Optimal. Leaf size=41 \[ \frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {5 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac {\text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.11, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5968, 3312, 3301} \[ \frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {5 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac {\text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5968
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^5(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\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 )}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac {5 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac {5 \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}\\ &=\frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {5 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{16 a}+\frac {\text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 31, normalized size = 0.76 \[ \frac {10 \text {Chi}\left (\tanh ^{-1}(a x)\right )+5 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )+\text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{8} x^{8} - 4 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}} \operatorname {artanh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 30, normalized size = 0.73 \[ \frac {10 \Chi \left (\arctanh \left (a x \right )\right )+5 \Chi \left (3 \arctanh \left (a x \right )\right )+\Chi \left (5 \arctanh \left (a x \right )\right )}{16 a} \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {7}{2}} \operatorname {artanh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}} \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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