Optimal. Leaf size=123 \[ \frac{2 b^2 \left (3 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{3/2}}-\frac{\left (a^2-2 b^2\right ) \cot (x)}{a^2 \left (a^2-b^2\right )}-\frac{b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac{2 b \tanh ^{-1}(\cos (x))}{a^3} \]
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Rubi [A] time = 0.334821, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {2802, 3055, 3001, 3770, 2660, 618, 204} \[ \frac{2 b^2 \left (3 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{3/2}}-\frac{\left (a^2-2 b^2\right ) \cot (x)}{a^2 \left (a^2-b^2\right )}-\frac{b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac{2 b \tanh ^{-1}(\cos (x))}{a^3} \]
Antiderivative was successfully verified.
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Rule 2802
Rule 3055
Rule 3001
Rule 3770
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\csc ^2(x)}{(a+b \sin (x))^2} \, dx &=-\frac{b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac{\int \frac{\csc ^2(x) \left (a^2-2 b^2-a b \sin (x)+b^2 \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac{\left (a^2-2 b^2\right ) \cot (x)}{a^2 \left (a^2-b^2\right )}-\frac{b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac{\int \frac{\csc (x) \left (-2 b \left (a^2-b^2\right )+a b^2 \sin (x)\right )}{a+b \sin (x)} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=-\frac{\left (a^2-2 b^2\right ) \cot (x)}{a^2 \left (a^2-b^2\right )}-\frac{b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}-\frac{(2 b) \int \csc (x) \, dx}{a^3}+\frac{\left (b^2 \left (3 a^2-2 b^2\right )\right ) \int \frac{1}{a+b \sin (x)} \, dx}{a^3 \left (a^2-b^2\right )}\\ &=\frac{2 b \tanh ^{-1}(\cos (x))}{a^3}-\frac{\left (a^2-2 b^2\right ) \cot (x)}{a^2 \left (a^2-b^2\right )}-\frac{b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac{\left (2 b^2 \left (3 a^2-2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{a^3 \left (a^2-b^2\right )}\\ &=\frac{2 b \tanh ^{-1}(\cos (x))}{a^3}-\frac{\left (a^2-2 b^2\right ) \cot (x)}{a^2 \left (a^2-b^2\right )}-\frac{b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}-\frac{\left (4 b^2 \left (3 a^2-2 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{x}{2}\right )\right )}{a^3 \left (a^2-b^2\right )}\\ &=\frac{2 b^2 \left (3 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{3/2}}+\frac{2 b \tanh ^{-1}(\cos (x))}{a^3}-\frac{\left (a^2-2 b^2\right ) \cot (x)}{a^2 \left (a^2-b^2\right )}-\frac{b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\\ \end{align*}
Mathematica [A] time = 0.675647, size = 127, normalized size = 1.03 \[ \frac{\frac{4 b^2 \left (3 a^2-2 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{x}{2}\right )+b}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac{2 a b^3 \cos (x)}{(a-b) (a+b) (a+b \sin (x))}+a \tan \left (\frac{x}{2}\right )-a \cot \left (\frac{x}{2}\right )-4 b \log \left (\sin \left (\frac{x}{2}\right )\right )+4 b \log \left (\cos \left (\frac{x}{2}\right )\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 201, normalized size = 1.6 \begin{align*}{\frac{1}{2\,{a}^{2}}\tan \left ({\frac{x}{2}} \right ) }+2\,{\frac{{b}^{4}\tan \left ( x/2 \right ) }{{a}^{3} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}a+2\,\tan \left ( x/2 \right ) b+a \right ) \left ({a}^{2}-{b}^{2} \right ) }}+2\,{\frac{{b}^{3}}{{a}^{2} \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{2}a+2\,\tan \left ( x/2 \right ) b+a \right ) \left ({a}^{2}-{b}^{2} \right ) }}+6\,{\frac{{b}^{2}}{a \left ({a}^{2}-{b}^{2} \right ) ^{3/2}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{4}}{{a}^{3} \left ({a}^{2}-{b}^{2} \right ) ^{3/2}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-{\frac{1}{2\,{a}^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-2\,{\frac{b\ln \left ( \tan \left ( x/2 \right ) \right ) }{{a}^{3}}} \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 = 4.12804, size = 1775, normalized size = 14.43 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{2}{\left (x \right )}}{\left (a + b \sin{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.70443, size = 316, normalized size = 2.57 \begin{align*} \frac{2 \,{\left (3 \, a^{2} b^{2} - 2 \, b^{4}\right )}{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, x\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\right )} \sqrt{a^{2} - b^{2}}} + \frac{4 \, a^{3} b \tan \left (\frac{1}{2} \, x\right )^{3} - 4 \, a b^{3} \tan \left (\frac{1}{2} \, x\right )^{3} - 3 \, a^{4} \tan \left (\frac{1}{2} \, x\right )^{2} + 11 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, x\right )^{2} + 4 \, b^{4} \tan \left (\frac{1}{2} \, x\right )^{2} - 2 \, a^{3} b \tan \left (\frac{1}{2} \, x\right ) + 14 \, a b^{3} \tan \left (\frac{1}{2} \, x\right ) - 3 \, a^{4} + 3 \, a^{2} b^{2}}{6 \,{\left (a^{5} - a^{3} b^{2}\right )}{\left (a \tan \left (\frac{1}{2} \, x\right )^{3} + 2 \, b \tan \left (\frac{1}{2} \, x\right )^{2} + a \tan \left (\frac{1}{2} \, x\right )\right )}} - \frac{2 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac{\tan \left (\frac{1}{2} \, x\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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