Optimal. Leaf size=126 \[ \frac {1}{4} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\frac {1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt {a+b \cot ^4(x)}-\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3751, 1262,
749, 829, 858, 223, 212, 739} \begin {gather*} -\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac {1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt {a+b \cot ^4(x)}+\frac {1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{4} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 223
Rule 739
Rule 749
Rule 829
Rule 858
Rule 1262
Rule 3751
Rubi steps
\begin {align*} \int \cot (x) \left (a+b \cot ^4(x)\right )^{3/2} \, dx &=-\text {Subst}\left (\int \frac {x \left (a+b x^4\right )^{3/2}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{1+x} \, dx,x,\cot ^2(x)\right )\right )\\ &=-\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac {1}{2} \text {Subst}\left (\int \frac {(a-b x) \sqrt {a+b x^2}}{1+x} \, dx,x,\cot ^2(x)\right )\\ &=-\frac {1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt {a+b \cot ^4(x)}-\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac {\text {Subst}\left (\int \frac {a b (2 a+b)-b^2 (3 a+2 b) x}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )}{4 b}\\ &=-\frac {1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt {a+b \cot ^4(x)}-\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2}-\frac {1}{2} (a+b)^2 \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )+\frac {1}{4} (b (3 a+2 b)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\cot ^2(x)\right )\\ &=-\frac {1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt {a+b \cot ^4(x)}-\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2}+\frac {1}{2} (a+b)^2 \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 )+\frac {1}{4} (b (3 a+2 b)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )\\ &=\frac {1}{4} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+\frac {1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\frac {1}{4} \left (2 (a+b)-b \cot ^2(x)\right ) \sqrt {a+b \cot ^4(x)}-\frac {1}{6} \left (a+b \cot ^4(x)\right )^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 4.63, size = 167, normalized size = 1.33 \begin {gather*} \frac {1}{12} \left (6 \sqrt {b} (a+b) \tanh ^{-1}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a+b \cot ^4(x)}}\right )+6 (a+b)^{3/2} \tanh ^{-1}\left (\frac {a-b \cot ^2(x)}{\sqrt {a+b} \sqrt {a+b \cot ^4(x)}}\right )-\sqrt {a+b \cot ^4(x)} \left (8 a+6 b-3 b \cot ^2(x)+2 b \cot ^4(x)\right )+\frac {3 \sqrt {a} \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {b} \cot ^2(x)}{\sqrt {a}}\right ) \sqrt {a+b \cot ^4(x)}}{\sqrt {1+\frac {b \cot ^4(x)}{a}}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(311\) vs.
\(2(103)=206\).
time = 0.24, size = 312, normalized size = 2.48
method | result | size |
derivativedivides | \(-\frac {b \left (\cot ^{4}\left (x \right )\right ) \sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}}{6}-\frac {2 a \sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}}{3}+\frac {b \left (\cot ^{2}\left (x \right )\right ) \sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}}{4}+\frac {3 a \sqrt {b}\, \ln \left (\sqrt {b}\, \left (\cot ^{2}\left (x \right )\right )+\sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}\right )}{4}-\frac {b \sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}}{2}+\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \left (\cot ^{2}\left (x \right )\right )+\sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}\right )}{2}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right ) a^{2}}{2 \sqrt {a +b}}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right ) a b}{\sqrt {a +b}}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right ) b^{2}}{2 \sqrt {a +b}}\) | \(312\) |
default | \(-\frac {b \left (\cot ^{4}\left (x \right )\right ) \sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}}{6}-\frac {2 a \sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}}{3}+\frac {b \left (\cot ^{2}\left (x \right )\right ) \sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}}{4}+\frac {3 a \sqrt {b}\, \ln \left (\sqrt {b}\, \left (\cot ^{2}\left (x \right )\right )+\sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}\right )}{4}-\frac {b \sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}}{2}+\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \left (\cot ^{2}\left (x \right )\right )+\sqrt {a +b \left (\cot ^{4}\left (x \right )\right )}\right )}{2}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right ) a^{2}}{2 \sqrt {a +b}}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right ) a b}{\sqrt {a +b}}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\cot ^{2}\left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\cot ^{2}\left (x \right )\right )^{2}-2 b \left (1+\cot ^{2}\left (x \right )\right )+a +b}}{1+\cot ^{2}\left (x \right )}\right ) b^{2}}{2 \sqrt {a +b}}\) | \(312\) |
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. 357 vs.
\(2 (104) = 208\).
time = 3.30, size = 1486, normalized size = 11.79 \begin {gather*} \text {Too large to display} \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 \left (a + b \cot ^{4}{\left (x \right )}\right )^{\frac {3}{2}} \cot {\left (x \right )}\, 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. 445 vs.
\(2 (104) = 208\).
time = 0.65, size = 445, normalized size = 3.53 \begin {gather*} -\frac {{\left (3 \, a b + 2 \, b^{2}\right )} \arctan \left (-\frac {\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}}{\sqrt {-b}}\right )}{2 \, \sqrt {-b}} - \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \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 )} {\left (a + b\right )} + \sqrt {a + b} b \right |}\right )}{2 \, \sqrt {a + b}} - \frac {3 \, {\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 )}^{5} {\left (5 \, a b + 6 \, b^{2}\right )} + 8 \, {\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 )}^{3} b^{3} - 12 \, {\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 )}^{4} {\left (a b + 3 \, b^{2}\right )} \sqrt {a + b} + 12 \, {\left (a b^{2} + b^{3}\right )} {\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 )}^{2} \sqrt {a + b} + 3 \, {\left (3 \, a b^{3} + 2 \, b^{4}\right )} {\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 )} - 8 \, {\left (a b^{3} + b^{4}\right )} \sqrt {a + b}}{6 \, {\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 )}^{2} - b\right )}^{3}} \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 \mathrm {cot}\left (x\right )\,{\left (b\,{\mathrm {cot}\left (x\right )}^4+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________