Optimal. Leaf size=93 \[ -\frac {1}{2 a \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}-\frac {5 x}{2 \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+\frac {5 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{16 a}+\frac {45 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{32 a}+\frac {25 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{32 a} \]
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Rubi [A]
time = 0.28, 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 = {6113, 6179,
6181, 5556, 3382, 6115, 3393} \begin {gather*} -\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} \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 )^{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 \text {Subst}\left (\int \frac {\cosh ^5(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {10 \text {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 \text {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 \text {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 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}+\frac {5 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}+\frac {5 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a}+\frac {25 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}-\frac {5 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {25 \text {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 = 0.17, size = 79, normalized size = 0.85 \begin {gather*} \frac {-\frac {16}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2}-\frac {80 a x}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}+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} \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. \(271\) vs.
\(2(79)=158\).
time = 2.86, size = 272, normalized size = 2.92
method | result | size |
default | \(\frac {45 \hyperbolicCosineIntegral \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2} a^{2} x^{2}+25 \hyperbolicCosineIntegral \left (5 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2} a^{2} x^{2}+10 \arctanh \left (a x \right )^{2} \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right ) a^{2} x^{2}-15 \arctanh \left (a x \right ) \sinh \left (3 \arctanh \left (a x \right )\right ) a^{2} x^{2}-5 \arctanh \left (a x \right ) \sinh \left (5 \arctanh \left (a x \right )\right ) a^{2} x^{2}-\cosh \left (5 \arctanh \left (a x \right )\right ) a^{2} x^{2}-5 \cosh \left (3 \arctanh \left (a x \right )\right ) a^{2} x^{2}+10 \sqrt {-a^{2} x^{2}+1}\, a x \arctanh \left (a x \right )-45 \hyperbolicCosineIntegral \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}-25 \hyperbolicCosineIntegral \left (5 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}-10 \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}+15 \sinh \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+5 \sinh \left (5 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+\cosh \left (5 \arctanh \left (a x \right )\right )+10 \sqrt {-a^{2} x^{2}+1}+5 \cosh \left (3 \arctanh \left (a x \right )\right )}{32 a \arctanh \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}\) | \(272\) |
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 {7}{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 )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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