Optimal. Leaf size=251 \[ -\frac{6 \left (a^2+b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^4 b^3 d}+\frac{2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^2 b^3 d}+\frac{\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac{3 \left (a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{2 a x}{b^3}+\frac{\cos (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.335801, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.379, Rules used = {2897, 3770, 3767, 8, 3768, 2638, 2664, 12, 2660, 618, 204} \[ -\frac{6 \left (a^2+b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^4 b^3 d}+\frac{2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^2 b^3 d}+\frac{\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac{3 \left (a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{2 a x}{b^3}+\frac{\cos (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 2897
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2638
Rule 2664
Rule 12
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\int \left (\frac{2 a}{b^3}-\frac{3 \left (a^2-b^2\right ) \csc (c+d x)}{a^4}-\frac{2 b \csc ^2(c+d x)}{a^3}+\frac{\csc ^3(c+d x)}{a^2}-\frac{\sin (c+d x)}{b^2}+\frac{\left (a^2-b^2\right )^3}{a^3 b^3 (a+b \sin (c+d x))^2}-\frac{3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )}{a^4 b^3 (a+b \sin (c+d x))}\right ) \, dx\\ &=\frac{2 a x}{b^3}+\frac{\int \csc ^3(c+d x) \, dx}{a^2}-\frac{\int \sin (c+d x) \, dx}{b^2}-\frac{(2 b) \int \csc ^2(c+d x) \, dx}{a^3}-\frac{\left (3 \left (a^2-b^2\right )\right ) \int \csc (c+d x) \, dx}{a^4}+\frac{\left (a^2-b^2\right )^3 \int \frac{1}{(a+b \sin (c+d x))^2} \, dx}{a^3 b^3}-\frac{\left (3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^4 b^3}\\ &=\frac{2 a x}{b^3}+\frac{3 \left (a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\cos (c+d x)}{b^2 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac{\int \csc (c+d x) \, dx}{2 a^2}+\frac{\left (a^2-b^2\right )^2 \int \frac{a}{a+b \sin (c+d x)} \, dx}{a^3 b^3}+\frac{(2 b) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}-\frac{\left (6 \left (a^2-b^2\right )^2 \left (a^2+b^2\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^4 b^3 d}\\ &=\frac{2 a x}{b^3}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac{3 \left (a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\cos (c+d x)}{b^2 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac{\left (a^2-b^2\right )^2 \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^2 b^3}+\frac{\left (12 \left (a^2-b^2\right )^2 \left (a^2+b^2\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^4 b^3 d}\\ &=\frac{2 a x}{b^3}-\frac{6 \left (a^2-b^2\right )^{3/2} \left (a^2+b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^4 b^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac{3 \left (a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\cos (c+d x)}{b^2 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac{\left (2 \left (a^2-b^2\right )^2\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^2 b^3 d}\\ &=\frac{2 a x}{b^3}-\frac{6 \left (a^2-b^2\right )^{3/2} \left (a^2+b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^4 b^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac{3 \left (a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\cos (c+d x)}{b^2 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}-\frac{\left (4 \left (a^2-b^2\right )^2\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^2 b^3 d}\\ &=\frac{2 a x}{b^3}+\frac{2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^2 b^3 d}-\frac{6 \left (a^2-b^2\right )^{3/2} \left (a^2+b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^4 b^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a^2 d}+\frac{3 \left (a^2-b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac{\cos (c+d x)}{b^2 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac{\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 6.19018, size = 315, normalized size = 1.25 \[ \frac{\left (6 b^2-5 a^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^4 d}+\frac{\left (5 a^2-6 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^4 d}+\frac{-2 a^2 b^2 \cos (c+d x)+a^4 \cos (c+d x)+b^4 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}-\frac{2 \left (a^2-b^2\right )^{3/2} \left (2 a^2+3 b^2\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (a \sin \left (\frac{1}{2} (c+d x)\right )+b \cos \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2-b^2}}\right )}{a^4 b^3 d}-\frac{b \tan \left (\frac{1}{2} (c+d x)\right )}{a^3 d}+\frac{b \cot \left (\frac{1}{2} (c+d x)\right )}{a^3 d}-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 a^2 d}+\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 a^2 d}+\frac{2 a (c+d x)}{b^3 d}+\frac{\cos (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.19, size = 618, 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 = 5.47255, size = 2700, normalized size = 10.76 \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 = 1.27648, size = 625, normalized size = 2.49 \begin{align*} \frac{\frac{16 \,{\left (d x + c\right )} a}{b^{3}} - \frac{4 \,{\left (5 \, a^{2} - 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{4}} + \frac{30 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 36 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{2}}{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} - \frac{16 \,{\left (2 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} + 3 \, b^{6}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{\sqrt{a^{2} - b^{2}} a^{4} b^{3}} + \frac{16 \,{\left (a^{4} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 2 \, a^{3} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a^{4} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, a^{2} b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a\right )} a^{4} b^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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