Optimal. Leaf size=79 \[ -\frac {3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}+\frac {3 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {9 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{8 a} \]
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Rubi [A] time = 0.36, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5966, 6032, 6034, 5448, 3301, 5968, 3312} \[ -\frac {3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}+\frac {3 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {9 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5448
Rule 5966
Rule 5968
Rule 6032
Rule 6034
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}+\frac {1}{2} (3 a) \int \frac {x}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac {3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3}{2} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)} \, dx+\left (3 a^2\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac {3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh ^3(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac {3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \left (\frac {3 \cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {3 \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac {3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}-\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac {3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {9 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{8 a}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 56, normalized size = 0.71 \[ \frac {-\frac {4 \left (3 a x \tanh ^{-1}(a x)+1\right )}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}+3 \text {Chi}\left (\tanh ^{-1}(a x)\right )+9 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{8 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 180, normalized size = 2.28 \[ \frac {3 \arctanh \left (a x \right )^{2} \Chi \left (\arctanh \left (a x \right )\right ) x^{2} a^{2}+9 \arctanh \left (a x \right )^{2} \Chi \left (3 \arctanh \left (a x \right )\right ) x^{2} a^{2}-3 \arctanh \left (a x \right ) \sinh \left (3 \arctanh \left (a x \right )\right ) x^{2} a^{2}-\cosh \left (3 \arctanh \left (a x \right )\right ) x^{2} a^{2}+3 \sqrt {-a^{2} x^{2}+1}\, a x \arctanh \left (a x \right )-3 \Chi \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}-9 \Chi \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}+3 \sinh \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}+\cosh \left (3 \arctanh \left (a x \right )\right )}{8 a \arctanh \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )} \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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