Optimal. Leaf size=61 \[ -\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{5/2}}-\frac {\cot ^3(x)}{3 (a+b)}-\frac {(a+2 b) \cot (x)}{(a+b)^2} \]
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Rubi [A] time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3191, 390, 205} \[ -\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{5/2}}-\frac {\cot ^3(x)}{3 (a+b)}-\frac {(a+2 b) \cot (x)}{(a+b)^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 390
Rule 3191
Rubi steps
\begin {align*} \int \frac {\csc ^4(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{a+(a+b) x^2} \, dx,x,\cot (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a+2 b}{(a+b)^2}+\frac {x^2}{a+b}+\frac {b^2}{(a+b)^2 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\cot (x)\right )\\ &=-\frac {(a+2 b) \cot (x)}{(a+b)^2}-\frac {\cot ^3(x)}{3 (a+b)}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{(a+b)^2}\\ &=-\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{5/2}}-\frac {(a+2 b) \cot (x)}{(a+b)^2}-\frac {\cot ^3(x)}{3 (a+b)}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 59, normalized size = 0.97 \[ \frac {b^2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{5/2}}-\frac {\cot (x) \left ((a+b) \csc ^2(x)+2 a+5 b\right )}{3 (a+b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 396, normalized size = 6.49 \[ \left [\frac {4 \, {\left (2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2}\right )} \cos \relax (x)^{3} + 3 \, {\left (b^{2} \cos \relax (x)^{2} - b^{2}\right )} \sqrt {-a^{2} - a b} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \relax (x)^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \relax (x)^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \relax (x)^{3} - a \cos \relax (x)\right )} \sqrt {-a^{2} - a b} \sin \relax (x) + a^{2}}{b^{2} \cos \relax (x)^{4} + 2 \, a b \cos \relax (x)^{2} + a^{2}}\right ) \sin \relax (x) - 12 \, {\left (a^{3} + 3 \, a^{2} b + 2 \, a b^{2}\right )} \cos \relax (x)}{12 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} - {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}, \frac {2 \, {\left (2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2}\right )} \cos \relax (x)^{3} + 3 \, {\left (b^{2} \cos \relax (x)^{2} - b^{2}\right )} \sqrt {a^{2} + a b} \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \relax (x)^{2} - a}{2 \, \sqrt {a^{2} + a b} \cos \relax (x) \sin \relax (x)}\right ) \sin \relax (x) - 6 \, {\left (a^{3} + 3 \, a^{2} b + 2 \, a b^{2}\right )} \cos \relax (x)}{6 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} - {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 90, normalized size = 1.48 \[ \frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \relax (x)}{\sqrt {a^{2} + a b}}\right )\right )} b^{2}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a^{2} + a b}} - \frac {3 \, a \tan \relax (x)^{2} + 6 \, b \tan \relax (x)^{2} + a + b}{3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 65, normalized size = 1.07 \[ -\frac {1}{3 \left (a +b \right ) \tan \relax (x )^{3}}-\frac {a}{\left (a +b \right )^{2} \tan \relax (x )}-\frac {2 b}{\left (a +b \right )^{2} \tan \relax (x )}+\frac {b^{2} \arctan \left (\frac {a \tan \relax (x )}{\sqrt {\left (a +b \right ) a}}\right )}{\left (a +b \right )^{2} \sqrt {\left (a +b \right ) a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 70, normalized size = 1.15 \[ \frac {b^{2} \arctan \left (\frac {a \tan \relax (x)}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {3 \, {\left (a + 2 \, b\right )} \tan \relax (x)^{2} + a + b}{3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.34, size = 67, normalized size = 1.10 \[ \frac {b^2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\mathrm {tan}\relax (x)\,\left (a^2+2\,a\,b+b^2\right )}{{\left (a+b\right )}^{5/2}}\right )}{\sqrt {a}\,{\left (a+b\right )}^{5/2}}-\frac {\frac {1}{3\,\left (a+b\right )}+\frac {{\mathrm {tan}\relax (x)}^2\,\left (a+2\,b\right )}{{\left (a+b\right )}^2}}{{\mathrm {tan}\relax (x)}^3} \]
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 \[ \int \frac {\csc ^{4}{\relax (x )}}{a + b \cos ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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