Optimal. Leaf size=476 \[ \frac{2 b^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^9 d}-\frac{b \left (-161 a^4 b^2+245 a^2 b^4+15 a^6-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac{\left (40 a^6 b^2-240 a^4 b^4+320 a^2 b^6+5 a^8-128 b^8\right ) \tanh ^{-1}(\cos (c+d x))}{128 a^9 d}-\frac{\left (-85 a^2 b^2+48 a^4+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{\left (-60 a^2 b^2+35 a^4+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}-\frac{\left (-104 a^2 b^2+59 a^4+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}+\frac{b \left (-77 a^2 b^2+45 a^4+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}+\frac{\left (-88 a^4 b^2+144 a^2 b^4+5 a^6-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d} \]
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Rubi [A] time = 2.21452, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2896, 3055, 3001, 3770, 2660, 618, 204} \[ \frac{2 b^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^9 d}-\frac{b \left (-161 a^4 b^2+245 a^2 b^4+15 a^6-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac{\left (40 a^6 b^2-240 a^4 b^4+320 a^2 b^6+5 a^8-128 b^8\right ) \tanh ^{-1}(\cos (c+d x))}{128 a^9 d}-\frac{\left (-85 a^2 b^2+48 a^4+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{\left (-60 a^2 b^2+35 a^4+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}-\frac{\left (-104 a^2 b^2+59 a^4+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}+\frac{b \left (-77 a^2 b^2+45 a^4+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}+\frac{\left (-88 a^4 b^2+144 a^2 b^4+5 a^6-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d} \]
Antiderivative was successfully verified.
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Rule 2896
Rule 3055
Rule 3001
Rule 3770
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac{\int \frac{\csc ^7(c+d x) \left (28 \left (48 a^4-85 a^2 b^2+40 b^4\right )-4 a b \left (14 a^2-5 b^2\right ) \sin (c+d x)-40 \left (28 a^4-49 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1120 a^2 b^2}\\ &=-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac{\int \frac{\csc ^6(c+d x) \left (-240 b \left (35 a^4-60 a^2 b^2+28 b^4\right )-20 a b^2 \left (7 a^2+8 b^2\right ) \sin (c+d x)+140 b \left (48 a^4-85 a^2 b^2+40 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6720 a^3 b^2}\\ &=-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac{\int \frac{\csc ^5(c+d x) \left (700 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right )-20 a b^3 \left (95 a^2-56 b^2\right ) \sin (c+d x)-960 b^2 \left (35 a^4-60 a^2 b^2+28 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{33600 a^4 b^2}\\ &=-\frac{\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac{\int \frac{\csc ^4(c+d x) \left (-3840 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right )-60 a b^2 \left (175 a^4-200 a^2 b^2+112 b^4\right ) \sin (c+d x)+2100 b^3 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{134400 a^5 b^2}\\ &=\frac{b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac{\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac{\int \frac{\csc ^3(c+d x) \left (-6300 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right )+60 a b^3 \left (435 a^4-1064 a^2 b^2+560 b^4\right ) \sin (c+d x)-7680 b^4 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{403200 a^6 b^2}\\ &=\frac{\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac{\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac{\int \frac{\csc ^2(c+d x) \left (7680 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right )-60 a b^2 \left (525 a^6+2280 a^4 b^2-4592 a^2 b^4+2240 b^6\right ) \sin (c+d x)-6300 b^3 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{806400 a^7 b^2}\\ &=-\frac{b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac{\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac{\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac{\int \frac{\csc (c+d x) \left (-6300 b^2 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right )-6300 a b^3 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{806400 a^8 b^2}\\ &=-\frac{b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac{\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac{\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac{\left (b^3 \left (a^2-b^2\right )^3\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^9}-\frac{\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \int \csc (c+d x) \, dx}{128 a^9}\\ &=\frac{\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \tanh ^{-1}(\cos (c+d x))}{128 a^9 d}-\frac{b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac{\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac{\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac{\left (2 b^3 \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^9 d}\\ &=\frac{\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \tanh ^{-1}(\cos (c+d x))}{128 a^9 d}-\frac{b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac{\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac{\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac{\left (4 b^3 \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^9 d}\\ &=\frac{2 b^3 \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^9 d}+\frac{\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \tanh ^{-1}(\cos (c+d x))}{128 a^9 d}-\frac{b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac{\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac{b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac{\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac{\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac{\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac{a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac{\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x)}{8 a d}\\ \end{align*}
Mathematica [A] time = 3.57399, size = 593, normalized size = 1.25 \[ \frac{1720320 b^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )-6720 \left (40 a^6 b^2-240 a^4 b^4+320 a^2 b^6+5 a^8-128 b^8\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+6720 \left (40 a^6 b^2-240 a^4 b^4+320 a^2 b^6+5 a^8-128 b^8\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+a \csc ^8(c+d x) \left (88704 a^4 b^3 \sin (2 (c+d x))-86912 a^4 b^3 \sin (4 (c+d x))+42112 a^4 b^3 \sin (6 (c+d x))-10304 a^4 b^3 \sin (8 (c+d x))-174720 a^2 b^5 \sin (2 (c+d x))+183680 a^2 b^5 \sin (4 (c+d x))-85120 a^2 b^5 \sin (6 (c+d x))+15680 a^2 b^5 \sin (8 (c+d x))-17080 a^5 b^2 \cos (5 (c+d x))+9240 a^5 b^2 \cos (7 (c+d x))+62160 a^3 b^4 \cos (5 (c+d x))-15120 a^3 b^4 \cos (7 (c+d x))-35 a \left (680 a^4 b^2-1392 a^2 b^4+1765 a^6+960 b^6\right ) \cos (c+d x)-35 \left (-904 a^5 b^2+2736 a^3 b^4+895 a^7-1728 a b^6\right ) \cos (3 (c+d x))+13440 a^6 b \sin (2 (c+d x))+13440 a^6 b \sin (4 (c+d x))+5760 a^6 b \sin (6 (c+d x))+960 a^6 b \sin (8 (c+d x))-13895 a^7 \cos (5 (c+d x))-525 a^7 \cos (7 (c+d x))-33600 a b^6 \cos (5 (c+d x))+6720 a b^6 \cos (7 (c+d x))+94080 b^7 \sin (2 (c+d x))-94080 b^7 \sin (4 (c+d x))+40320 b^7 \sin (6 (c+d x))-6720 b^7 \sin (8 (c+d x))\right )}{860160 a^9 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.135, size = 1143, normalized size = 2.4 \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 = 8.86559, size = 4911, normalized size = 10.32 \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 [B] time = 1.30762, size = 1280, normalized size = 2.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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