Optimal. Leaf size=100 \[ \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {3}{2 a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {2}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.46, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6032, 6028, 5966, 6034, 5448, 12, 3298} \[ \frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{a^2}-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {3}{2 a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {2}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3298
Rule 5448
Rule 5966
Rule 6028
Rule 6032
Rule 6034
Rubi steps
\begin {align*} \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx &=-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\frac {1}{2} (3 a) \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {1}{2 a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+2 \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\frac {3 \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx}{2 a}-\frac {3 \int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-3 \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+6 \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {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 )}{a^2}-\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}+\frac {6 \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^2}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^2}-\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}+\frac {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 )}{a^2}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{4 a^2}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^2}\\ &=-\frac {x}{2 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {2}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {3}{2 a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{a^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 96, normalized size = 0.96 \[ -\frac {-\left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )-2 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2 \text {Shi}\left (4 \tanh ^{-1}(a x)\right )+3 a^2 x^2 \tanh ^{-1}(a x)+a x+\tanh ^{-1}(a x)}{2 a^2 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.56, size = 256, normalized size = 2.56 \[ \frac {{\left (2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 8 \, a x - 4 \, {\left (3 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{4 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {x}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.18, size = 82, normalized size = 0.82 \[ \frac {-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{8 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )}+\frac {\Shi \left (2 \arctanh \left (a x \right )\right )}{2}-\frac {\sinh \left (4 \arctanh \left (a x \right )\right )}{16 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )}+\Shi \left (4 \arctanh \left (a x \right )\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, a x + {\left (3 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) - {\left (3 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )}{{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (-a x + 1\right )^{2}} + \int -\frac {2 \, {\left (3 \, a^{2} x^{3} + 5 \, x\right )}}{{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) - {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {x}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {x}{a^{6} x^{6} \operatorname {atanh}^{3}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{3}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{3}{\left (a x \right )} - \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________