Optimal. Leaf size=55 \[ \frac {35 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{64 a}+\frac {21 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{64 a}+\frac {7 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{64 a}+\frac {\text {Chi}\left (7 \tanh ^{-1}(a x)\right )}{64 a} \]
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Rubi [A]
time = 0.09, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6115, 3393,
3382} \begin {gather*} \frac {35 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{64 a}+\frac {21 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{64 a}+\frac {7 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{64 a}+\frac {\text {Chi}\left (7 \tanh ^{-1}(a x)\right )}{64 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 6115
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh ^7(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \left (\frac {35 \cosh (x)}{64 x}+\frac {21 \cosh (3 x)}{64 x}+\frac {7 \cosh (5 x)}{64 x}+\frac {\cosh (7 x)}{64 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \frac {\cosh (7 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {7 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {21 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {35 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}\\ &=\frac {35 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{64 a}+\frac {21 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{64 a}+\frac {7 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{64 a}+\frac {\text {Chi}\left (7 \tanh ^{-1}(a x)\right )}{64 a}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 40, normalized size = 0.73 \begin {gather*} \frac {35 \text {Chi}\left (\tanh ^{-1}(a x)\right )+21 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )+7 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )+\text {Chi}\left (7 \tanh ^{-1}(a x)\right )}{64 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.89, size = 39, normalized size = 0.71
method | result | size |
default | \(\frac {35 \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right )+21 \hyperbolicCosineIntegral \left (3 \arctanh \left (a x \right )\right )+7 \hyperbolicCosineIntegral \left (5 \arctanh \left (a x \right )\right )+\hyperbolicCosineIntegral \left (7 \arctanh \left (a x \right )\right )}{64 a}\) | \(39\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}} \operatorname {atanh}{\left (a x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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