Optimal. Leaf size=286 \[ \frac {(2 A-5 B) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{a^{5/2} d}-\frac {(43 A-115 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{16 \sqrt {2} a^{5/2} d}+\frac {(A-B) \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}+\frac {(7 A-15 B) \sin (c+d x)}{16 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}-\frac {(11 A-35 B) \sin (c+d x)}{16 a^2 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}} \]
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Rubi [A]
time = 0.61, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {3034, 4104,
4106, 4108, 3893, 212, 3886, 221} \begin {gather*} -\frac {(43 A-115 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x) \sqrt {\sec (c+d x)}}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {(2 A-5 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{5/2} d}-\frac {(11 A-35 B) \sin (c+d x)}{16 a^2 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {(7 A-15 B) \sin (c+d x)}{16 a d \cos ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac {(A-B) \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 221
Rule 3034
Rule 3886
Rule 3893
Rule 4104
Rule 4106
Rule 4108
Rubi steps
\begin {align*} \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx\\ &=\frac {(A-B) \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (\frac {5}{2} a (A-B)-a (A-5 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=\frac {(A-B) \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}+\frac {(7 A-15 B) \sin (c+d x)}{16 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (\frac {3}{4} a^2 (7 A-15 B)-\frac {1}{2} a^2 (11 A-35 B) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{8 a^4}\\ &=\frac {(A-B) \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}+\frac {(7 A-15 B) \sin (c+d x)}{16 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}-\frac {(11 A-35 B) \sin (c+d x)}{16 a^2 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (-\frac {1}{4} a^3 (11 A-35 B)+4 a^3 (2 A-5 B) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{8 a^5}\\ &=\frac {(A-B) \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}+\frac {(7 A-15 B) \sin (c+d x)}{16 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}-\frac {(11 A-35 B) \sin (c+d x)}{16 a^2 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}-\frac {\left ((43 A-115 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx}{32 a^2}+\frac {\left ((2 A-5 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \, dx}{2 a^3}\\ &=\frac {(A-B) \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}+\frac {(7 A-15 B) \sin (c+d x)}{16 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}-\frac {(11 A-35 B) \sin (c+d x)}{16 a^2 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {\left ((43 A-115 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{16 a^2 d}-\frac {\left ((2 A-5 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^3 d}\\ &=\frac {(2 A-5 B) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{a^{5/2} d}-\frac {(43 A-115 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{16 \sqrt {2} a^{5/2} d}+\frac {(A-B) \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}}+\frac {(7 A-15 B) \sin (c+d x)}{16 a d \cos ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}}-\frac {(11 A-35 B) \sin (c+d x)}{16 a^2 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1061\) vs. \(2(286)=572\).
time = 6.18, size = 1061, normalized size = 3.71 \begin {gather*} -\frac {B \sin (c+d x)}{4 d \cos ^{\frac {11}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}-\frac {A \sin (c+d x)}{4 d \cos ^{\frac {9}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}+\frac {7 B (1+\sec (c+d x)) \sin (c+d x)}{16 d \cos ^{\frac {11}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}+\frac {3 A (1+\sec (c+d x)) \sin (c+d x)}{16 d \cos ^{\frac {9}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}-\frac {7 B (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \cos ^{\frac {9}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}-\frac {3 A (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \cos ^{\frac {7}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}+\frac {11 B (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \cos ^{\frac {7}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}+\frac {7 A (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \cos ^{\frac {5}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}-\frac {15 B (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \cos ^{\frac {5}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}-\frac {11 A (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \cos ^{\frac {3}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}+\frac {35 B (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \cos ^{\frac {3}{2}}(c+d x) (a (1+\sec (c+d x)))^{5/2}}-\frac {11 A \text {ArcSin}\left (\sqrt {1-\sec (c+d x)}\right ) \sqrt {\cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \sqrt {1-\sec (c+d x)} (a (1+\sec (c+d x)))^{5/2}}+\frac {35 B \text {ArcSin}\left (\sqrt {1-\sec (c+d x)}\right ) \sqrt {\cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \sqrt {1-\sec (c+d x)} (a (1+\sec (c+d x)))^{5/2}}-\frac {43 A \text {ArcSin}\left (\sqrt {\sec (c+d x)}\right ) \sqrt {\cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \sqrt {1-\sec (c+d x)} (a (1+\sec (c+d x)))^{5/2}}+\frac {115 B \text {ArcSin}\left (\sqrt {\sec (c+d x)}\right ) \sqrt {\cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) (1+\sec (c+d x))^2 \sin (c+d x)}{16 d \sqrt {1-\sec (c+d x)} (a (1+\sec (c+d x)))^{5/2}}+\frac {43 A \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {\sec (c+d x)}}{\sqrt {1-\sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) (1+\sec (c+d x))^2 \sin (c+d x)}{16 \sqrt {2} d \sqrt {1-\sec (c+d x)} (a (1+\sec (c+d x)))^{5/2}}-\frac {115 B \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {\sec (c+d x)}}{\sqrt {1-\sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x) (1+\sec (c+d x))^2 \sin (c+d x)}{16 \sqrt {2} d \sqrt {1-\sec (c+d x)} (a (1+\sec (c+d x)))^{5/2}} \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. \(820\) vs.
\(2(241)=482\).
time = 14.12, size = 821, normalized size = 2.87
method | result | size |
default | \(\frac {\left (-1+\cos \left (d x +c \right )\right )^{2} \left (16 A \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (-1-\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+16 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right )-40 B \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (-1-\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )-40 B \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right )+16 A \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (-1-\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sin \left (d x +c \right ) \cos \left (d x +c \right )+16 A \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sin \left (d x +c \right ) \cos \left (d x +c \right )-43 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right )+11 A \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-40 B \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (-1-\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sin \left (d x +c \right ) \cos \left (d x +c \right )-40 B \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sqrt {2}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )+115 B \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right )-35 B \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-43 A \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sin \left (d x +c \right ) \cos \left (d x +c \right )+4 A \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+115 B \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-20 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )-15 A \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+39 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+16 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{16 d \sqrt {\cos \left (d x +c \right )}\, \sin \left (d x +c \right )^{5} \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, a^{3}}\) | \(821\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 14037 vs.
\(2 (241) = 482\).
time = 4.95, size = 14037, normalized size = 49.08 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.43, size = 850, normalized size = 2.97 \begin {gather*} \left [-\frac {\sqrt {2} {\left ({\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \, {\left ({\left (11 \, A - 35 \, B\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (3 \, A - 11 \, B\right )} \cos \left (d x + c\right ) - 16 \, B\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 16 \, {\left ({\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} + 4 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 2\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right )^{2} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right )}{64 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right )\right )}}, \frac {\sqrt {2} {\left ({\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (43 \, A - 115 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )}}{a \sin \left (d x + c\right )}\right ) - 2 \, {\left ({\left (11 \, A - 35 \, B\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (3 \, A - 11 \, B\right )} \cos \left (d x + c\right ) - 16 \, B\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 16 \, {\left ({\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, A - 5 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right )}{32 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right )\right )}}\right ] \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 \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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