Optimal. Leaf size=17 \[ \frac {\tanh ^{-1}(a x)^{1+p}}{a (1+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6095}
\begin {gather*} \frac {\tanh ^{-1}(a x)^{p+1}}{a (p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6095
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^p}{1-a^2 x^2} \, dx &=\frac {\tanh ^{-1}(a x)^{1+p}}{a (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}(a x)^{1+p}}{a (1+p)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.28, size = 18, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\arctanh \left (a x \right )^{p +1}}{a \left (p +1\right )}\) | \(18\) |
default | \(\frac {\arctanh \left (a x \right )^{p +1}}{a \left (p +1\right )}\) | \(18\) |
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. 84 vs.
\(2 (17) = 34\).
time = 0.36, size = 84, normalized size = 4.94 \begin {gather*} \frac {\cosh \left (p \log \left (\frac {1}{2} \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )\right ) \log \left (-\frac {a x + 1}{a x - 1}\right ) + \log \left (-\frac {a x + 1}{a x - 1}\right ) \sinh \left (p \log \left (\frac {1}{2} \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )\right )}{2 \, {\left (a p + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (12) = 24\).
time = 0.87, size = 26, normalized size = 1.53 \begin {gather*} \begin {cases} \frac {\begin {cases} \frac {\operatorname {atanh}^{p + 1}{\left (a x \right )}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (\operatorname {atanh}{\left (a x \right )} \right )} & \text {otherwise} \end {cases}}{a} & \text {for}\: a \neq 0 \\0^{p} x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 30, normalized size = 1.76 \begin {gather*} \frac {\left (\frac {1}{2} \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )^{p + 1}}{a {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.05, size = 33, normalized size = 1.94 \begin {gather*} \left \{\begin {array}{cl} \frac {\ln \left (\mathrm {atanh}\left (a\,x\right )\right )}{a} & \text {\ if\ \ }p=-1\\ \frac {{\mathrm {atanh}\left (a\,x\right )}^{p+1}}{a\,\left (p+1\right )} & \text {\ if\ \ }p\neq -1 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________