Optimal. Leaf size=303 \[ -\frac{a \left (2 a^2+21 b^2\right ) \cot (c+d x)}{35 d}-\frac{3 b \left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{\left (-19 a^2 b^2+4 a^4+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}-\frac{b \left (-116 a^2 b^2+105 a^4+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}+\frac{\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d} \]
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Rubi [A] time = 0.860202, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2893, 3047, 3031, 3021, 2748, 3767, 8, 3770} \[ -\frac{a \left (2 a^2+21 b^2\right ) \cot (c+d x)}{35 d}-\frac{3 b \left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{\left (-19 a^2 b^2+4 a^4+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}-\frac{b \left (-116 a^2 b^2+105 a^4+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}+\frac{\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d} \]
Antiderivative was successfully verified.
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Rule 2893
Rule 3047
Rule 3031
Rule 3021
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \cot ^4(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}-\frac{\int \csc ^6(c+d x) (a+b \sin (c+d x))^3 \left (6 \left (8 a^2-b^2\right )+3 a b \sin (c+d x)-3 \left (14 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{42 a^2}\\ &=\frac{\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}-\frac{\int \csc ^5(c+d x) (a+b \sin (c+d x))^2 \left (3 b \left (53 a^2-6 b^2\right )-6 a \left (3 a^2-b^2\right ) \sin (c+d x)-9 b \left (18 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{210 a^2}\\ &=\frac{b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}-\frac{\int \csc ^4(c+d x) (a+b \sin (c+d x)) \left (-18 \left (4 a^4-19 a^2 b^2+2 b^4\right )-3 a b \left (81 a^2-2 b^2\right ) \sin (c+d x)-3 b^2 \left (163 a^2-6 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{840 a^2}\\ &=-\frac{\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac{b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}+\frac{\int \csc ^3(c+d x) \left (9 b \left (105 a^4-116 a^2 b^2+12 b^4\right )+72 a^3 \left (2 a^2+21 b^2\right ) \sin (c+d x)+9 b^3 \left (163 a^2-6 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{2520 a^2}\\ &=-\frac{b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac{\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac{b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}+\frac{\int \csc ^2(c+d x) \left (144 a^3 \left (2 a^2+21 b^2\right )+945 a^2 b \left (a^2+2 b^2\right ) \sin (c+d x)\right ) \, dx}{5040 a^2}\\ &=-\frac{b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac{\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac{b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}+\frac{1}{16} \left (3 b \left (a^2+2 b^2\right )\right ) \int \csc (c+d x) \, dx+\frac{1}{35} \left (a \left (2 a^2+21 b^2\right )\right ) \int \csc ^2(c+d x) \, dx\\ &=-\frac{3 b \left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac{\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac{b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}-\frac{\left (a \left (2 a^2+21 b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{35 d}\\ &=-\frac{3 b \left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \left (2 a^2+21 b^2\right ) \cot (c+d x)}{35 d}-\frac{b \left (105 a^4-116 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{560 a^2 d}-\frac{\left (4 a^4-19 a^2 b^2+2 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{140 a d}+\frac{b \left (53 a^2-6 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{280 a^2 d}+\frac{\left (8 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^3}{35 a^2 d}+\frac{b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^4}{14 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^4}{7 a d}\\ \end{align*}
Mathematica [A] time = 0.934619, size = 324, normalized size = 1.07 \[ -\frac{56 a \left (14 a^2-3 b^2\right ) \cos (3 (c+d x)) \csc ^7(c+d x)+70 \cot (c+d x) \csc ^6(c+d x) \left (b \left (31 a^2-18 b^2\right ) \sin (c+d x)+12 a \left (2 a^2+b^2\right )\right )-3360 a^2 b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+3360 a^2 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+1540 a^2 b \sin (4 (c+d x)) \csc ^7(c+d x)+105 a^2 b \sin (6 (c+d x)) \csc ^7(c+d x)+112 a^3 \cos (5 (c+d x)) \csc ^7(c+d x)-16 a^3 \cos (7 (c+d x)) \csc ^7(c+d x)-504 a b^2 \cos (5 (c+d x)) \csc ^7(c+d x)-168 a b^2 \cos (7 (c+d x)) \csc ^7(c+d x)-6720 b^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+6720 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+840 b^3 \sin (4 (c+d x)) \csc ^7(c+d x)-350 b^3 \sin (6 (c+d x)) \csc ^7(c+d x)}{17920 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.116, size = 309, normalized size = 1. \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{2\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{16\,d}}+{\frac{3\,{a}^{2}b\cos \left ( dx+c \right ) }{16\,d}}+{\frac{3\,{a}^{2}b\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{3\,a{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{8\,d}}+{\frac{3\,{b}^{3}\cos \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{b}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05677, size = 281, normalized size = 0.93 \begin{align*} \frac{35 \, a^{2} b{\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 )} - 70 \, b^{3}{\left (\frac{2 \,{\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \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{672 \, a b^{2}}{\tan \left (d x + c\right )^{5}} - \frac{32 \,{\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{1120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88397, size = 837, normalized size = 2.76 \begin{align*} -\frac{32 \,{\left (2 \, a^{3} + 21 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} - 224 \,{\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \,{\left ({\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{2} b - 2 \, b^{3} + 3 \,{\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 105 \,{\left ({\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - a^{2} b - 2 \, b^{3} + 3 \,{\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 70 \,{\left ({\left (3 \, a^{2} b - 10 \, b^{3}\right )} \cos \left (d x + c\right )^{5} + 8 \,{\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\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 = 1.29045, size = 616, normalized size = 2.03 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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