Optimal. Leaf size=38 \[ -\frac {\left (a^2+b^2\right ) \log (a+b \cot (x))}{b^3}+\frac {a \cot (x)}{b^2}-\frac {\cot ^2(x)}{2 b} \]
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Rubi [A] time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 697} \[ -\frac {\left (a^2+b^2\right ) \log (a+b \cot (x))}{b^3}+\frac {a \cot (x)}{b^2}-\frac {\cot ^2(x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 697
Rule 3506
Rubi steps
\begin {align*} \int \frac {\csc ^4(x)}{a+b \cot (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1+\frac {x^2}{b^2}}{a+x} \, dx,x,b \cot (x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b^2}+\frac {a^2+b^2}{b^2 (a+x)}\right ) \, dx,x,b \cot (x)\right )}{b}\\ &=\frac {a \cot (x)}{b^2}-\frac {\cot ^2(x)}{2 b}-\frac {\left (a^2+b^2\right ) \log (a+b \cot (x))}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 48, normalized size = 1.26 \[ \frac {2 \left (a^2+b^2\right ) (\log (\sin (x))-\log (a \sin (x)+b \cos (x)))+2 a b \cot (x)-b^2 \csc ^2(x)}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 118, normalized size = 3.11 \[ -\frac {2 \, a b \cos \relax (x) \sin \relax (x) - b^{2} + {\left ({\left (a^{2} + b^{2}\right )} \cos \relax (x)^{2} - a^{2} - b^{2}\right )} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}\right ) - {\left ({\left (a^{2} + b^{2}\right )} \cos \relax (x)^{2} - a^{2} - b^{2}\right )} \log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right )}{2 \, {\left (b^{3} \cos \relax (x)^{2} - b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.51, size = 78, normalized size = 2.05 \[ \frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | \tan \relax (x) \right |}\right )}{b^{3}} - \frac {{\left (a^{3} + a b^{2}\right )} \log \left ({\left | a \tan \relax (x) + b \right |}\right )}{a b^{3}} - \frac {3 \, a^{2} \tan \relax (x)^{2} + 3 \, b^{2} \tan \relax (x)^{2} - 2 \, a b \tan \relax (x) + b^{2}}{2 \, b^{3} \tan \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 64, normalized size = 1.68 \[ -\frac {\ln \left (a \tan \relax (x )+b \right ) a^{2}}{b^{3}}-\frac {\ln \left (a \tan \relax (x )+b \right )}{b}-\frac {1}{2 b \tan \relax (x )^{2}}+\frac {\ln \left (\tan \relax (x )\right ) a^{2}}{b^{3}}+\frac {\ln \left (\tan \relax (x )\right )}{b}+\frac {a}{b^{2} \tan \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 52, normalized size = 1.37 \[ -\frac {{\left (a^{2} + b^{2}\right )} \log \left (a \tan \relax (x) + b\right )}{b^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (\tan \relax (x)\right )}{b^{3}} + \frac {2 \, a \tan \relax (x) - b}{2 \, b^{2} \tan \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 44, normalized size = 1.16 \[ -\frac {\frac {1}{2\,b}-\frac {a\,\mathrm {tan}\relax (x)}{b^2}}{{\mathrm {tan}\relax (x)}^2}-\frac {2\,\mathrm {atanh}\left (\frac {2\,a\,\mathrm {tan}\relax (x)}{b}+1\right )\,\left (a^2+b^2\right )}{b^3} \]
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 \[ \int \frac {\csc ^{4}{\relax (x )}}{a + b \cot {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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