Optimal. Leaf size=353 \[ \frac {a^{5/2} (1304 A+1132 B+1015 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{512 d}+\frac {a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a (12 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A]
time = 0.74, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {4350, 4173,
4103, 4101, 3888, 3886, 221} \begin {gather*} \frac {a^{5/2} (1304 A+1132 B+1015 C) \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 )}{512 d}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (120 A+156 B+115 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{480 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a (12 B+5 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d \cos ^{\frac {7}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 3886
Rule 3888
Rule 4101
Rule 4103
Rule 4173
Rule 4350
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (12 A+5 C)+\frac {1}{2} a (12 B+5 C) \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac {a (12 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {15}{4} a^2 (8 A+4 B+5 C)+\frac {1}{4} a^2 (120 A+156 B+115 C) \sec (c+d x)\right ) \, dx}{30 a}\\ &=\frac {a^2 (120 A+156 B+115 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a (12 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {5}{8} a^3 (312 A+252 B+235 C)+\frac {3}{8} a^3 (680 A+628 B+545 C) \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a (12 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{384} \left (a^2 (1304 A+1132 B+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a (12 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{512} \left (a^2 (1304 A+1132 B+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a (12 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (a^2 (1304 A+1132 B+1015 C) \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}{1024}\\ &=\frac {a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a (12 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\left (a^2 (1304 A+1132 B+1015 C) \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 )}{512 d}\\ &=\frac {a^{5/2} (1304 A+1132 B+1015 C) \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)}}{512 d}+\frac {a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (120 A+156 B+115 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a (12 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(947\) vs. \(2(353)=706\).
time = 6.58, size = 947, normalized size = 2.68 \begin {gather*} \frac {4 \sec ^5\left (\frac {1}{2} (c+d x)\right ) (a (1+\sec (c+d x)))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sqrt {\frac {1}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}} \sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )} \left (\frac {C \sin \left (\frac {1}{2} (c+d x)\right )}{48 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^6}+\frac {(B+2 C) \sin \left (\frac {1}{2} (c+d x)\right )}{40 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {(A+2 B+C) \sin \left (\frac {1}{2} (c+d x)\right )}{32 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {(2 A+B) \sin \left (\frac {1}{2} (c+d x)\right )}{24 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {A \sin \left (\frac {1}{2} (c+d x)\right )}{16 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {3}{64} A \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )+\frac {5}{384} (2 A+B) \left (\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+3 \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right )+\frac {7 (A+2 B+C) \left (\frac {16 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+5 \left (\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+3 \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right )\right )}{3072}+\frac {3 (B+2 C) \left (\frac {96 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+7 \left (\frac {16 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+5 \left (\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+3 \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right )\right )\right )}{10240}+\frac {11 C \left (\frac {256 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^5}+3 \left (\frac {96 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^4}+7 \left (\frac {16 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^3}+5 \left (\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}+3 \left (\sqrt {2} \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right )\right )\right )\right )}{122880}\right )}{d (A+2 C+2 B \cos (c+d x)+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. \(814\) vs.
\(2(305)=610\).
time = 0.20, size = 815, normalized size = 2.31
method | result | size |
default | \(-\frac {a^{2} \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (-1+\cos \left (d x +c \right )\right ) \left (19560 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 ) \left (\cos ^{6}\left (d x +c \right )\right )+19560 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 ) \left (\cos ^{6}\left (d x +c \right )\right )+16980 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 ) \left (\cos ^{6}\left (d x +c \right )\right )+16980 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 ) \left (\cos ^{6}\left (d x +c \right )\right )+15225 C \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 ) \left (\cos ^{6}\left (d x +c \right )\right )+15225 C \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 ) \left (\cos ^{6}\left (d x +c \right )\right )+39120 A \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )+33960 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )+30450 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )+26080 A \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+22640 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+20300 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+14720 A \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+18112 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+16240 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+3840 A \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+11136 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+13920 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+3072 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+8960 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+2560 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )\right )}{15360 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{\frac {11}{2}} \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}\) | \(815\) |
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. 16461 vs.
\(2 (305) = 610\).
time = 3.29, size = 16461, normalized size = 46.63 \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 = 5.73, size = 625, normalized size = 1.77 \begin {gather*} \left [\frac {4 \, {\left (15 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 10 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (920 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (40 \, A + 116 \, B + 145 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 128 \, {\left (12 \, B + 35 \, C\right )} a^{2} \cos \left (d x + c\right ) + 1280 \, C a^{2}\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 ) + 15 \, {\left ({\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{7} + {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6}\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 )}{30720 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}, \frac {2 \, {\left (15 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 10 \, {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (920 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (40 \, A + 116 \, B + 145 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 128 \, {\left (12 \, B + 35 \, C\right )} a^{2} \cos \left (d x + c\right ) + 1280 \, C a^{2}\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 ) + 15 \, {\left ({\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{7} + {\left (1304 \, A + 1132 \, B + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6}\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 )}{15360 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\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 {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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