Optimal. Leaf size=14 \[ \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6181, 5556, 12,
3379} \begin {gather*} \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3379
Rule 5556
Rule 6181
Rubi steps
\begin {align*} \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^2}\\ &=\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 5.72, size = 13, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {\hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )}{2 a^{2}}\) | \(13\) |
default | \(\frac {\hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )}{2 a^{2}}\) | \(13\) |
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. 38 vs.
\(2 (12) = 24\).
time = 0.35, size = 38, normalized size = 2.71 \begin {gather*} \frac {\operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )}{4 \, a^{2}} \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 {x}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {x}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________