Optimal. Leaf size=500 \[ \frac {2 \left (99 a^2 A-209 a b B-113 A b^2\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}+\frac {2 \left (231 a^3 B+495 a^2 A b-275 a b^2 B-5 A b^3\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{1155 a d}-\frac {2 \left (1155 a^4 A-2541 a^3 b B-1485 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3465 a^2 d}-\frac {2 \left (3465 a^5 B+8085 a^4 A b-5313 a^3 b^2 B-495 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3465 a^3 d}-\frac {2 a (11 a B+14 A b) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {(-b+i a)^{5/2} (-B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{5/2} (B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d} \]
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Rubi [A] time = 2.47, antiderivative size = 500, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4241, 3605, 3645, 3649, 3616, 3615, 93, 203, 206} \[ \frac {2 \left (99 a^2 A-209 a b B-113 A b^2\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}+\frac {2 \left (495 a^2 A b+231 a^3 B-275 a b^2 B-5 A b^3\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{1155 a d}-\frac {2 \left (-1485 a^2 A b^2+1155 a^4 A-2541 a^3 b B+55 a b^3 B-20 A b^4\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3465 a^2 d}-\frac {2 \left (-495 a^2 A b^3+8085 a^4 A b-5313 a^3 b^2 B+3465 a^5 B-110 a b^4 B+40 A b^5\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3465 a^3 d}-\frac {2 a (11 a B+14 A b) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {(-b+i a)^{5/2} (-B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(b+i a)^{5/2} (B+i A) \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d} \]
Antiderivative was successfully verified.
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Rule 93
Rule 203
Rule 206
Rule 3605
Rule 3615
Rule 3616
Rule 3645
Rule 3649
Rule 4241
Rubi steps
\begin {align*} \int \cot ^{\frac {13}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\tan ^{\frac {13}{2}}(c+d x)} \, dx\\ &=-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}+\frac {1}{11} \left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sqrt {a+b \tan (c+d x)} \left (\frac {1}{2} a (14 A b+11 a B)-\frac {11}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac {1}{2} b (8 a A-11 b B) \tan ^2(c+d x)\right )}{\tan ^{\frac {11}{2}}(c+d x)} \, dx\\ &=-\frac {2 a (14 A b+11 a B) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}+\frac {1}{99} \left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {1}{4} a \left (99 a^2 A-113 A b^2-209 a b B\right )-\frac {99}{4} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-\frac {1}{4} b \left (184 a A b+88 a^2 B-99 b^2 B\right ) \tan ^2(c+d x)}{\tan ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 \left (99 a^2 A-113 A b^2-209 a b B\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}-\frac {2 a (14 A b+11 a B) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}-\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {3}{8} a \left (495 a^2 A b-5 A b^3+231 a^3 B-275 a b^2 B\right )-\frac {693}{8} a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)-\frac {3}{4} a b \left (99 a^2 A-113 A b^2-209 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{693 a}\\ &=\frac {2 \left (495 a^2 A b-5 A b^3+231 a^3 B-275 a b^2 B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{1155 a d}+\frac {2 \left (99 a^2 A-113 A b^2-209 a b B\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}-\frac {2 a (14 A b+11 a B) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}+\frac {\left (16 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {3}{16} a \left (1155 a^4 A-1485 a^2 A b^2-20 A b^4-2541 a^3 b B+55 a b^3 B\right )+\frac {3465}{16} a^2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+\frac {3}{4} a b \left (495 a^2 A b-5 A b^3+231 a^3 B-275 a b^2 B\right ) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{3465 a^2}\\ &=-\frac {2 \left (1155 a^4 A-1485 a^2 A b^2-20 A b^4-2541 a^3 b B+55 a b^3 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3465 a^2 d}+\frac {2 \left (495 a^2 A b-5 A b^3+231 a^3 B-275 a b^2 B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{1155 a d}+\frac {2 \left (99 a^2 A-113 A b^2-209 a b B\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}-\frac {2 a (14 A b+11 a B) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}-\frac {\left (32 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{32} a \left (8085 a^4 A b-495 a^2 A b^3+40 A b^5+3465 a^5 B-5313 a^3 b^2 B-110 a b^4 B\right )+\frac {10395}{32} a^3 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)+\frac {3}{16} a b \left (1155 a^4 A-1485 a^2 A b^2-20 A b^4-2541 a^3 b B+55 a b^3 B\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{10395 a^3}\\ &=-\frac {2 \left (8085 a^4 A b-495 a^2 A b^3+40 A b^5+3465 a^5 B-5313 a^3 b^2 B-110 a b^4 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3465 a^3 d}-\frac {2 \left (1155 a^4 A-1485 a^2 A b^2-20 A b^4-2541 a^3 b B+55 a b^3 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3465 a^2 d}+\frac {2 \left (495 a^2 A b-5 A b^3+231 a^3 B-275 a b^2 B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{1155 a d}+\frac {2 \left (99 a^2 A-113 A b^2-209 a b B\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}-\frac {2 a (14 A b+11 a B) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}+\frac {\left (64 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {10395}{64} a^4 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )-\frac {10395}{64} a^4 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{10395 a^4}\\ &=-\frac {2 \left (8085 a^4 A b-495 a^2 A b^3+40 A b^5+3465 a^5 B-5313 a^3 b^2 B-110 a b^4 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3465 a^3 d}-\frac {2 \left (1155 a^4 A-1485 a^2 A b^2-20 A b^4-2541 a^3 b B+55 a b^3 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3465 a^2 d}+\frac {2 \left (495 a^2 A b-5 A b^3+231 a^3 B-275 a b^2 B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{1155 a d}+\frac {2 \left (99 a^2 A-113 A b^2-209 a b B\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}-\frac {2 a (14 A b+11 a B) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}-\frac {1}{2} \left ((a-i b)^3 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx-\frac {1}{2} \left ((a+i b)^3 (A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 \left (8085 a^4 A b-495 a^2 A b^3+40 A b^5+3465 a^5 B-5313 a^3 b^2 B-110 a b^4 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3465 a^3 d}-\frac {2 \left (1155 a^4 A-1485 a^2 A b^2-20 A b^4-2541 a^3 b B+55 a b^3 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3465 a^2 d}+\frac {2 \left (495 a^2 A b-5 A b^3+231 a^3 B-275 a b^2 B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{1155 a d}+\frac {2 \left (99 a^2 A-113 A b^2-209 a b B\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}-\frac {2 a (14 A b+11 a B) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}-\frac {\left ((a-i b)^3 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {\left ((a+i b)^3 (A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac {2 \left (8085 a^4 A b-495 a^2 A b^3+40 A b^5+3465 a^5 B-5313 a^3 b^2 B-110 a b^4 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3465 a^3 d}-\frac {2 \left (1155 a^4 A-1485 a^2 A b^2-20 A b^4-2541 a^3 b B+55 a b^3 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3465 a^2 d}+\frac {2 \left (495 a^2 A b-5 A b^3+231 a^3 B-275 a b^2 B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{1155 a d}+\frac {2 \left (99 a^2 A-113 A b^2-209 a b B\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}-\frac {2 a (14 A b+11 a B) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}-\frac {\left ((a-i b)^3 (A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\left ((a+i b)^3 (A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\\ &=-\frac {(i a-b)^{5/2} (i A-B) \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {(i a+b)^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 \left (8085 a^4 A b-495 a^2 A b^3+40 A b^5+3465 a^5 B-5313 a^3 b^2 B-110 a b^4 B\right ) \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}{3465 a^3 d}-\frac {2 \left (1155 a^4 A-1485 a^2 A b^2-20 A b^4-2541 a^3 b B+55 a b^3 B\right ) \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{3465 a^2 d}+\frac {2 \left (495 a^2 A b-5 A b^3+231 a^3 B-275 a b^2 B\right ) \cot ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{1155 a d}+\frac {2 \left (99 a^2 A-113 A b^2-209 a b B\right ) \cot ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{693 d}-\frac {2 a (14 A b+11 a B) \cot ^{\frac {9}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}}{99 d}-\frac {2 a A \cot ^{\frac {11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}{11 d}\\ \end {align*}
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Mathematica [A] time = 6.97, size = 653, normalized size = 1.31 \[ \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {b B (a+b \tan (c+d x))^{3/2}}{4 d \tan ^{\frac {11}{2}}(c+d x)}+\frac {1}{4} \left (-\frac {b (5 a B+8 A b) \sqrt {a+b \tan (c+d x)}}{10 d \tan ^{\frac {11}{2}}(c+d x)}+\frac {1}{5} \left (-\frac {\left (80 a^2 A-165 a b B-88 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{22 d \tan ^{\frac {11}{2}}(c+d x)}-\frac {2 \left (\frac {5 a \left (88 a^2 B+184 a A b-99 b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{18 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2 \left (\frac {10 a^2 \left (99 a^2 A-209 a b B-113 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 \left (-\frac {3 a^2 \left (231 a^3 B+495 a^2 A b-275 a b^2 B-5 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 \left (-\frac {5 a^2 \left (1155 a^4 A-2541 a^3 b B-1485 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sqrt {a+b \tan (c+d x)}}{2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (\frac {15 a^2 \left (3465 a^5 B+8085 a^4 A b-5313 a^3 b^2 B-495 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \tan (c+d x)}}{4 d \sqrt {\tan (c+d x)}}+\frac {51975 a^5 \left ((-1)^{3/4} (-a+i b)^{5/2} (A-i B) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-(-1)^{3/4} (a+i b)^{5/2} (A+i B) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right )}{8 d}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 7.01, size = 103764, normalized size = 207.53 \[ \text {output too large to display} \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{\frac {13}{2}}\,{d x} \]
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 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{13/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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