Optimal. Leaf size=217 \[ -\frac {2 a (5 A+3 (B+C)) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a (7 A+7 B+5 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a (5 A+3 (B+C)) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (7 A+7 B+5 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a (B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a C \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d} \]
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Rubi [A]
time = 0.18, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4161, 4132,
3853, 3856, 2720, 4131, 2719} \begin {gather*} \frac {2 a (7 A+7 B+5 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{21 d}+\frac {2 a (5 A+3 (B+C)) \sin (c+d x) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 a (7 A+7 B+5 C) \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 {2 a (5 A+3 (B+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 {2 a (B+C) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {2 a C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 3853
Rule 3856
Rule 4131
Rule 4132
Rule 4161
Rubi steps
\begin {align*} \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 a C \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2}{7} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {7 a A}{2}+\frac {1}{2} a (7 A+7 B+5 C) \sec (c+d x)+\frac {7}{2} a (B+C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 a C \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2}{7} \int \sec ^{\frac {3}{2}}(c+d x) \left (\frac {7 a A}{2}+\frac {7}{2} a (B+C) \sec ^2(c+d x)\right ) \, dx+\frac {1}{7} (a (7 A+7 B+5 C)) \int \sec ^{\frac {5}{2}}(c+d x) \, dx\\ &=\frac {2 a (7 A+7 B+5 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a (B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a C \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{21} (a (7 A+7 B+5 C)) \int \sqrt {\sec (c+d x)} \, dx+\frac {1}{5} (a (5 A+3 (B+C))) \int \sec ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 a (5 A+3 (B+C)) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (7 A+7 B+5 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a (B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a C \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{5} (a (5 A+3 (B+C))) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{21} \left (a (7 A+7 B+5 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 a (7 A+7 B+5 C) \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 {2 a (5 A+3 (B+C)) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (7 A+7 B+5 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a (B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a C \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{5} \left (a (5 A+3 (B+C)) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=-\frac {2 a (5 A+3 (B+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 {2 a (7 A+7 B+5 C) \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 {2 a (5 A+3 (B+C)) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 a (7 A+7 B+5 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 a (B+C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 a C \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 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 = 7.01, size = 1029, normalized size = 4.74 \begin {gather*} a \left (\frac {2 \sqrt {2} 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 ^2(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 ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {2 \sqrt {2} B 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 ^2(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 ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {2 \sqrt {2} 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 ^2(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 ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {4 A \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {4 B \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {20 C \cos ^{\frac {5}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{21 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {\left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4 (5 A+3 B+3 C) \cos (d x) \csc (c)}{5 d}+\frac {4 C \sec (c) \sec ^3(c+d x) \sin (d x)}{7 d}+\frac {4 \sec (c) \sec ^2(c+d x) (5 C \sin (c)+7 B \sin (d x)+7 C \sin (d x))}{35 d}+\frac {4 \sec (c) \sec (c+d x) (21 B \sin (c)+21 C \sin (c)+35 A \sin (d x)+35 B \sin (d x)+25 C \sin (d x))}{105 d}+\frac {4 (7 A+7 B+5 C) \tan (c)}{21 d}\right )}{(A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {3}{2}}(c+d x)}\right ) \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. \(822\) vs.
\(2(243)=486\).
time = 0.20, size = 823, normalized size = 3.79
method | result | size |
default | \(\text {Expression too large to display}\) | \(823\) |
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 = 1.12, size = 263, normalized size = 1.21 \begin {gather*} \frac {-5 i \, \sqrt {2} {\left (7 \, A + 7 \, B + 5 \, C\right )} a \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 (7 \, A + 7 \, B + 5 \, C\right )} a \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 (5 \, A + 3 \, B + 3 \, C\right )} a \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 (5 \, A + 3 \, B + 3 \, C\right )} a \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 \, A + 3 \, B + 3 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, A + 7 \, B + 5 \, C\right )} a \cos \left (d x + c\right )^{2} + 21 \, {\left (B + C\right )} a \cos \left (d x + c\right ) + 15 \, C a\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {a}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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