Optimal. Leaf size=92 \[ -\frac{2 b \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^2 d \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.242228, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {2889, 3056, 12, 2747, 3770, 2660, 618, 204} \[ -\frac{2 b \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^2 d \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2889
Rule 3056
Rule 12
Rule 2747
Rule 3770
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\int \frac{\csc (c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx\\ &=\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))}+\frac{\int \frac{\left (a^2-b^2\right ) \csc (c+d x)}{a+b \sin (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))}+\frac{\int \frac{\csc (c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))}+\frac{\int \csc (c+d x) \, dx}{a^2}-\frac{b \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^2}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))}-\frac{(2 b) \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 d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))}+\frac{(4 b) \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 d}\\ &=-\frac{2 b \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^2 \sqrt{a^2-b^2} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^2 d}+\frac{\cos (c+d x)}{a d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.211391, size = 97, normalized size = 1.05 \[ \frac{-\frac{2 b \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}+\frac{a \cos (c+d x)}{a+b \sin (c+d x)}+\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.141, size = 153, normalized size = 1.7 \begin{align*} 2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) b}{d{a}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}+2\,{\frac{1}{da \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}-2\,{\frac{b}{d{a}^{2}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }+{\frac{1}{d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \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 = 2.02589, size = 1111, normalized size = 12.08 \begin{align*} \left [-\frac{{\left (b^{2} \sin \left (d x + c\right ) + a b\right )} \sqrt{-a^{2} + b^{2}} \log \left (-\frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \,{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \,{\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right ) +{\left (a^{3} - a b^{2} +{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (a^{3} - a b^{2} +{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left ({\left (a^{4} b - a^{2} b^{3}\right )} d \sin \left (d x + c\right ) +{\left (a^{5} - a^{3} b^{2}\right )} d\right )}}, \frac{2 \,{\left (b^{2} \sin \left (d x + c\right ) + a b\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \sin \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 2 \,{\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right ) -{\left (a^{3} - a b^{2} +{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (a^{3} - a b^{2} +{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left ({\left (a^{4} b - a^{2} b^{3}\right )} d \sin \left (d x + c\right ) +{\left (a^{5} - a^{3} b^{2}\right )} d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos ^{2}{\left (c + d x \right )} \csc{\left (c + d x \right )}}{\left (a + b \sin{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22913, size = 176, normalized size = 1.91 \begin{align*} -\frac{\frac{2 \,{\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 )} b}{\sqrt{a^{2} - b^{2}} a^{2}} - \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{2 \,{\left (b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a\right )}}{{\left (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^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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