Optimal. Leaf size=17 \[ -8 \cot (2 x)-\frac {8}{3} \cot ^3(2 x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.47, number of steps
used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2700, 276}
\begin {gather*} \frac {\tan ^3(x)}{3}+3 \tan (x)-\frac {1}{3} \cot ^3(x)-3 \cot (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 276
Rule 2700
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^4} \, dx,x,\tan (x)\right )\\ &=\text {Subst}\left (\int \left (3+\frac {1}{x^4}+\frac {3}{x^2}+x^2\right ) \, dx,x,\tan (x)\right )\\ &=-3 \cot (x)-\frac {\cot ^3(x)}{3}+3 \tan (x)+\frac {\tan ^3(x)}{3}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 33, normalized size = 1.94 \begin {gather*} -\frac {8 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x)+\frac {8 \tan (x)}{3}+\frac {1}{3} \sec ^2(x) \tan (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs.
\(2(15)=30\).
time = 0.22, size = 36, normalized size = 2.12
method | result | size |
parallelrisch | \(-\frac {\left (-\cos \left (6 x \right )+3 \cos \left (2 x \right )\right ) \left (\sec ^{3}\left (x \right )\right ) \left (\csc ^{3}\left (x \right )\right )}{6}\) | \(24\) |
risch | \(\frac {32 i \left (3 \,{\mathrm e}^{4 i x}-1\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3} \left ({\mathrm e}^{2 i x}+1\right )^{3}}\) | \(31\) |
default | \(\frac {1}{3 \sin \left (x \right )^{3} \cos \left (x \right )^{3}}-\frac {2}{3 \sin \left (x \right )^{3} \cos \left (x \right )}+\frac {8}{3 \cos \left (x \right ) \sin \left (x \right )}-\frac {16 \cot \left (x \right )}{3}\) | \(36\) |
norman | \(\frac {\frac {1}{24}+\frac {5 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4}-\frac {91 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}+\frac {35 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2}-\frac {91 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{8}+\frac {5 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{4}+\frac {\left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{24}}{\tan \left (\frac {x}{2}\right )^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.38, size = 25, normalized size = 1.47 \begin {gather*} \frac {1}{3} \, \tan \left (x\right )^{3} - \frac {9 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} + 3 \, \tan \left (x\right ) \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. 39 vs.
\(2 (15) = 30\).
time = 0.60, size = 39, normalized size = 2.29 \begin {gather*} -\frac {16 \, \cos \left (x\right )^{6} - 24 \, \cos \left (x\right )^{4} + 6 \, \cos \left (x\right )^{2} + 1}{3 \, {\left (\cos \left (x\right )^{5} - \cos \left (x\right )^{3}\right )} \sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.02, size = 29, normalized size = 1.71 \begin {gather*} - \frac {16 \cos {\left (2 x \right )}}{3 \sin {\left (2 x \right )}} - \frac {8 \cos {\left (2 x \right )}}{3 \sin ^{3}{\left (2 x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.58, size = 18, normalized size = 1.06 \begin {gather*} -\frac {8 \, {\left (3 \, \tan \left (2 \, x\right )^{2} + 1\right )}}{3 \, \tan \left (2 \, x\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.21, size = 38, normalized size = 2.24 \begin {gather*} -\frac {24\,\cos \left (2\,x\right )-16\,{\cos \left (2\,x\right )}^3}{3\,\sin \left (2\,x\right )-3\,{\cos \left (2\,x\right )}^2\,\sin \left (2\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Chatgpt [F] Failed to verify
time = 1.00, size = 25, normalized size = 1.47 \begin {gather*} -\frac {\tan \left (x \right ) \cot \left (x \right )}{3}+\frac {\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )}{3}+\frac {\ln \left (\tan \left (x \right )-\sec \left (x \right )\right )}{3} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________