Optimal. Leaf size=424 \[ \frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {2 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^7 d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}+\frac {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))} \]
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Rubi [A] time = 1.52, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2726, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac {2 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^7 d}-\frac {\left (-135 a^2 b^2+38 a^4+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {b \left (-40 a^2 b^2+15 a^4+24 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac {\left (-82 a^2 b^2+15 a^4+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}+\frac {\left (-17 a^2 b^2+4 a^4+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}+\frac {\left (-12 a^2 b^2+2 a^4+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2726
Rule 3001
Rule 3055
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^4(c+d x) \left (12 \left (5 a^4-22 a^2 b^2+15 b^4\right )-4 a b \left (10 a^2-3 b^2\right ) \sin (c+d x)-4 \left (10 a^4-45 a^2 b^2+36 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{120 a^2 b^2}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^4(c+d x) \left (12 \left (15 a^6-97 a^4 b^2+142 a^2 b^4-60 b^6\right )-12 a b \left (5 a^4-8 a^2 b^2+3 b^4\right ) \sin (c+d x)-60 \left (2 a^6-14 a^4 b^2+21 a^2 b^4-9 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 a^3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (-180 b \left (4 a^6-21 a^4 b^2+29 a^2 b^4-12 b^6\right )+12 a b^2 \left (16 a^4-31 a^2 b^2+15 b^4\right ) \sin (c+d x)+24 b \left (15 a^6-97 a^4 b^2+142 a^2 b^4-60 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{360 a^4 b^2 \left (a^2-b^2\right )}\\ &=\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (48 b^2 \left (38 a^6-173 a^4 b^2+225 a^2 b^4-90 b^6\right )-12 a b^3 \left (73 a^4-133 a^2 b^2+60 b^4\right ) \sin (c+d x)-180 b^2 \left (4 a^6-21 a^4 b^2+29 a^2 b^4-12 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{720 a^5 b^2 \left (a^2-b^2\right )}\\ &=-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-180 b^3 \left (15 a^6-55 a^4 b^2+64 a^2 b^4-24 b^6\right )-180 a b^2 \left (4 a^6-21 a^4 b^2+29 a^2 b^4-12 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{720 a^6 b^2 \left (a^2-b^2\right )}\\ &=-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac {\left (\left (a^2-6 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^7}-\frac {\left (b \left (15 a^6-55 a^4 b^2+64 a^2 b^4-24 b^6\right )\right ) \int \csc (c+d x) \, dx}{4 a^7 \left (a^2-b^2\right )}\\ &=\frac {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac {\left (2 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^2\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 {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\left (4 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^2\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 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^{3/2} \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 {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 1.61, size = 361, normalized size = 0.85 \[ -\frac {1920 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )+240 b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-240 b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \cot (c+d x) \csc ^5(c+d x) \left (196 a^5+1162 a^4 b \sin (c+d x)-562 a^4 b \sin (3 (c+d x))+76 a^4 b \sin (5 (c+d x))-735 a^3 b^2-3060 a^2 b^3 \sin (c+d x)+1470 a^2 b^3 \sin (3 (c+d x))-270 a^2 b^3 \sin (5 (c+d x))-12 \left (16 a^5-85 a^3 b^2+60 a b^4\right ) \cos (2 (c+d x))+\left (92 a^5-285 a^3 b^2+180 a b^4\right ) \cos (4 (c+d x))+540 a b^4+1800 b^5 \sin (c+d x)-900 b^5 \sin (3 (c+d x))+180 b^5 \sin (5 (c+d x))\right )}{a \csc (c+d x)+b}}{960 a^7 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.05, size = 2011, normalized size = 4.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 596, normalized size = 1.41 \[ -\frac {\frac {120 \, {\left (15 \, a^{4} b - 40 \, a^{2} b^{3} + 24 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{7}} + \frac {960 \, {\left (a^{6} - 8 \, a^{4} b^{2} + 13 \, a^{2} b^{4} - 6 \, b^{6}\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 {960 \, {\left (a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{7}} - \frac {3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 240 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 330 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1620 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1200 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{10}} - \frac {4110 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 10960 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6576 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 330 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1620 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1200 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{5}}{a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.82, size = 897, normalized size = 2.12 \[ \text {result too large to display} \]
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 = 12.16, size = 1424, normalized size = 3.36 \[ \text {result too large to display} \]
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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