Optimal. Leaf size=244 \[ -\frac {4 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d} \]
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Rubi [A]
time = 0.32, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3039, 4103,
4082, 3872, 3856, 2720, 3853, 2719} \begin {gather*} \frac {4 a^3 (41 A+42 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{105 d}+\frac {2 (11 A+7 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{35 d}+\frac {4 a^3 (7 A+9 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {4 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {4 a^3 (7 A+9 B) \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 {2 a A \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3039
Rule 3853
Rule 3856
Rule 3872
Rule 4082
Rule 4103
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac {9}{2}}(c+d x) \, dx &=\int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^3 (B+A \sec (c+d x)) \, dx\\ &=\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2}{7} \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^2 \left (\frac {1}{2} a (A+7 B)+\frac {1}{2} a (11 A+7 B) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {4}{35} \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x)) \left (\frac {1}{2} a^2 (8 A+21 B)+\frac {1}{2} a^2 (41 A+42 B) \sec (c+d x)\right ) \, dx\\ &=\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {8}{105} \int \sqrt {\sec (c+d x)} \left (\frac {5}{4} a^3 (13 A+21 B)+\frac {21}{4} a^3 (7 A+9 B) \sec (c+d x)\right ) \, dx\\ &=\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}+\frac {1}{5} \left (2 a^3 (7 A+9 B)\right ) \int \sec ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{21} \left (2 a^3 (13 A+21 B)\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {4 a^3 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^3 (7 A+9 B)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (2 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {4 a^3 (13 A+21 B) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 d}-\frac {1}{5} \left (2 a^3 (7 A+9 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {4 a^3 (7 A+9 B) \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 (13 A+21 B) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (7 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {4 a^3 (41 A+42 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 a A \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{7 d}+\frac {2 (11 A+7 B) \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{35 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 = 4.49, size = 435, normalized size = 1.78 \begin {gather*} \frac {a^3 e^{-i d x} (1+\cos (c+d x))^3 \csc (c) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (7 \sqrt {2} (7 A+9 B) e^{2 i d x} \left (-1+e^{2 i c}\right ) \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-\frac {e^{-i (c-d x)} \left (-1+e^{2 i c}\right ) \left (21 B \left (-5+16 e^{i (c+d x)}-5 e^{2 i (c+d x)}+54 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+56 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+18 e^{7 i (c+d x)}\right )+2 A \left (-65+84 e^{i (c+d x)}-95 e^{2 i (c+d x)}+441 e^{3 i (c+d x)}+95 e^{4 i (c+d x)}+504 e^{5 i (c+d x)}+65 e^{6 i (c+d x)}+147 e^{7 i (c+d x)}\right )+10 i (13 A+21 B) \left (1+e^{2 i (c+d x)}\right )^3 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right ) \sqrt {\sec (c+d x)}}{2 \left (1+e^{2 i (c+d x)}\right )^3}\right )}{420 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(901\) vs.
\(2(268)=536\).
time = 1.04, size = 902, normalized size = 3.70
method | result | size |
default | \(\text {Expression too large to display}\) | \(902\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 = 0.13, size = 263, normalized size = 1.08 \begin {gather*} -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (13 \, A + 21 \, B\right )} a^{3} \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 i \, \sqrt {2} {\left (13 \, A + 21 \, B\right )} a^{3} \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 i \, \sqrt {2} {\left (7 \, A + 9 \, B\right )} a^{3} \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 i \, \sqrt {2} {\left (7 \, A + 9 \, B\right )} a^{3} \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 {{\left (42 \, {\left (7 \, A + 9 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 5 \, {\left (26 \, A + 21 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 21 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 15 \, A a^{3}\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{105 \, d \cos \left (d x + c\right )^{3}} \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+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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