Optimal. Leaf size=492 \[ -\frac{\sqrt{a^2-b^2} \left (-29 a^2 b^2+2 a^4+42 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^8 d}-\frac{\left (-645 a^2 b^2+91 a^4+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac{b \left (-200 a^2 b^2+45 a^4+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left (-187 a^2 b^2+15 a^4+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac{\left (-79 a^2 b^2+8 a^4+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}+\frac{\left (-54 a^2 b^2+4 a^4+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left (-60 a^2 b^2+5 a^4+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 2.12688, antiderivative size = 492, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{\sqrt{a^2-b^2} \left (-29 a^2 b^2+2 a^4+42 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^8 d}-\frac{\left (-645 a^2 b^2+91 a^4+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac{b \left (-200 a^2 b^2+45 a^4+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left (-187 a^2 b^2+15 a^4+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac{\left (-79 a^2 b^2+8 a^4+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}+\frac{\left (-54 a^2 b^2+4 a^4+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left (-60 a^2 b^2+5 a^4+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2726
Rule 3055
Rule 3001
Rule 3770
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\int \frac{\csc ^4(c+d x) \left (12 \left (5 a^4-44 a^2 b^2+42 b^4\right )-12 a b \left (5 a^2-3 b^2\right ) \sin (c+d x)-20 \left (2 a^4-20 a^2 b^2+21 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{240 a^2 b^2}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\int \frac{\csc ^4(c+d x) \left (24 \left (10 a^6-114 a^4 b^2+209 a^2 b^4-105 b^6\right )-8 a b \left (20 a^4-41 a^2 b^2+21 b^4\right ) \sin (c+d x)-32 \left (5 a^6-65 a^4 b^2+123 a^2 b^4-63 b^6\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{480 a^3 b^2 \left (a^2-b^2\right )}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^4(c+d x) \left (48 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right )-24 a b \left (10 a^2-21 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-120 \left (a^2-b^2\right )^2 \left (4 a^4-54 a^2 b^2+63 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{480 a^4 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^3(c+d x) \left (-360 b \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right )+24 a b^2 \left (62 a^2-105 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+96 b \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1440 a^5 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac{\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^2(c+d x) \left (96 b^2 \left (a^2-b^2\right )^2 \left (91 a^4-645 a^2 b^2+630 b^4\right )-24 a b^3 \left (311 a^2-420 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-360 b^2 \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2880 a^6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac{\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac{\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc (c+d x) \left (-360 b^3 \left (a^2-b^2\right )^2 \left (45 a^4-200 a^2 b^2+168 b^4\right )-360 a b^2 \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2880 a^7 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac{\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac{\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}-\frac{\left (\left (a^2-b^2\right ) \left (2 a^4-29 a^2 b^2+42 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 a^8}-\frac{\left (b \left (45 a^4-200 a^2 b^2+168 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^8}\\ &=\frac{b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac{\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac{\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}-\frac{\left (\left (a^2-b^2\right ) \left (2 a^4-29 a^2 b^2+42 b^4\right )\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^8 d}\\ &=\frac{b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac{\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac{\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left (2 \left (a^2-b^2\right ) \left (2 a^4-29 a^2 b^2+42 b^4\right )\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^8 d}\\ &=-\frac{\sqrt{a^2-b^2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^8 d}+\frac{b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac{\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac{\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac{\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac{\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.70473, size = 448, normalized size = 0.91 \[ \frac{-\frac{3840 \left (-31 a^4 b^2+71 a^2 b^4+2 a^6-42 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}-480 b \left (-200 a^2 b^2+45 a^4+168 b^4\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+480 b \left (-200 a^2 b^2+45 a^4+168 b^4\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{2 a \cot (c+d x) \csc ^6(c+d x) \left (42270 a^3 b^3 \sin (c+d x)-20715 a^3 b^3 \sin (3 (c+d x))+3975 a^3 b^3 \sin (5 (c+d x))+182 a^4 b^2 \cos (6 (c+d x))-1290 a^2 b^4 \cos (6 (c+d x))+2 \left (-2131 a^4 b^2-6315 a^2 b^4+384 a^6+9450 b^6\right ) \cos (2 (c+d x))+\left (824 a^4 b^2+6060 a^2 b^4-368 a^6-7560 b^6\right ) \cos (4 (c+d x))+3256 a^4 b^2+7860 a^2 b^4-8156 a^5 b \sin (c+d x)+3956 a^5 b \sin (3 (c+d x))-608 a^5 b \sin (5 (c+d x))-784 a^6-37800 a b^5 \sin (c+d x)+18900 a b^5 \sin (3 (c+d x))-3780 a b^5 \sin (5 (c+d x))+1260 b^6 \cos (6 (c+d x))-12600 b^6\right )}{(a \csc (c+d x)+b)^2}}{3840 a^8 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.138, size = 1252, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 6.51664, size = 6091, normalized size = 12.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.85019, size = 987, normalized size = 2.01 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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