Optimal. Leaf size=58 \[ -\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6113, 6179,
6181, 3393, 3382, 6115} \begin {gather*} -\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3382
Rule 3393
Rule 6113
Rule 6115
Rule 6179
Rule 6181
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}+a \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+a^2 \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {\text {Subst}\left (\int \frac {\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 58, normalized size = 1.00 \begin {gather*} \frac {1+2 a x \tanh ^{-1}(a x)+2 \left (-1+a^2 x^2\right ) \tanh ^{-1}(a x)^2 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a \left (-1+a^2 x^2\right ) \tanh ^{-1}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 6.82, size = 51, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {-\frac {1}{4 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\hyperbolicCosineIntegral \left (2 \arctanh \left (a x \right )\right )}{a}\) | \(51\) |
default | \(\frac {-\frac {1}{4 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\hyperbolicCosineIntegral \left (2 \arctanh \left (a x \right )\right )}{a}\) | \(51\) |
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. 122 vs.
\(2 (53) = 106\).
time = 0.38, size = 122, normalized size = 2.10 \begin {gather*} \frac {4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + {\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4}{2 \, {\left (a^{3} x^{2} - a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{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 {1}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{3}{\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 = -0.02 \begin {gather*} \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________