Optimal. Leaf size=334 \[ -\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d} \]
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Rubi [A] time = 0.91, antiderivative size = 334, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2893, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8} \[ -\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {\left (-148 a^2 b^2+35 a^4+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}-\frac {b \left (-25 a^2 b^2+24 a^4+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2893
Rule 3021
Rule 3031
Rule 3047
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}-\frac {\int \csc ^7(c+d x) (a+b \sin (c+d x))^3 \left (3 \left (21 a^2-4 b^2\right )+3 a b \sin (c+d x)-8 \left (7 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{56 a^2}\\ &=\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}-\frac {\int \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (9 b \left (23 a^2-4 b^2\right )-3 a \left (7 a^2-2 b^2\right ) \sin (c+d x)-6 b \left (35 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{336 a^2}\\ &=\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}-\frac {\int \csc ^5(c+d x) (a+b \sin (c+d x)) \left (-3 \left (35 a^4-148 a^2 b^2+24 b^4\right )-3 a b \left (109 a^2-2 b^2\right ) \sin (c+d x)-12 b^2 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{1680 a^2}\\ &=-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {\int \csc ^4(c+d x) \left (72 b \left (24 a^4-25 a^2 b^2+4 b^4\right )+315 a^3 \left (a^2+8 b^2\right ) \sin (c+d x)+48 b^3 \left (53 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6720 a^2}\\ &=-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {\int \csc ^3(c+d x) \left (945 a^3 \left (a^2+8 b^2\right )+576 a^2 b \left (6 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx}{20160 a^2}\\ &=-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {1}{35} \left (b \left (6 a^2+7 b^2\right )\right ) \int \csc ^2(c+d x) \, dx+\frac {1}{64} \left (3 a \left (a^2+8 b^2\right )\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}+\frac {1}{128} \left (3 a \left (a^2+8 b^2\right )\right ) \int \csc (c+d x) \, dx-\frac {\left (b \left (6 a^2+7 b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{35 d}\\ &=-\frac {3 a \left (a^2+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac {3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}-\frac {b \left (24 a^4-25 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac {\left (35 a^4-148 a^2 b^2+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}+\frac {3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac {\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d}\\ \end {align*}
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Mathematica [A] time = 1.60, size = 268, normalized size = 0.80 \[ -\frac {-6720 a \left (a^2+8 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6720 a \left (a^2+8 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^8(c+d x) \left (35 \left (333 a^3+104 a b^2\right ) \cos (3 (c+d x))+805 a^3 \cos (5 (c+d x))-105 a^3 \cos (7 (c+d x))+35 a \left (671 a^2+248 b^2\right ) \cos (c+d x)+21504 a^2 b \sin (2 (c+d x))+16128 a^2 b \sin (4 (c+d x))+3072 a^2 b \sin (6 (c+d x))-384 a^2 b \sin (8 (c+d x))-11480 a b^2 \cos (5 (c+d x))-840 a b^2 \cos (7 (c+d x))+2688 b^3 \sin (2 (c+d x))+896 b^3 \sin (4 (c+d x))-896 b^3 \sin (6 (c+d x))-448 b^3 \sin (8 (c+d x))\right )}{286720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 384, normalized size = 1.15 \[ \frac {210 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} - 70 \, {\left (11 \, a^{3} - 40 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 770 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 210 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right ) - 105 \, {\left ({\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 8 \, a b^{2} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left ({\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} + 8 \, a b^{2} - 4 \, {\left (a^{3} + 8 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 256 \, {\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{7} - 7 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{8960 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 457, normalized size = 1.37 \[ \frac {35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 240 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 336 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 448 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5040 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4480 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1680 \, {\left (a^{3} + 8 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {4566 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 36528 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 5040 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4480 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1680 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1680 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 448 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 35 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{71680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 358, normalized size = 1.07 \[ -\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{6}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{64 d \sin \left (d x +c \right )^{4}}+\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}+\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{128 d}+\frac {3 a^{3} \cos \left (d x +c \right )}{128 d}+\frac {3 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {3 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}-\frac {6 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{35 d \sin \left (d x +c \right )^{5}}-\frac {a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{6}}-\frac {a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{4}}+\frac {a \,b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{2}}+\frac {a \,b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{16 d}+\frac {3 a \,b^{2} \cos \left (d x +c \right )}{16 d}+\frac {3 a \,b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {b^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 248, normalized size = 0.74 \[ \frac {35 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 280 \, a b^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1792 \, b^{3}}{\tan \left (d x + c\right )^{5}} - \frac {768 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2} b}{\tan \left (d x + c\right )^{7}}}{8960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.95, size = 381, normalized size = 1.14 \[ \frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3}{256}+\frac {3\,a\,b^2}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {3\,a^2\,b}{128}+\frac {b^3}{32}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {3\,a^2\,b}{640}-\frac {b^3}{160}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^3}{128}+\frac {3\,a\,b^2}{16}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^3+6\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (6\,a^2\,b+8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {6\,a^2\,b}{5}-\frac {8\,b^3}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (18\,a^2\,b+16\,b^3\right )-\frac {a^3}{8}-\frac {6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{7}-2\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{256\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {9\,a^2\,b}{128}+\frac {b^3}{16}\right )}{d}-\frac {3\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d} \]
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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