Optimal. Leaf size=59 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {a}{(a-b) b \sqrt {a+b \cot ^2(x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3751, 457, 79,
65, 214} \begin {gather*} \frac {a}{b (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 79
Rule 214
Rule 457
Rule 3751
Rubi steps
\begin {align*} \int \frac {\cot ^3(x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx &=-\text {Subst}\left (\int \frac {x^3}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {x}{(1+x) (a+b x)^{3/2}} \, dx,x,\cot ^2(x)\right )\right )\\ &=\frac {a}{(a-b) b \sqrt {a+b \cot ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )}{2 (a-b)}\\ &=\frac {a}{(a-b) b \sqrt {a+b \cot ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cot ^2(x)}\right )}{(a-b) b}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {a}{(a-b) b \sqrt {a+b \cot ^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 59, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {a}{(a-b) b \sqrt {a+b \cot ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 68, normalized size = 1.15
method | result | size |
derivativedivides | \(\frac {1}{b \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}+\frac {\arctan \left (\frac {\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{\sqrt {-a +b}}\right )}{\left (a -b \right ) \sqrt {-a +b}}+\frac {1}{\left (a -b \right ) \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}\) | \(68\) |
default | \(\frac {1}{b \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}+\frac {\arctan \left (\frac {\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{\sqrt {-a +b}}\right )}{\left (a -b \right ) \sqrt {-a +b}}+\frac {1}{\left (a -b \right ) \sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}\) | \(68\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs.
\(2 (51) = 102\).
time = 2.49, size = 385, normalized size = 6.53 \begin {gather*} \left [-\frac {{\left (a b + b^{2} - {\left (a b - 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 ) - 2 \, {\left (a^{2} - a b - {\left (a^{2} - a b\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}}}{2 \, {\left (a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )\right )}}, -\frac {{\left (a b + b^{2} - {\left (a b - 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 ) - {\left (a^{2} - a b - {\left (a^{2} - a b\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}}}{a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\cot ^{3}{\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs.
\(2 (51) = 102\).
time = 0.47, size = 109, normalized size = 1.85 \begin {gather*} -\frac {\log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {a - b} a - \sqrt {a - b} b\right )}} + \frac {\frac {a \sin \left (x\right )}{\sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} {\left (a b - b^{2}\right )}} + \frac {\log \left ({\left | -\sqrt {a - b} \sin \left (x\right ) + \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} \right |}\right )}{{\left (a - b\right )}^{\frac {3}{2}}}}{\mathrm {sgn}\left (\sin \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.92, size = 52, normalized size = 0.88 \begin {gather*} \frac {a}{\left (a\,b-b^2\right )\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}{\sqrt {a-b}}\right )}{{\left (a-b\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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