Optimal. Leaf size=319 \[ -\frac {4 a^3 (7 A+5 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (143 A+105 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (7 A+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (143 A+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d} \]
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Rubi [A]
time = 0.41, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4174, 4103,
4082, 3872, 3853, 3856, 2719, 2720} \begin {gather*} \frac {8 a^3 (44 A+35 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{385 d}+\frac {4 a^3 (143 A+105 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{231 d}+\frac {2 (33 A+35 C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{231 d}+\frac {4 a^3 (7 A+5 C) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (143 A+105 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}-\frac {4 a^3 (7 A+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {4 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{33 a d}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^3}{11 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 3872
Rule 4082
Rule 4103
Rule 4174
Rubi steps
\begin {align*} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (\frac {1}{2} a (11 A+3 C)+3 a C \sec (c+d x)\right ) \, dx}{11 a}\\ &=\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {4 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (\frac {9}{4} a^2 (11 A+5 C)+\frac {3}{4} a^2 (33 A+35 C) \sec (c+d x)\right ) \, dx}{99 a}\\ &=\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}+\frac {8 \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) \left (\frac {45}{4} a^3 (11 A+7 C)+\frac {9}{2} a^3 (44 A+35 C) \sec (c+d x)\right ) \, dx}{693 a}\\ &=\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}+\frac {16 \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {693}{8} a^4 (7 A+5 C)+\frac {45}{8} a^4 (143 A+105 C) \sec (c+d x)\right ) \, dx}{3465 a}\\ &=\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}+\frac {1}{5} \left (2 a^3 (7 A+5 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{77} \left (2 a^3 (143 A+105 C)\right ) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {4 a^3 (7 A+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (143 A+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}-\frac {1}{5} \left (2 a^3 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{231} \left (2 a^3 (143 A+105 C)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {4 a^3 (7 A+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (143 A+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}-\frac {1}{5} \left (2 a^3 (7 A+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (2 a^3 (143 A+105 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 (7 A+5 C) \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 {4 a^3 (143 A+105 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a^3 (7 A+5 C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (143 A+105 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{231 d}+\frac {8 a^3 (44 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{385 d}+\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{11 d}+\frac {4 C \sec ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{33 a d}+\frac {2 (33 A+35 C) \sec ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{231 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.14, size = 863, normalized size = 2.71 \begin {gather*} \frac {7 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 ) \, _2F_1\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 )}{15 \sqrt {2} d (A+2 C+A \cos (2 c+2 d x))}+\frac {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 ) \, _2F_1\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 )}{3 \sqrt {2} d (A+2 C+A \cos (2 c+2 d x))}+\frac {13 A \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |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 {5 C \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |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 )}{11 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 {(7 A+5 C) \cos (d x) \csc (c)}{5 d}+\frac {C \sec (c) \sec ^5(c+d x) \sin (d x)}{22 d}+\frac {\sec (c) \sec ^4(c+d x) (3 C \sin (c)+11 C \sin (d x))}{66 d}+\frac {\sec (c) \sec ^3(c+d x) (77 C \sin (c)+33 A \sin (d x)+126 C \sin (d x))}{462 d}+\frac {\sec (c) \sec ^2(c+d x) (165 A \sin (c)+630 C \sin (c)+693 A \sin (d x)+770 C \sin (d x))}{2310 d}+\frac {\sec (c) \sec (c+d x) (693 A \sin (c)+770 C \sin (c)+1430 A \sin (d x)+1050 C \sin (d x))}{2310 d}+\frac {(143 A+105 C) \tan (c)}{231 d}\right )}{(A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1381\) vs.
\(2(335)=670\).
time = 16.21, size = 1382, normalized size = 4.33
method | result | size |
default | \(\text {Expression too large to display}\) | \(1382\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.18, size = 299, normalized size = 0.94 \begin {gather*} -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\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 ) - 231 i \, \sqrt {2} {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\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 (462 \, {\left (7 \, A + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 10 \, {\left (143 \, A + 105 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (9 \, A + 10 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 15 \, {\left (11 \, A + 42 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, C a^{3} \cos \left (d x + c\right ) + 105 \, C a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{1155 \, d \cos \left (d x + c\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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