Optimal. Leaf size=170 \[ \frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac {4 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{3 a} \]
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Rubi [A] time = 0.96, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5966, 6032, 6028, 6034, 3312, 3301, 5968, 5448} \[ \frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac {4 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{3 a} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5448
Rule 5966
Rule 5968
Rule 6028
Rule 6032
Rule 6034
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^5} \, dx &=-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}+a \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^4} \, dx\\ &=-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac {1}{3} \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx+a^2 \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac {1}{6 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {1}{3} (2 a) \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+\int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx-\int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac {2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx-a \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+(2 a) \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+\left (2 a^2\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac {2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}-a^2 \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \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-\int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac {2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {2 \operatorname {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 )}{3 a}-\frac {\operatorname {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}-\frac {\operatorname {Subst}\left (\int \frac {\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {2 \operatorname {Subst}\left (\int \frac {\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {2 \operatorname {Subst}\left (\int \left (-\frac {1}{8 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {6 \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac {2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{12 a}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {2 \operatorname {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 \operatorname {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}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac {2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac {\text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{3 a}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}-2 \frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac {x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac {2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac {4 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 132, normalized size = 0.78 \[ -\frac {12 a^3 x^3 \tanh ^{-1}(a x)^3-4 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )-16 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4 \text {Chi}\left (4 \tanh ^{-1}(a x)\right )+6 a^2 x^2 \tanh ^{-1}(a x)^2+20 a x \tanh ^{-1}(a x)^3+2 \tanh ^{-1}(a x)^2+4 a x \tanh ^{-1}(a x)+3}{12 a \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 303, normalized size = 1.78 \[ \frac {{\left (4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} - 4 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - 16 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - 4 \, {\left (3 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} - 24}{6 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname {artanh}\left (a x\right )^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 152, normalized size = 0.89 \[ \frac {-\frac {3}{32 \arctanh \left (a x \right )^{4}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{8 \arctanh \left (a x \right )^{4}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{12 \arctanh \left (a x \right )^{3}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{12 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{6 \arctanh \left (a x \right )}+\frac {\Chi \left (2 \arctanh \left (a x \right )\right )}{3}-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{32 \arctanh \left (a x \right )^{4}}-\frac {\sinh \left (4 \arctanh \left (a x \right )\right )}{24 \arctanh \left (a x \right )^{3}}-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{12 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (4 \arctanh \left (a x \right )\right )}{3 \arctanh \left (a x \right )}+\frac {4 \Chi \left (4 \arctanh \left (a x \right )\right )}{3}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left ({\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (a x + 1\right )^{3} - {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (-a x + 1\right )^{3} + 4 \, a x \log \left (a x + 1\right ) + {\left (3 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + {\left (3 \, a^{2} x^{2} + 3 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (a x + 1\right ) + 1\right )} \log \left (-a x + 1\right )^{2} - {\left (3 \, {\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (a x + 1\right )^{2} + 4 \, a x + 2 \, {\left (3 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right ) + 6\right )}}{3 \, {\left ({\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right )^{4} - 4 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right )^{3} \log \left (-a x + 1\right ) + 6 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right )^{2} - 4 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{3} + {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )^{4}\right )}} + \int -\frac {2 \, {\left (3 \, a^{4} x^{4} + 24 \, a^{2} x^{2} + 5\right )}}{3 \, {\left ({\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) - {\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^5\,{\left (a^2\,x^2-1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{a^{6} x^{6} \operatorname {atanh}^{5}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{5}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{5}{\left (a x \right )} - \operatorname {atanh}^{5}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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