Optimal. Leaf size=334 \[ -\frac{b \left (6 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac{3 a \left (a^2+8 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{\left (-148 a^2 b^2+35 a^4+24 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{2240 a d}-\frac{b \left (-25 a^2 b^2+24 a^4+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{280 a^2 d}-\frac{3 a \left (a^2+8 b^2\right ) \cot (c+d x) \csc (c+d x)}{128 d}+\frac{\left (21 a^2-4 b^2\right ) \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{112 a^2 d}+\frac{3 b \left (23 a^2-4 b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{560 a^2 d}+\frac{b \cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^7(c+d x) (a+b \sin (c+d x))^4}{8 a d} \]
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Rubi [A] time = 0.907272, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, 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 2893
Rule 3047
Rule 3031
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
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*}
Mathematica [A] time = 1.53012, size = 268, normalized size = 0.8 \[ -\frac{-6720 a \left (a^2+8 b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+6720 a \left (a^2+8 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\csc ^8(c+d x) \left (35 a \left (671 a^2+248 b^2\right ) \cos (c+d x)+35 \left (333 a^3+104 a b^2\right ) \cos (3 (c+d x))+21504 a^2 b \sin (2 (c+d x))+16128 a^2 b \sin (4 (c+d x))+3072 a^2 b \sin (6 (c+d x))-384 a^2 b \sin (8 (c+d x))+805 a^3 \cos (5 (c+d x))-105 a^3 \cos (7 (c+d x))-11480 a b^2 \cos (5 (c+d x))-840 a b^2 \cos (7 (c+d x))+2688 b^3 \sin (2 (c+d x))+896 b^3 \sin (4 (c+d x))-896 b^3 \sin (6 (c+d x))-448 b^3 \sin (8 (c+d x))\right )}{286720 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.117, size = 358, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{64\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{128\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{128\,d}}+{\frac{3\,{a}^{3}\cos \left ( dx+c \right ) }{128\,d}}+{\frac{3\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{3\,{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{6\,{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{16\,d}}+{\frac{3\,a{b}^{2}\cos \left ( dx+c \right ) }{16\,d}}+{\frac{3\,a{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05045, size = 335, normalized size = 1. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91738, size = 957, normalized size = 2.87 \begin{align*} \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 )}} \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 = 1.37819, size = 617, normalized size = 1.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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