Integrand size = 35, antiderivative size = 286 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 a^3 (27 A+17 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (21 A+11 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (27 A+17 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a^3 (21 A+16 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {4 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+73 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d} \]
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Time = 0.87 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4174, 4103, 4082, 3872, 3856, 2720, 3853, 2719} \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^3 (21 A+16 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {2 (63 A+73 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{315 d}+\frac {4 a^3 (27 A+17 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (21 A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {4 a^3 (27 A+17 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{21 a d}+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d} \]
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3872
Rule 4082
Rule 4103
Rule 4174
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (\frac {1}{2} a (9 A+C)+3 a C \sec (c+d x)\right ) \, dx}{9 a} \\ & = \frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {4 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {4 \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (\frac {1}{4} a^2 (63 A+13 C)+\frac {1}{4} a^2 (63 A+73 C) \sec (c+d x)\right ) \, dx}{63 a} \\ & = \frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {4 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+73 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {8 \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x)) \left (\frac {3}{4} a^3 (63 A+23 C)+\frac {9}{2} a^3 (21 A+16 C) \sec (c+d x)\right ) \, dx}{315 a} \\ & = \frac {8 a^3 (21 A+16 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {4 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+73 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {16 \int \sqrt {\sec (c+d x)} \left (\frac {45}{8} a^4 (21 A+11 C)+\frac {63}{8} a^4 (27 A+17 C) \sec (c+d x)\right ) \, dx}{945 a} \\ & = \frac {8 a^3 (21 A+16 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {4 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+73 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {1}{21} \left (2 a^3 (21 A+11 C)\right ) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{15} \left (2 a^3 (27 A+17 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {4 a^3 (27 A+17 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a^3 (21 A+16 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {4 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+73 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}-\frac {1}{15} \left (2 a^3 (27 A+17 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^3 (21 A+11 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {4 a^3 (21 A+11 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 {4 a^3 (27 A+17 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a^3 (21 A+16 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {4 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+73 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d}-\frac {1}{15} \left (2 a^3 (27 A+17 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx \\ & = -\frac {4 a^3 (27 A+17 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 {4 a^3 (21 A+11 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 {4 a^3 (27 A+17 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {8 a^3 (21 A+16 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{9 d}+\frac {4 C \sec ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+73 C) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{315 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.82 (sec) , antiderivative size = 818, normalized size of antiderivative = 2.86 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 A 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 ^5(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 ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{5 \sqrt {2} d (A+2 C+A \cos (2 c+2 d x))}+\frac {17 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 ^5(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 ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{45 \sqrt {2} d (A+2 C+A \cos (2 c+2 d x))}+\frac {A \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{d (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {11 C \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right )}{21 d (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)}+\frac {\sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \left (\frac {(27 A+17 C) \cos (d x) \csc (c)}{15 d}+\frac {C \sec (c) \sec ^4(c+d x) \sin (d x)}{18 d}+\frac {\sec (c) \sec ^3(c+d x) (7 C \sin (c)+27 C \sin (d x))}{126 d}+\frac {\sec (c) \sec ^2(c+d x) (135 C \sin (c)+63 A \sin (d x)+238 C \sin (d x))}{630 d}+\frac {\sec (c) \sec (c+d x) (63 A \sin (c)+238 C \sin (c)+315 A \sin (d x)+330 C \sin (d x))}{630 d}+\frac {(21 A+22 C) \tan (c)}{42 d}\right )}{(A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1219\) vs. \(2(306)=612\).
Time = 5.23 (sec) , antiderivative size = 1220, normalized size of antiderivative = 4.27
method | result | size |
default | \(\text {Expression too large to display}\) | \(1220\) |
parts | \(\text {Expression too large to display}\) | \(1548\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.98 \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (21 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (21 \, A + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (27 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (27 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (42 \, {\left (27 \, A + 17 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (21 \, A + 22 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (9 \, A + 34 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 135 \, C a^{3} \cos \left (d x + c\right ) + 35 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )} \,d x } \]
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\[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )} \,d x } \]
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Timed out. \[ \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}} \,d x \]
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