Optimal. Leaf size=37 \[ -\frac{1}{3 a^4 x^3}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^7} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0122327, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {325, 212, 206, 203} \[ -\frac{1}{3 a^4 x^3}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^7} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 325
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a^4-x^4\right )} \, dx &=-\frac{1}{3 a^4 x^3}+\frac{\int \frac{1}{a^4-x^4} \, dx}{a^4}\\ &=-\frac{1}{3 a^4 x^3}+\frac{\int \frac{1}{a^2-x^2} \, dx}{2 a^6}+\frac{\int \frac{1}{a^2+x^2} \, dx}{2 a^6}\\ &=-\frac{1}{3 a^4 x^3}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7}+\frac{\tanh ^{-1}\left (\frac{x}{a}\right )}{2 a^7}\\ \end{align*}
Mathematica [A] time = 0.0068022, size = 48, normalized size = 1.3 \[ -\frac{1}{3 a^4 x^3}-\frac{\log (a-x)}{4 a^7}+\frac{\log (a+x)}{4 a^7}+\frac{\tan ^{-1}\left (\frac{x}{a}\right )}{2 a^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 41, normalized size = 1.1 \begin{align*} -{\frac{1}{3\,{a}^{4}{x}^{3}}}-{\frac{\ln \left ( -a+x \right ) }{4\,{a}^{7}}}+{\frac{1}{2\,{a}^{7}}\arctan \left ({\frac{x}{a}} \right ) }+{\frac{\ln \left ( a+x \right ) }{4\,{a}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.40574, size = 54, normalized size = 1.46 \begin{align*} \frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{7}} + \frac{\log \left (a + x\right )}{4 \, a^{7}} - \frac{\log \left (-a + x\right )}{4 \, a^{7}} - \frac{1}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.84476, size = 112, normalized size = 3.03 \begin{align*} \frac{6 \, x^{3} \arctan \left (\frac{x}{a}\right ) + 3 \, x^{3} \log \left (a + x\right ) - 3 \, x^{3} \log \left (-a + x\right ) - 4 \, a^{3}}{12 \, a^{7} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.368821, size = 48, normalized size = 1.3 \begin{align*} - \frac{1}{3 a^{4} x^{3}} - \frac{\frac{\log{\left (- a + x \right )}}{4} - \frac{\log{\left (a + x \right )}}{4} + \frac{i \log{\left (- i a + x \right )}}{4} - \frac{i \log{\left (i a + x \right )}}{4}}{a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.04898, size = 57, normalized size = 1.54 \begin{align*} \frac{\arctan \left (\frac{x}{a}\right )}{2 \, a^{7}} + \frac{\log \left ({\left | a + x \right |}\right )}{4 \, a^{7}} - \frac{\log \left ({\left | -a + x \right |}\right )}{4 \, a^{7}} - \frac{1}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]