Optimal. Leaf size=78 \[ -\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3670, 444, 51, 63, 208} \[ -\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 444
Rule 3670
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\left (a+b \cot ^2(x)\right )^{5/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{5/2}} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac {1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x) (a+b x)^{3/2}} \, dx,x,\cot ^2(x)\right )}{2 (a-b)}\\ &=-\frac {1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )}{2 (a-b)^2}\\ &=-\frac {1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cot ^2(x)}\right )}{(a-b)^2 b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{5/2}}-\frac {1}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}}-\frac {1}{(a-b)^2 \sqrt {a+b \cot ^2(x)}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 47, normalized size = 0.60 \[ -\frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b \cot ^2(x)+a}{a-b}\right )}{3 (a-b) \left (a+b \cot ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 627, normalized size = 8.04 \[ \left [\frac {3 \, {\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {a - b} \log \left (-\sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )} - {\left (a - b\right )} \cos \left (2 \, x\right ) + a\right ) - 4 \, {\left (2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + 2 \, a^{2} - a b - b^{2} - {\left (4 \, a^{2} - 5 \, a b + b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{6 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} \cos \left (2 \, x\right )\right )}}, \frac {3 \, {\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{a - b}\right ) - 2 \, {\left (2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + 2 \, a^{2} - a b - b^{2} - {\left (4 \, a^{2} - 5 \, a b + b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{3 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right )} \cos \left (2 \, x\right )\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 75, normalized size = 0.96 \[ -\frac {1}{\left (a -b \right )^{2} \sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}-\frac {1}{3 \left (a -b \right ) \left (a +b \left (\cot ^{2}\relax (x )\right )\right )^{\frac {3}{2}}}-\frac {\arctan \left (\frac {\sqrt {a +b \left (\cot ^{2}\relax (x )\right )}}{\sqrt {-a +b}}\right )}{\left (a -b \right )^{2} \sqrt {-a +b}} \]
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 = 4.46, size = 82, normalized size = 1.05 \[ \frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\relax (x)}^2+a}\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}{2\,{\left (a-b\right )}^{5/2}}\right )}{{\left (a-b\right )}^{5/2}}-\frac {\frac {1}{3\,\left (a-b\right )}+\frac {b\,{\mathrm {cot}\relax (x)}^2+a}{{\left (a-b\right )}^2}}{{\left (b\,{\mathrm {cot}\relax (x)}^2+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.77, size = 70, normalized size = 0.90 \[ - \frac {1}{3 \left (a - b\right ) \left (a + b \cot ^{2}{\relax (x )}\right )^{\frac {3}{2}}} - \frac {1}{\left (a - b\right )^{2} \sqrt {a + b \cot ^{2}{\relax (x )}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {a + b \cot ^{2}{\relax (x )}}}{\sqrt {- a + b}} \right )}}{\sqrt {- a + b} \left (a - b\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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