Optimal. Leaf size=84 \[ -\frac {\left (2 a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}+\frac {2 a^3 \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2}}+\frac {a \cot (x)}{b^2}-\frac {\cot (x) \csc (x)}{2 b} \]
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Rubi [A] time = 0.25, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3851, 4082, 3998, 3770, 3831, 2660, 618, 206} \[ -\frac {\left (2 a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}+\frac {2 a^3 \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2}}+\frac {a \cot (x)}{b^2}-\frac {\cot (x) \csc (x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 3770
Rule 3831
Rule 3851
Rule 3998
Rule 4082
Rubi steps
\begin {align*} \int \frac {\csc ^4(x)}{a+b \csc (x)} \, dx &=-\frac {\cot (x) \csc (x)}{2 b}+\frac {\int \frac {\csc (x) \left (a+b \csc (x)-2 a \csc ^2(x)\right )}{a+b \csc (x)} \, dx}{2 b}\\ &=\frac {a \cot (x)}{b^2}-\frac {\cot (x) \csc (x)}{2 b}+\frac {\int \frac {\csc (x) \left (a b+\left (2 a^2+b^2\right ) \csc (x)\right )}{a+b \csc (x)} \, dx}{2 b^2}\\ &=\frac {a \cot (x)}{b^2}-\frac {\cot (x) \csc (x)}{2 b}-\frac {a^3 \int \frac {\csc (x)}{a+b \csc (x)} \, dx}{b^3}+\frac {\left (2 a^2+b^2\right ) \int \csc (x) \, dx}{2 b^3}\\ &=-\frac {\left (2 a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}+\frac {a \cot (x)}{b^2}-\frac {\cot (x) \csc (x)}{2 b}-\frac {a^3 \int \frac {1}{1+\frac {a \sin (x)}{b}} \, dx}{b^4}\\ &=-\frac {\left (2 a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}+\frac {a \cot (x)}{b^2}-\frac {\cot (x) \csc (x)}{2 b}-\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^4}\\ &=-\frac {\left (2 a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}+\frac {a \cot (x)}{b^2}-\frac {\cot (x) \csc (x)}{2 b}+\frac {\left (4 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (1-\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}+2 \tan \left (\frac {x}{2}\right )\right )}{b^4}\\ &=-\frac {\left (2 a^2+b^2\right ) \tanh ^{-1}(\cos (x))}{2 b^3}+\frac {2 a^3 \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2}}+\frac {a \cot (x)}{b^2}-\frac {\cot (x) \csc (x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 144, normalized size = 1.71 \[ \frac {8 a^2 \log \left (\sin \left (\frac {x}{2}\right )\right )-8 a^2 \log \left (\cos \left (\frac {x}{2}\right )\right )-\frac {16 a^3 \tan ^{-1}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}-4 a b \tan \left (\frac {x}{2}\right )+4 a b \cot \left (\frac {x}{2}\right )-b^2 \csc ^2\left (\frac {x}{2}\right )+b^2 \sec ^2\left (\frac {x}{2}\right )+4 b^2 \log \left (\sin \left (\frac {x}{2}\right )\right )-4 b^2 \log \left (\cos \left (\frac {x}{2}\right )\right )}{8 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 524, normalized size = 6.24 \[ \left [\frac {4 \, {\left (a^{3} b - a b^{3}\right )} \cos \relax (x) \sin \relax (x) - 2 \, {\left (a^{3} \cos \relax (x)^{2} - a^{3}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \relax (x)^{2} + 2 \, a b \sin \relax (x) + a^{2} + b^{2} + 2 \, {\left (b \cos \relax (x) \sin \relax (x) + a \cos \relax (x)\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{2} b^{2} - b^{4}\right )} \cos \relax (x) - {\left (2 \, a^{4} - a^{2} b^{2} - b^{4} - {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4} - {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{4 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{2} b^{3} - b^{5}\right )} \cos \relax (x)^{2}\right )}}, \frac {4 \, {\left (a^{3} b - a b^{3}\right )} \cos \relax (x) \sin \relax (x) - 4 \, {\left (a^{3} \cos \relax (x)^{2} - a^{3}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \relax (x) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \relax (x)}\right ) - 2 \, {\left (a^{2} b^{2} - b^{4}\right )} \cos \relax (x) - {\left (2 \, a^{4} - a^{2} b^{2} - b^{4} - {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4} - {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} \cos \relax (x)^{2}\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right )}{4 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{2} b^{3} - b^{5}\right )} \cos \relax (x)^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 141, normalized size = 1.68 \[ -\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, x\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )} a^{3}}{\sqrt {-a^{2} + b^{2}} b^{3}} + \frac {b \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, a \tan \left (\frac {1}{2} \, x\right )}{8 \, b^{2}} + \frac {{\left (2 \, a^{2} + b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{2 \, b^{3}} - \frac {12 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 6 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, a b \tan \left (\frac {1}{2} \, x\right ) + b^{2}}{8 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 112, normalized size = 1.33 \[ \frac {\tan ^{2}\left (\frac {x}{2}\right )}{8 b}-\frac {a \tan \left (\frac {x}{2}\right )}{2 b^{2}}-\frac {2 a^{3} \arctan \left (\frac {2 \tan \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{b^{3} \sqrt {-a^{2}+b^{2}}}-\frac {1}{8 b \tan \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right ) a^{2}}{b^{3}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{2 b}+\frac {a}{2 b^{2} \tan \left (\frac {x}{2}\right )} \]
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 = 0.58, size = 515, normalized size = 6.13 \[ -\frac {b^2\,\left (\frac {\cos \relax (x)\,\sqrt {a^2-b^2}}{2}-\frac {\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )\,\sqrt {a^2-b^2}}{4}+\frac {\cos \left (2\,x\right )\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )\,\sqrt {a^2-b^2}}{4}\right )-\frac {a^2\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )\,\sqrt {a^2-b^2}}{2}-\frac {a\,b\,\sin \left (2\,x\right )\,\sqrt {a^2-b^2}}{2}+\frac {a^2\,\cos \left (2\,x\right )\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )\,\sqrt {a^2-b^2}}{2}+a^3\,\mathrm {atan}\left (\frac {a^4\,\sin \left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,8{}\mathrm {i}-b^4\,\sin \left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,1{}\mathrm {i}+a\,b^3\,\cos \left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,1{}\mathrm {i}+a^3\,b\,\cos \left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,4{}\mathrm {i}}{-8\,\sin \left (\frac {x}{2}\right )\,a^5-4\,\cos \left (\frac {x}{2}\right )\,a^4\,b+4\,\sin \left (\frac {x}{2}\right )\,a^3\,b^2+\cos \left (\frac {x}{2}\right )\,a^2\,b^3+2\,\sin \left (\frac {x}{2}\right )\,a\,b^4+\cos \left (\frac {x}{2}\right )\,b^5}\right )\,1{}\mathrm {i}-a^3\,\cos \left (2\,x\right )\,\mathrm {atan}\left (\frac {a^4\,\sin \left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,8{}\mathrm {i}-b^4\,\sin \left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,1{}\mathrm {i}+a\,b^3\,\cos \left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,1{}\mathrm {i}+a^3\,b\,\cos \left (\frac {x}{2}\right )\,\sqrt {a^2-b^2}\,4{}\mathrm {i}}{-8\,\sin \left (\frac {x}{2}\right )\,a^5-4\,\cos \left (\frac {x}{2}\right )\,a^4\,b+4\,\sin \left (\frac {x}{2}\right )\,a^3\,b^2+\cos \left (\frac {x}{2}\right )\,a^2\,b^3+2\,\sin \left (\frac {x}{2}\right )\,a\,b^4+\cos \left (\frac {x}{2}\right )\,b^5}\right )\,1{}\mathrm {i}}{\frac {b^3\,\sqrt {a^2-b^2}}{2}-\frac {b^3\,\cos \left (2\,x\right )\,\sqrt {a^2-b^2}}{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 \[ \int \frac {\csc ^{4}{\relax (x )}}{a + b \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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