Optimal. Leaf size=93 \[ -\frac {5 x}{2 \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}+\frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{16 a}+\frac {45 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{32 a}+\frac {25 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{32 a} \]
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Rubi [A] time = 0.40, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ -\frac {5 x}{2 \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}+\frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{16 a}+\frac {45 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{32 a}+\frac {25 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{32 a} \]
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 )^{7/2} \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}+\frac {1}{2} (5 a) \int \frac {x}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}-\frac {5 x}{2 \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac {5}{2} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)} \, dx+\left (10 a^2\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}-\frac {5 x}{2 \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac {5 \operatorname {Subst}\left (\int \frac {\cosh ^5(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {10 \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}-\frac {5 x}{2 \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac {5 \operatorname {Subst}\left (\int \left (\frac {5 \cosh (x)}{8 x}+\frac {5 \cosh (3 x)}{16 x}+\frac {\cosh (5 x)}{16 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {10 \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{8 x}+\frac {\cosh (3 x)}{16 x}+\frac {\cosh (5 x)}{16 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}-\frac {5 x}{2 \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac {5 \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}+\frac {5 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}+\frac {5 \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}+\frac {25 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}-\frac {5 \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {25 \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}-\frac {5 x}{2 \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{16 a}+\frac {45 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{32 a}+\frac {25 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{32 a}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 79, normalized size = 0.85 \[ \frac {-\frac {80 a x}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}-\frac {16}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}+10 \text {Chi}\left (\tanh ^{-1}(a x)\right )+45 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )+25 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{32 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{8} x^{8} - 4 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{3}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.56, size = 272, normalized size = 2.92 \[ \frac {10 \arctanh \left (a x \right )^{2} \Chi \left (\arctanh \left (a x \right )\right ) x^{2} a^{2}+45 \arctanh \left (a x \right )^{2} \Chi \left (3 \arctanh \left (a x \right )\right ) x^{2} a^{2}+25 \arctanh \left (a x \right )^{2} \Chi \left (5 \arctanh \left (a x \right )\right ) x^{2} a^{2}-15 \arctanh \left (a x \right ) \sinh \left (3 \arctanh \left (a x \right )\right ) x^{2} a^{2}-5 \arctanh \left (a x \right ) \sinh \left (5 \arctanh \left (a x \right )\right ) x^{2} a^{2}-5 \cosh \left (3 \arctanh \left (a x \right )\right ) x^{2} a^{2}-\cosh \left (5 \arctanh \left (a x \right )\right ) x^{2} a^{2}+10 \sqrt {-a^{2} x^{2}+1}\, a x \arctanh \left (a x \right )-10 \Chi \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}-45 \Chi \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}-25 \Chi \left (5 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}+15 \sinh \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+5 \sinh \left (5 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+10 \sqrt {-a^{2} x^{2}+1}+5 \cosh \left (3 \arctanh \left (a x \right )\right )+\cosh \left (5 \arctanh \left (a x \right )\right )}{32 a \arctanh \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )} \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (1-a^2\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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