Optimal. Leaf size=68 \[ \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{2 a^2}-\frac {x}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {1}{2 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6006, 5966, 6034, 3298} \[ \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{2 a^2}-\frac {x}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {1}{2 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3298
Rule 5966
Rule 6006
Rule 6034
Rubi steps
\begin {align*} \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx &=-\frac {x}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {x}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {1}{2 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {1}{2} \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac {x}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {1}{2 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac {x}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {1}{2 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{2 a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 43, normalized size = 0.63 \[ \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )-\frac {a x+\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} x}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.36, size = 154, normalized size = 2.26 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{4 a^{2} \left (a x -1\right ) \arctanh \left (a x \right )^{2}}+\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{4 a^{2} \left (a x -1\right ) \arctanh \left (a x \right )}-\frac {\Ei \left (1, -\arctanh \left (a x \right )\right )}{4 a^{2}}+\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{4 a^{2} \left (a x +1\right ) \arctanh \left (a x \right )^{2}}-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{4 a^{2} \left (a x +1\right ) \arctanh \left (a x \right )}+\frac {\Ei \left (1, \arctanh \left (a x \right )\right )}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________