Optimal. Leaf size=84 \[ \frac {b \cot (x)}{a^2}-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^3}-\frac {2 b^3 \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 \sqrt {a^2-b^2}}-\frac {\cot (x) \csc (x)}{2 a} \]
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Rubi [A] time = 0.27, antiderivative size = 84, 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} \[ -\frac {2 b^3 \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 \sqrt {a^2-b^2}}-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^3}+\frac {b \cot (x)}{a^2}-\frac {\cot (x) \csc (x)}{2 a} \]
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 ^3(x)}{a+b \sin (x)} \, dx &=-\frac {\cot (x) \csc (x)}{2 a}+\frac {\int \frac {\csc ^2(x) \left (-2 b+a \sin (x)+b \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{2 a}\\ &=\frac {b \cot (x)}{a^2}-\frac {\cot (x) \csc (x)}{2 a}+\frac {\int \frac {\csc (x) \left (a^2+2 b^2+a b \sin (x)\right )}{a+b \sin (x)} \, dx}{2 a^2}\\ &=\frac {b \cot (x)}{a^2}-\frac {\cot (x) \csc (x)}{2 a}-\frac {b^3 \int \frac {1}{a+b \sin (x)} \, dx}{a^3}+\frac {\left (a^2+2 b^2\right ) \int \csc (x) \, dx}{2 a^3}\\ &=-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^3}+\frac {b \cot (x)}{a^2}-\frac {\cot (x) \csc (x)}{2 a}-\frac {\left (2 b^3\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}\\ &=-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^3}+\frac {b \cot (x)}{a^2}-\frac {\cot (x) \csc (x)}{2 a}+\frac {\left (4 b^3\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}\\ &=-\frac {2 b^3 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \sqrt {a^2-b^2}}-\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (x))}{2 a^3}+\frac {b \cot (x)}{a^2}-\frac {\cot (x) \csc (x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 144, normalized size = 1.71 \[ \frac {-\frac {16 b^3 \tan ^{-1}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-a^2 \csc ^2\left (\frac {x}{2}\right )+a^2 \sec ^2\left (\frac {x}{2}\right )+4 a^2 \log \left (\sin \left (\frac {x}{2}\right )\right )-4 a^2 \log \left (\cos \left (\frac {x}{2}\right )\right )-4 a b \tan \left (\frac {x}{2}\right )+4 a b \cot \left (\frac {x}{2}\right )+8 b^2 \log \left (\sin \left (\frac {x}{2}\right )\right )-8 b^2 \log \left (\cos \left (\frac {x}{2}\right )\right )}{8 a^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 490, normalized size = 5.83 \[ \left [\frac {4 \, {\left (a^{3} b - a b^{3}\right )} \cos \relax (x) \sin \relax (x) + 2 \, {\left (b^{3} \cos \relax (x)^{2} - b^{3}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2} - 2 \, {\left (a \cos \relax (x) \sin \relax (x) + b \cos \relax (x)\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \relax (x) - {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4} - {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4} - {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{4 \, {\left (a^{5} - a^{3} b^{2} - {\left (a^{5} - a^{3} b^{2}\right )} \cos \relax (x)^{2}\right )}}, \frac {4 \, {\left (a^{3} b - a b^{3}\right )} \cos \relax (x) \sin \relax (x) - 4 \, {\left (b^{3} \cos \relax (x)^{2} - b^{3}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \relax (x) + b}{\sqrt {a^{2} - b^{2}} \cos \relax (x)}\right ) - 2 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \relax (x) - {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4} - {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4} - {\left (a^{4} + a^{2} b^{2} - 2 \, b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{4 \, {\left (a^{5} - a^{3} b^{2} - {\left (a^{5} - a^{3} b^{2}\right )} \cos \relax (x)^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 141, normalized size = 1.68 \[ -\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} b^{3}}{\sqrt {a^{2} - b^{2}} a^{3}} + \frac {a \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, x\right )}{8 \, a^{2}} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, a^{3}} - \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 12 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, a b \tan \left (\frac {1}{2} \, x\right ) + a^{2}}{8 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 112, normalized size = 1.33 \[ \frac {\tan ^{2}\left (\frac {x}{2}\right )}{8 a}-\frac {\tan \left (\frac {x}{2}\right ) b}{2 a^{2}}-\frac {1}{8 a \tan \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 a}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right ) b^{2}}{a^{3}}+\frac {b}{2 a^{2} \tan \left (\frac {x}{2}\right )}-\frac {2 b^{3} \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{3} \sqrt {a^{2}-b^{2}}} \]
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 = 7.39, size = 531, normalized size = 6.32 \[ \frac {a^4\,\left (\frac {\cos \relax (x)}{2}-\frac {\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{4}+\frac {\cos \left (2\,x\right )\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{4}\right )-a^2\,\left (\frac {b^2\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{4}+\frac {b^2\,\cos \relax (x)}{2}-\frac {b^2\,\cos \left (2\,x\right )\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{4}\right )+\frac {b^4\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{2}-b^3\,\mathrm {atan}\left (\frac {-a^4\,\sin \left (\frac {x}{2}\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}+b^4\,\sin \left (\frac {x}{2}\right )\,\sqrt {b^2-a^2}\,8{}\mathrm {i}+a\,b^3\,\cos \left (\frac {x}{2}\right )\,\sqrt {b^2-a^2}\,4{}\mathrm {i}+a^3\,b\,\cos \left (\frac {x}{2}\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}}{\cos \left (\frac {x}{2}\right )\,a^5+2\,\sin \left (\frac {x}{2}\right )\,a^4\,b+\cos \left (\frac {x}{2}\right )\,a^3\,b^2+4\,\sin \left (\frac {x}{2}\right )\,a^2\,b^3-4\,\cos \left (\frac {x}{2}\right )\,a\,b^4-8\,\sin \left (\frac {x}{2}\right )\,b^5}\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}-\frac {b^4\,\cos \left (2\,x\right )\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{2}+\frac {a\,b^3\,\sin \left (2\,x\right )}{2}-\frac {a^3\,b\,\sin \left (2\,x\right )}{2}+b^3\,\cos \left (2\,x\right )\,\mathrm {atan}\left (\frac {-a^4\,\sin \left (\frac {x}{2}\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}+b^4\,\sin \left (\frac {x}{2}\right )\,\sqrt {b^2-a^2}\,8{}\mathrm {i}+a\,b^3\,\cos \left (\frac {x}{2}\right )\,\sqrt {b^2-a^2}\,4{}\mathrm {i}+a^3\,b\,\cos \left (\frac {x}{2}\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}}{\cos \left (\frac {x}{2}\right )\,a^5+2\,\sin \left (\frac {x}{2}\right )\,a^4\,b+\cos \left (\frac {x}{2}\right )\,a^3\,b^2+4\,\sin \left (\frac {x}{2}\right )\,a^2\,b^3-4\,\cos \left (\frac {x}{2}\right )\,a\,b^4-8\,\sin \left (\frac {x}{2}\right )\,b^5}\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}}{\frac {a^5\,\cos \left (2\,x\right )}{2}-\frac {a^5}{2}+\frac {a^3\,b^2}{2}-\frac {a^3\,b^2\,\cos \left (2\,x\right )}{2}} \]
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 \[ \int \frac {\csc ^{3}{\relax (x )}}{a + b \sin {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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