Optimal. Leaf size=123 \[ \frac {2 b \tanh ^{-1}(\cos (x))}{a^3}-\frac {\left (a^2-2 b^2\right ) \cot (x)}{a^2 \left (a^2-b^2\right )}-\frac {b^2 \cot (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac {2 b^2 \left (3 a^2-2 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{3/2}} \]
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Rubi [A] time = 0.33, antiderivative size = 123, normalized size of antiderivative = 1.00, 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 204
Rule 618
Rule 2660
Rule 2802
Rule 3001
Rule 3055
Rule 3770
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*}
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Mathematica [A] time = 0.72, 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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fricas [B] time = 0.84, size = 784, normalized size = 6.37 \[ \left [-\frac {2 \, {\left (a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \cos \relax (x) \sin \relax (x) + {\left (3 \, a^{2} b^{3} - 2 \, b^{5} - {\left (3 \, a^{2} b^{3} - 2 \, b^{5}\right )} \cos \relax (x)^{2} + {\left (3 \, a^{3} b^{2} - 2 \, a b^{4}\right )} \sin \relax (x)\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2} + 2 \, {\left (a \cos \relax (x) \sin \relax (x) + b \cos \relax (x)\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2}}\right ) + 2 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \relax (x) - 2 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6} - {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \relax (x)^{2} + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \relax (x)\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 2 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6} - {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \relax (x)^{2} + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \relax (x)\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{2 \, {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5} - {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \relax (x)^{2} + {\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \sin \relax (x)\right )}}, -\frac {{\left (a^{5} b - 3 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \cos \relax (x) \sin \relax (x) + {\left (3 \, a^{2} b^{3} - 2 \, b^{5} - {\left (3 \, a^{2} b^{3} - 2 \, b^{5}\right )} \cos \relax (x)^{2} + {\left (3 \, a^{3} b^{2} - 2 \, a b^{4}\right )} \sin \relax (x)\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \relax (x) + b}{\sqrt {a^{2} - b^{2}} \cos \relax (x)}\right ) + {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \cos \relax (x) - {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6} - {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \relax (x)^{2} + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \relax (x)\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6} - {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \cos \relax (x)^{2} + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \relax (x)\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5} - {\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \relax (x)^{2} + {\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \sin \relax (x)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 234, normalized size = 1.90 \[ \frac {2 \, {\left (3 \, a^{2} b^{2} - 2 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {4 \, a^{3} b \tan \left (\frac {1}{2} \, x\right )^{3} - 4 \, a b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} - 3 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 11 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 4 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, a^{3} b \tan \left (\frac {1}{2} \, x\right ) + 14 \, a b^{3} \tan \left (\frac {1}{2} \, x\right ) - 3 \, a^{4} + 3 \, a^{2} b^{2}}{6 \, {\left (a^{5} - a^{3} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, x\right )^{2} + a \tan \left (\frac {1}{2} \, x\right )\right )}} - \frac {2 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} + \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 201, normalized size = 1.63 \[ \frac {\tan \left (\frac {x}{2}\right )}{2 a^{2}}-\frac {1}{2 a^{2} \tan \left (\frac {x}{2}\right )}-\frac {2 b \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}+\frac {2 b^{4} \tan \left (\frac {x}{2}\right )}{a^{3} \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right ) \left (a^{2}-b^{2}\right )}+\frac {2 b^{3}}{a^{2} \left (\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) a +2 \tan \left (\frac {x}{2}\right ) b +a \right ) \left (a^{2}-b^{2}\right )}+\frac {6 b^{2} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}-\frac {4 b^{4} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{3} \left (a^{2}-b^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.63, size = 1471, normalized size = 11.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\relax (x )}}{\left (a + b \sin {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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