Optimal. Leaf size=49 \[ -\frac {2 b \log (\tan (x))}{a^3}+\frac {2 b \log (a+b \tan (x))}{a^3}-\frac {\frac {b}{a^2}+\frac {1}{b}}{a+b \tan (x)}-\frac {\cot (x)}{a^2} \]
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Rubi [A] time = 0.08, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3087, 894} \[ -\frac {\frac {b}{a^2}+\frac {1}{b}}{a+b \tan (x)}-\frac {2 b \log (\tan (x))}{a^3}+\frac {2 b \log (a+b \tan (x))}{a^3}-\frac {\cot (x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3087
Rubi steps
\begin {align*} \int \frac {\csc ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2}{x^2 (a+b x)^2} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {a^2+b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx,x,\tan (x)\right )\\ &=-\frac {\cot (x)}{a^2}-\frac {2 b \log (\tan (x))}{a^3}+\frac {2 b \log (a+b \tan (x))}{a^3}-\frac {\frac {1}{b}+\frac {b}{a^2}}{a+b \tan (x)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 76, normalized size = 1.55 \[ \frac {a^2 \left (-\cot ^2(x)\right )+a^2+2 b^2 \log (a \cos (x)+b \sin (x))-a b \cot (x) (-2 \log (a \cos (x)+b \sin (x))+2 \log (\sin (x))+1)-2 b^2 \log (\sin (x))+b^2}{a^3 (a \cot (x)+b)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 134, normalized size = 2.73 \[ \frac {2 \, a^{2} \cos \relax (x)^{2} + 2 \, a b \cos \relax (x) \sin \relax (x) - a^{2} + {\left (b^{2} \cos \relax (x)^{2} - a b \cos \relax (x) \sin \relax (x) - b^{2}\right )} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right ) - {\left (b^{2} \cos \relax (x)^{2} - a b \cos \relax (x) \sin \relax (x) - b^{2}\right )} \log \left (-\frac {1}{4} \, \cos \relax (x)^{2} + \frac {1}{4}\right )}{a^{3} b \cos \relax (x)^{2} - a^{4} \cos \relax (x) \sin \relax (x) - a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 63, normalized size = 1.29 \[ \frac {2 \, b \log \left ({\left | b \tan \relax (x) + a \right |}\right )}{a^{3}} - \frac {2 \, b \log \left ({\left | \tan \relax (x) \right |}\right )}{a^{3}} - \frac {a^{2} \tan \relax (x) + 2 \, b^{2} \tan \relax (x) + a b}{{\left (b \tan \relax (x)^{2} + a \tan \relax (x)\right )} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 60, normalized size = 1.22 \[ -\frac {1}{b \left (a +b \tan \relax (x )\right )}-\frac {b}{a^{2} \left (a +b \tan \relax (x )\right )}+\frac {2 b \ln \left (a +b \tan \relax (x )\right )}{a^{3}}-\frac {1}{a^{2} \tan \relax (x )}-\frac {2 b \ln \left (\tan \relax (x )\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 62, normalized size = 1.27 \[ -\frac {a b + {\left (a^{2} + 2 \, b^{2}\right )} \tan \relax (x)}{a^{2} b^{2} \tan \relax (x)^{2} + a^{3} b \tan \relax (x)} + \frac {2 \, b \log \left (b \tan \relax (x) + a\right )}{a^{3}} - \frac {2 \, b \log \left (\tan \relax (x)\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 114, normalized size = 2.33 \[ \frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a^2}-\frac {a+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (5\,a^2+4\,b^2\right )}{a}}{-2\,a^3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+2\,a^3\,\mathrm {tan}\left (\frac {x}{2}\right )+4\,b\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}+\frac {2\,b\,\ln \left (-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a\right )}{a^3}-\frac {2\,b\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^3} \]
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 \[ \int \frac {\csc ^{2}{\relax (x )}}{\left (a \cos {\relax (x )} + b \sin {\relax (x )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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