \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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