Optimal. Leaf size=176 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)^2},x\right )}{2 a}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a^2 x^2+1}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2} \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\text {Shi}\left (4 \tanh ^{-1}(a x)\right )-\frac {1}{2 a x \tanh ^{-1}(a x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.77, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx &=a^2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx+\int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {1}{2} a \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx+\frac {1}{2} \left (3 a^3\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+\int \frac {1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\frac {1}{2} (3 a) \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx-\frac {1}{2} (3 a) \int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+\left (2 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\left (2 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+2 \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}-\left (3 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\left (6 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )+2 \operatorname {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )-3 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+6 \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-3 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )+6 \operatorname {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2} \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{4} \text {Shi}\left (4 \tanh ^{-1}(a x)\right )+\frac {3}{4} \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {3}{2} \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\text {Shi}\left (4 \tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 5.69, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )} \operatorname {artanh}\left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{3} \arctanh \left (a x \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, a x + {\left (5 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) - {\left (5 \, a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )}{{\left (a^{6} x^{6} - 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{6} x^{6} - 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + {\left (a^{6} x^{6} - 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} \log \left (-a x + 1\right )^{2}} + \int -\frac {2 \, {\left (10 \, a^{4} x^{4} - 3 \, a^{2} x^{2} + 1\right )}}{{\left (a^{8} x^{9} - 3 \, a^{6} x^{7} + 3 \, a^{4} x^{5} - a^{2} x^{3}\right )} \log \left (a x + 1\right ) - {\left (a^{8} x^{9} - 3 \, a^{6} x^{7} + 3 \, a^{4} x^{5} - a^{2} x^{3}\right )} \log \left (-a x + 1\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a^{6} x^{7} \operatorname {atanh}^{3}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname {atanh}^{3}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atanh}^{3}{\left (a x \right )} - x \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________