Optimal. Leaf size=69 \[ -\frac {1}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2 x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a}+\frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6113, 6179,
6181, 5556, 3382, 6115, 3393} \begin {gather*} -\frac {2 x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a}+\frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3382
Rule 3393
Rule 5556
Rule 6113
Rule 6115
Rule 6179
Rule 6181
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+(2 a) \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2 x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+2 \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\left (6 a^2\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2 x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {6 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2 x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cosh (2 x)}{2 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {6 \text {Subst}\left (\int \left (-\frac {1}{8 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2 x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2 x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a}+\frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 86, normalized size = 1.25 \begin {gather*} \frac {-1-4 a x \tanh ^{-1}(a x)+2 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )+2 \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^2 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a \left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 2.92, size = 88, normalized size = 1.28
method | result | size |
derivativedivides | \(\frac {-\frac {3}{16 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\hyperbolicCosineIntegral \left (2 \arctanh \left (a x \right )\right )-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{16 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (4 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )}+\hyperbolicCosineIntegral \left (4 \arctanh \left (a x \right )\right )}{a}\) | \(88\) |
default | \(\frac {-\frac {3}{16 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\hyperbolicCosineIntegral \left (2 \arctanh \left (a x \right )\right )-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{16 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (4 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )}+\hyperbolicCosineIntegral \left (4 \arctanh \left (a x \right )\right )}{a}\) | \(88\) |
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. 241 vs.
\(2 (65) = 130\).
time = 0.39, size = 241, normalized size = 3.49 \begin {gather*} -\frac {8 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - {\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{4} x^{4} - 2 \, 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^{5} x^{4} - 2 \, 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}{a^{6} x^{6} \operatorname {atanh}^{3}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{3}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{3}{\left (a x \right )} - \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 = -0.01 \begin {gather*} -\int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________