Optimal. Leaf size=117 \[ \frac {\tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3751, 1262,
755, 837, 12, 739, 212} \begin {gather*} -\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}}-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 212
Rule 739
Rule 755
Rule 837
Rule 1262
Rule 3751
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\left (a+b \cot ^4(x)\right )^{5/2}} \, dx &=-\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x^4\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-3 a-2 b-2 b x}{(1+x) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot ^2(x)\right )}{6 a (a+b)}\\ &=-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}}-\frac {\text {Subst}\left (\int \frac {3 a^2 b}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )}{6 a^2 b (a+b)^2}\\ &=-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}}-\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )}{2 (a+b)^2}\\ &=-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}}+\frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {a-b \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^2}\\ &=\frac {\tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a+b \cot ^2(x)}{6 a (a+b) \left (a+b \cot ^4(x)\right )^{3/2}}-\frac {3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \cot ^4(x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.78, size = 114, normalized size = 0.97 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {a^2 (4 a+b)+3 a b (2 a+b) \cot ^2(x)+3 a^2 b \cot ^4(x)+b^2 (5 a+2 b) \cot ^6(x)}{6 a^2 (a+b)^2 \left (a+b \cot ^4(x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(601\) vs.
\(2(105)=210\).
time = 3.33, size = 602, normalized size = 5.15
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}-b \right ) a \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {\sqrt {b \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}-b \right ) a^{2} \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}+\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right )}{2 \left (\sqrt {-a b}+b \right )^{2} \left (\sqrt {-a b}-b \right )^{2} \sqrt {a +b}}-\frac {\sqrt {b \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}+b \right ) a \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}+b \right ) a^{2} \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {b \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right )^{2} a^{2} \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}+\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {b \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right )^{2} a^{2} \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}\) | \(602\) |
default | \(-\frac {\sqrt {b \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}-b \right ) a \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {\sqrt {b \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}-b \right ) a^{2} \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}+\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right )}{2 \left (\sqrt {-a b}+b \right )^{2} \left (\sqrt {-a b}-b \right )^{2} \sqrt {a +b}}-\frac {\sqrt {b \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}+b \right ) a \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}}{24 \left (\sqrt {-a b}+b \right ) a^{2} \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {b \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right )^{2} a^{2} \left (\cot ^{2}\left (x \right )-\frac {\sqrt {-a b}}{b}\right )}+\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {b \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right )^{2} a^{2} \left (\cot ^{2}\left (x \right )+\frac {\sqrt {-a b}}{b}\right )}\) | \(602\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 686 vs.
\(2 (103) = 206\).
time = 3.87, size = 1365, normalized size = 11.67 \begin {gather*} \left [\frac {3 \, {\left ({\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (2 \, x\right )^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2} - 4 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (2 \, x\right )^{3} + 2 \, {\left (3 \, a^{4} - 2 \, a^{3} b + 3 \, a^{2} b^{2}\right )} \cos \left (2 \, x\right )^{2} - 4 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {a + b} \log \left (\frac {1}{2} \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + \frac {1}{2} \, a^{2} + \frac {1}{2} \, b^{2} + \frac {1}{2} \, {\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {a + b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} - {\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\right ) - 4 \, {\left ({\left (2 \, a^{4} + a^{3} b - 5 \, a^{2} b^{2} - 5 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )^{4} + 2 \, a^{4} + 7 \, a^{3} b + 9 \, a^{2} b^{2} + 5 \, a b^{3} + b^{4} - 2 \, {\left (4 \, a^{4} + 2 \, a^{3} b - a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (2 \, x\right )^{3} + 12 \, {\left (a^{4} + a^{3} b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (4 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{12 \, {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5} + {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (2 \, x\right )^{4} - 4 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} \cos \left (2 \, x\right )^{3} + 2 \, {\left (3 \, a^{7} + 7 \, a^{6} b + 6 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 7 \, a^{3} b^{4} + 3 \, a^{2} b^{5}\right )} \cos \left (2 \, x\right )^{2} - 4 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} \cos \left (2 \, x\right )\right )}}, -\frac {3 \, {\left ({\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (2 \, x\right )^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2} - 4 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (2 \, x\right )^{3} + 2 \, {\left (3 \, a^{4} - 2 \, a^{3} b + 3 \, a^{2} b^{2}\right )} \cos \left (2 \, x\right )^{2} - 4 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (2 \, x\right )\right )} \sqrt {-a - b} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt {-a - b} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{{\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 ) + 2 \, {\left ({\left (2 \, a^{4} + a^{3} b - 5 \, a^{2} b^{2} - 5 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )^{4} + 2 \, a^{4} + 7 \, a^{3} b + 9 \, a^{2} b^{2} + 5 \, a b^{3} + b^{4} - 2 \, {\left (4 \, a^{4} + 2 \, a^{3} b - a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (2 \, x\right )^{3} + 12 \, {\left (a^{4} + a^{3} b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (4 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right )} \cos \left (2 \, x\right )\right )} \sqrt {\frac {{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, {\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{6 \, {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5} + {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (2 \, x\right )^{4} - 4 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} \cos \left (2 \, x\right )^{3} + 2 \, {\left (3 \, a^{7} + 7 \, a^{6} b + 6 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 7 \, a^{3} b^{4} + 3 \, a^{2} b^{5}\right )} \cos \left (2 \, x\right )^{2} - 4 \, {\left (a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} \cos \left (2 \, x\right )\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot {\left (x \right )}}{\left (a + b \cot ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 276 vs.
\(2 (103) = 206\).
time = 0.46, size = 276, normalized size = 2.36 \begin {gather*} -\frac {{\left (2 \, {\left (\frac {{\left (2 \, a^{3} b - a^{2} b^{2} - 4 \, a b^{3} - b^{4}\right )} \sin \left (x\right )^{2}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}} + \frac {3 \, {\left (3 \, a b^{3} + b^{4}\right )}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}}\right )} \sin \left (x\right )^{2} + \frac {3 \, {\left (a^{2} b^{2} - 5 \, a b^{3} - 2 \, b^{4}\right )}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}}\right )} \sin \left (x\right )^{2} + \frac {5 \, a b^{3} + 2 \, b^{4}}{a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}}}{6 \, {\left (a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b\right )}^{\frac {3}{2}}} - \frac {\log \left ({\left | -{\left (\sqrt {a + b} \sin \left (x\right )^{2} - \sqrt {a \sin \left (x\right )^{4} + b \sin \left (x\right )^{4} - 2 \, b \sin \left (x\right )^{2} + b}\right )} \sqrt {a + b} + b \right |}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a + b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {cot}\left (x\right )}{{\left (b\,{\mathrm {cot}\left (x\right )}^4+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________