Optimal. Leaf size=363 \[ \frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {2 b \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^7 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d} \]
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Rubi [A] time = 1.48, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2896, 3055, 3001, 3770, 2660, 618, 204} \[ \frac {2 b \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {b \left (-35 a^2 b^2+23 a^4+15 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (-30 a^4 b^2+40 a^2 b^4+5 a^6-16 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^7 d}-\frac {\left (-13 a^2 b^2+8 a^4+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {\left (-22 a^2 b^2+15 a^4+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}-\frac {\left (-18 a^2 b^2+11 a^4+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2896
Rule 3001
Rule 3055
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^6(c+d x) \csc (c+d x)}{a+b \sin (c+d x)} \, dx &=-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc ^5(c+d x) \left (30 \left (8 a^4-13 a^2 b^2+6 b^4\right )-6 a b \left (5 a^2-b^2\right ) \sin (c+d x)-18 \left (10 a^4-15 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{180 a^2 b^2}\\ &=-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc ^4(c+d x) \left (-72 b \left (15 a^4-22 a^2 b^2+10 b^4\right )-18 a b^2 \left (5 a^2+2 b^2\right ) \sin (c+d x)+90 b \left (8 a^4-13 a^2 b^2+6 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{720 a^3 b^2}\\ &=-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc ^3(c+d x) \left (270 b^2 \left (11 a^4-18 a^2 b^2+8 b^4\right )-18 a b^3 \left (19 a^2-10 b^2\right ) \sin (c+d x)-144 b^2 \left (15 a^4-22 a^2 b^2+10 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2160 a^4 b^2}\\ &=-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc ^2(c+d x) \left (-288 b^3 \left (23 a^4-35 a^2 b^2+15 b^4\right )-18 a b^2 \left (75 a^4-82 a^2 b^2+40 b^4\right ) \sin (c+d x)+270 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4320 a^5 b^2}\\ &=\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \frac {\csc (c+d x) \left (-270 b^2 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right )+270 a b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4320 a^6 b^2}\\ &=\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\left (b \left (a^2-b^2\right )^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^7}-\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \int \csc (c+d x) \, dx}{16 a^7}\\ &=\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^7 d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\left (2 b \left (a^2-b^2\right )^3\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^7 d}\\ &=\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^7 d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\left (4 b \left (a^2-b^2\right )^3\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^7 d}\\ &=\frac {2 b \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {\left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^7 d}+\frac {b \left (23 a^4-35 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^6 d}-\frac {\left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^5 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{2 b d}+\frac {\left (15 a^4-22 a^2 b^2+10 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{3 b^2 d}-\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^4(c+d x)}{5 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}\\ \end {align*}
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Mathematica [A] time = 1.52, size = 356, normalized size = 0.98 \[ \frac {7680 b \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )+240 \left (-5 a^6+30 a^4 b^2-40 a^2 b^4+16 b^6\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+240 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 a \cot (c+d x) \csc ^5(c+d x) \left (-295 a^5+1168 a^4 b \sin (c+d x)-568 a^4 b \sin (3 (c+d x))+184 a^4 b \sin (5 (c+d x))+570 a^3 b^2-2320 a^2 b^3 \sin (c+d x)+1240 a^2 b^3 \sin (3 (c+d x))-280 a^2 b^3 \sin (5 (c+d x))+20 \left (7 a^5-42 a^3 b^2+24 a b^4\right ) \cos (2 (c+d x))-15 \left (11 a^5-18 a^3 b^2+8 a b^4\right ) \cos (4 (c+d x))-360 a b^4+1200 b^5 \sin (c+d x)-600 b^5 \sin (3 (c+d x))+120 b^5 \sin (5 (c+d x))\right )}{3840 a^7 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.15, size = 1462, normalized size = 4.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 627, normalized size = 1.73 \[ \frac {\frac {5 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 80 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 480 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1320 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2160 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 960 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {120 \, {\left (5 \, a^{6} - 30 \, a^{4} b^{2} + 40 \, a^{2} b^{4} - 16 \, b^{6}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{7}} + \frac {3840 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{7}} + \frac {1470 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 8820 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 11760 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4704 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1320 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2160 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 960 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 80 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{6}}{a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.51, size = 780, normalized size = 2.15 \[ -\frac {b^{2}}{64 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{6}}{d \,a^{7}}+\frac {b}{160 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {b^{5}}{2 d \,a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b^{3}}{24 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b^{4}}{8 d \,a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2 b \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d a \sqrt {a^{2}-b^{2}}}-\frac {15}{128 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {6 b^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{3} \sqrt {a^{2}-b^{2}}}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a d}-\frac {b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d \,a^{2}}+\frac {9 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}-\frac {7 b}{96 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {b^{2}}{4 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {9 b^{3}}{8 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{64 d \,a^{3}}+\frac {7 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d \,a^{2}}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{16 d \,a^{2}}+\frac {15 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{8 d \,a^{3}}+\frac {11 b}{16 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}}{8 d \,a^{5}}-\frac {5 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}}{2 d \,a^{5}}-\frac {b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \,a^{3}}-\frac {b^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{6}}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}}{24 d \,a^{4}}+\frac {3}{128 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d a}+\frac {6 b^{5} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{5} \sqrt {a^{2}-b^{2}}}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d a}-\frac {2 b^{7} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{7} \sqrt {a^{2}-b^{2}}}+\frac {15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d}-\frac {1}{384 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.34, size = 1289, normalized size = 3.55 \[ \text {result too large to display} \]
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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