Optimal. Leaf size=261 \[ -\frac{\left (2 a^2+7 b^2\right ) \cot (c+d x)}{35 d}+\frac{b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}-\frac{\left (-18 a^2 b^2+3 a^4+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac{2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d} \]
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Rubi [A] time = 0.628624, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{\left (2 a^2+7 b^2\right ) \cot (c+d x)}{35 d}+\frac{b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}-\frac{\left (-18 a^2 b^2+3 a^4+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac{2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d} \]
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 ^4(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}-\frac{\int \csc ^6(c+d x) (a+b \sin (c+d x))^2 \left (12 \left (4 a^2-b^2\right )+2 a b \sin (c+d x)-2 \left (21 a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{42 a^2}\\ &=\frac{2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}-\frac{\int \csc ^5(c+d x) (a+b \sin (c+d x)) \left (2 b \left (53 a^2-12 b^2\right )-2 a \left (9 a^2-b^2\right ) \sin (c+d x)-2 b \left (57 a^2-8 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{210 a^2}\\ &=\frac{b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac{2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}+\frac{\int \csc ^4(c+d x) \left (24 \left (3 a^4-18 a^2 b^2+4 b^4\right )+210 a^3 b \sin (c+d x)+8 b^2 \left (57 a^2-8 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{840 a^2}\\ &=-\frac{\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac{b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac{2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}+\frac{\int \csc ^3(c+d x) \left (630 a^3 b+72 a^2 \left (2 a^2+7 b^2\right ) \sin (c+d x)\right ) \, dx}{2520 a^2}\\ &=-\frac{\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac{b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac{2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}+\frac{1}{4} (a b) \int \csc ^3(c+d x) \, dx+\frac{1}{35} \left (2 a^2+7 b^2\right ) \int \csc ^2(c+d x) \, dx\\ &=-\frac{a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac{b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac{2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}+\frac{1}{8} (a b) \int \csc (c+d x) \, dx-\frac{\left (2 a^2+7 b^2\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{35 d}\\ &=-\frac{a b \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{\left (2 a^2+7 b^2\right ) \cot (c+d x)}{35 d}-\frac{a b \cot (c+d x) \csc (c+d x)}{8 d}-\frac{\left (3 a^4-18 a^2 b^2+4 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^2 d}+\frac{b \left (53 a^2-12 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{420 a d}+\frac{2 \left (4 a^2-b^2\right ) \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2}{35 a^2 d}+\frac{2 b \cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^3}{21 a^2 d}-\frac{\cot (c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^3}{7 a d}\\ \end{align*}
Mathematica [A] time = 1.29396, size = 322, normalized size = 1.23 \[ -\frac{\csc ^7(c+d x) \left (840 \left (6 a^2+b^2\right ) \cos (c+d x)+168 \left (14 a^2-b^2\right ) \cos (3 (c+d x))+336 a^2 \cos (5 (c+d x))-48 a^2 \cos (7 (c+d x))+2170 a b \sin (2 (c+d x))+3080 a b \sin (4 (c+d x))+210 a b \sin (6 (c+d x))-3675 a b \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+2205 a b \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-735 a b \sin (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+105 a b \sin (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+3675 a b \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2205 a b \sin (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+735 a b \sin (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-105 a b \sin (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-504 b^2 \cos (5 (c+d x))-168 b^2 \cos (7 (c+d x))\right )}{53760 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 194, normalized size = 0.7 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{12\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{ab\cos \left ( dx+c \right ) }{8\,d}}+{\frac{ab\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}}-{\frac{{b}^{2} \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 = 1.112, size = 181, normalized size = 0.69 \begin{align*} \frac{35 \, a 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 )} - \frac{336 \, b^{2}}{\tan \left (d x + c\right )^{5}} - \frac{48 \,{\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}}}{1680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85664, size = 653, normalized size = 2.5 \begin{align*} -\frac{48 \,{\left (2 \, a^{2} + 7 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 336 \,{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \,{\left (a b \cos \left (d x + c\right )^{6} - 3 \, a b \cos \left (d x + c\right )^{4} + 3 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 105 \,{\left (a b \cos \left (d x + c\right )^{6} - 3 \, a b \cos \left (d x + c\right )^{4} + 3 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 70 \,{\left (3 \, a b \cos \left (d x + c\right )^{5} + 8 \, a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \,{\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 = 1.32978, size = 468, normalized size = 1.79 \begin{align*} \frac{15 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 70 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 21 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 84 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 210 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 105 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 420 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 210 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1680 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 315 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 840 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{4356 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 315 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 840 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 210 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 105 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 420 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 210 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 21 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 84 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 70 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7}}}{13440 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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