Optimal. Leaf size=79 \[ -\frac {1}{2 a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}-\frac {3 x}{2 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{8 a}+\frac {9 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{8 a} \]
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Rubi [A]
time = 0.25, 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 = {6113, 6179,
6181, 5556, 3382, 6115, 3393} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 5556
Rule 6113
Rule 6115
Rule 6179
Rule 6181
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 \text {Subst}\left (\int \frac {\cosh ^3(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {3 \text {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 \text {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 \text {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 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}-\frac {3 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {9 \text {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 = 0.15, size = 56, normalized size = 0.71 \begin {gather*} \frac {-\frac {4 \left (1+3 a x \tanh ^{-1}(a x)\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs.
\(2(67)=134\).
time = 3.42, size = 180, normalized size = 2.28
method | result | size |
default | \(\frac {9 \hyperbolicCosineIntegral \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2} a^{2} x^{2}+3 \arctanh \left (a x \right )^{2} \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right ) a^{2} x^{2}-3 \arctanh \left (a x \right ) \sinh \left (3 \arctanh \left (a x \right )\right ) a^{2} x^{2}-\cosh \left (3 \arctanh \left (a x \right )\right ) a^{2} x^{2}+3 \sqrt {-a^{2} x^{2}+1}\, a x \arctanh \left (a x \right )-9 \hyperbolicCosineIntegral \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}-3 \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}+3 \sinh \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}+\cosh \left (3 \arctanh \left (a x \right )\right )}{8 a \arctanh \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}\) | \(180\) |
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 {5}{2}} \operatorname {atanh}^{3}{\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 = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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